FTX3044F
Finance IIA
Chapter 1
Investment Ethics
Unethical examples:
• Bernard Madoff
• Enron
Section 1
Standard 1: Professionalism
A: Knowledge of the Law
B: Independence & Objectivity
• Understand and comply will all CFA, government, etc laws
• Take reasonable care to maintain independence and objectivity
• Where in conflict, the strictest law must be adhered to
• May not offer, solicit or accept gift, compensation, offer, etc
which can compromise own or others’ independence and objectivity
• May not knowingly violate or assist others in violating laws
• Must dissociate from such violation - reporting of violations to
governmental or regulatory bodies may be advisable, but it is
not a requirement under CFA institute rules - recommend legal
opinion be obtained
• Companies should develop and adopt a code of ethics - put in
place written procedures for the reporting of suspected violations
• Benefits: small gifts permitted, shares from IPO not allowed,
all gifts must be disclosed to employer, no gifts which can be
perceived to possibly influence investment decisions
• Research: must not be done because of pressure from companies or to attract potential clients, paid research must not influence the researcher, company paying for expenses on research
visit acceptable if not excessive or out of the ordinary, must be
no pressure to issue favorable reports
• Integrity of opinions - protect, sate in reports, remunerate
• Limit payment of expenses by outside parties
• Restrict personal investment by investment professionals
• Ensure supervision of activities of analysts and portfolio managers
2
C: Misrepresentation
D: Misconduct
• Must not knowingly make misrepresentation relating to investment analysis, recommendations, sources, actions, etc
• May not engage in professional conduct involving dishonesty,
fraud, deceit, etc
• Give all info relevant to avoid possible misrep
• May not commit act which reflects adversely on professional
reputation, integrity, competence
• Do not misrepresent investment performance, qualifications,
abilities, track record or services that can be performed
• Attempt to prevent distribution of misrepresentative literature
• Plagiarism: citing quotations supposedly attributable to experts, etc without being specific about who they are; presenting opinions, data, etc of others with acknowledgement of
source but without original caveats and qualifying statements
• Understand limitations of firm or investment strategy
• Maintenance of professional integrity, good reputation and
competence
• Not necessarily only illegal activities, but any that can affect
above negatively
• Do not commit acts that makes the profession look bad e.g.
Drunkenness on the job
• Avoid deceit, dishonesty and fraud
• Be accurate and complete in presentations
• Keep copies of all sources, background notes, etc relevant to or
used in research reports
• Attribute all quotations, projections, tables, statistics, models
and ideas. Exceptions: when prepared by recognized financial
or statistical reporting service e.g. Reuters
3
Section 2
Standard 2: Integrity of Capital Markets
A: Material non-public information
• Keep records of info as proof and basis and save all research
• When in possession of material non-public information that
could affect the value of an investment you may not use or
cause others to act on this information
• Influential analyst’s opinions may in itself affect security price,
but not being a full insider the analyst does not need to make
reports etc known to all
• Material: can affect a reasonable investor’s investment decision
or is likely to affect the value of an investment
• If info is material, non-public encourage source company to disseminate it widely - if not possible, report to internal supervisory function and do not trade on it
• Non-public: not widely disseminated to the broader public
• Mosaic Theory: you are allowed to trade on a combination of
public material information and non-public, non-material information
• Material non-public info: earnings, tender offers, mergers and
acquisitions, significant asset changes, management changes,
patents, licences, significant supplier and customer changes, legal issues, financial position, cashflow issues, etc
• Materiality determined by: substance and specificity of info, reliability of source, strength of link to price of security, time
value of info
• Do not knowingly induce company insiders to disclose material non-public info
• Implement compliance procedures e.g. Monitoring of employee and proprietary trading and documentation
• “Chinese walls” between relevant departments
• Internal info barriers to ensure info only goes to those who
need it
• Companies should provide same info to all and not discriminate
• Beware of selective disclosure by management of company insiders to analysts
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B: Market Manipulation
• May not engage in practices that distort prices, mislead markets or artificially inflate trading volumes
• Artificially influencing prices of trading volumes on the market
is prohibited
• False rumours and misleading info
• Legitimate transactions include trades for tax purposes (e.g.
Selling and immediately re-buying a share) or affecting the
price when trading in an illiquid share
• Intent is the key. If activities are meant to mislead others, they
are likely prohibited
5
Section 3
Standard 3: Duties to Clients
A: Loyalty, prudence and care
B: Fair Dealing
• Duty of loyalty, must act with reasonable care and exercise prudent judgement
• Must deal fairly and objectively with all clients in investment
analysis recommendations, actions, etc
• Must act for benefit of clients (above own and employer’s interests)
• Objective and fair dealing with all clients - no discrimination
• Must determine applicable fiduciary duty and comply herewith
to persons/ interest to whom owed
• Prudent Man Rule: exercise care and diligence a prudent person familiar with the investment situation/ needs would exercise
• Investment decisions should be made with reference to the total portfolio
• At least a quarterly transaction and holdings statement to clients with all info pertaining to assets held on their behalf
• Periodically review investments to check compliance and mandates
• Applies to dissemination of info or investment decisions/ actions
• Investment recommendation should be communicated to all
clients
• Timing: dissemination to all at approximately same time
• Allocate on an impartial basis
• Limit no of people aware of pending recommendation and
make a policy that people in the know cannot act
• Written trade allocation procedures
• Systematic account review to ensure no client treated preferentially
• Diversify to minimize risk of loss
• Treat all clients equally
6
C: Suitability
D: Performance presentation
• In advisory relationship: investigate and understand client’s investment experience, knowledge, risk-return objectives, financial needs, financial constraints, etc before making investment
recommendations or taking investment actions
• In communicating investment results, take care to ensure it is
fair, accurate and complete
• Determine that investment is suitable for client needs, constraint, written objectives and mandate before making investment recommendations or taking investment actions
• When managing a portfolio or specific mandate to a mandate
or given style: make only investment recommendations or take
investment actions consistent with objectives, constraints and
mandate
• Investment Policy Statement (IPS): client description, investment objectives, investor constraints - liquidity, tax, cashflows,
time horizons, legal requirements, preferences, etc - benchmarks for performance management --- base asset allocation
on this and review regularly
• GIPS (Global Investment Performance Standards)
• No claims about ability to achieve future performance can be
made
• Do not selectively omit - include all relevant info
• Keep records/ data
• Consider level of understanding of clients
• Include terminated accounts in presentation
• Include other relevant disclosures to explain context of results
obtained
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E: Preservation of Confidentiality
• Keep info on current, former and prospective clients confidential unless:
• Info concerns illegal activities
• Disclosure required by law
• Client/ prospective client permits disclosure
• Applies to current, past and prospective clients
• Disclosure required by law overrides this standard
• Disclosure to co-workers involved with client only on a need to
know basis
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Section 4
Standard 4: Duties to Employers
A: Loyalty
B: Additional compensation arrangements
• Must act in interests of employer and not deprive employer of
benefits of skills and efforts, divulge info or cause harm to employer
• May not accept gifts, benefits, compensation, etc that competes (or may be reasonably expected to create a conflict of interest) with employer’s interest, unless written permission obtained from all parties involved
• Confidentiality
• Outside work for compensation: allowed if employer fully informed of all aspects and agrees to all aspects
• Leaving employer: must act in employer’s best interests right
up to point of leaving. No taking of confidential info, soliciting
or clients prior to leaving, misappropriation of IP (models, reports, etc) and client databases
• Includes monetary and non-monetary compensation
• Compliance: written report to employer specifying nature of
proposed outside service, parties involved, compensation, etc
• Use of skills, experience and contacts gained at previous employer not prohibited
• Personal and employer’s interests secondary to protection of
clients interest and integrity of capital markets - “whistleblowing” allowed if for above purpose
• A whistleblower is a person who tells the public or someone in
authority about alleged dishonest or illegal activities (misconduct) occurring in a government department or private company or organization
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C: Responsibilities of supervisors
• Make reasonable efforts to detect and prevent violations of applicable rules, laws, regulations or Code of Standards by anyone
under their authority or supervision
• Must also take steps to detect this (investigate where suspicion)
• Delegation of supervisory duties does not absolve supervisor
from responsibility
• Firm must have adequate compliance procedures, otherwise
must decline supervisory responsibility in writing until in place
• Members should take initiative and recommend that employer
adopt code of ethics
• Ideal is a clear code of ethics, supported by an efficient compliance system and processes
• Identify situations in which violations are likely to occur and
establish procedures that will address these
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Section 5
Standard 5: Investment analysis, recommendations & Action
A: Diligence and reasonable basis
B: Communication with clients and prospective clients
• Exercise diligence, independence and thoroughness in analyzing investments, making investment recommendations and taking investment actions
• Disclose to clients format and basic principles of investment
processes used to analyse investments, select securities and
build portfolios
• Have reasonable and adequate basis (supported by appropriate
research and investigation) for investment analysis, recommendations and actions
• Promptly disclose any changes that may materially affect these
processes
• Secondary and 3rd party research acceptable, but must be assessed thoroughly
• Group research and decision making: if you are not in agreement, you may dissociate from decision, etc but do not have to
if you believe the rest of the group used independent and informed judgement - document differences
• Distinguish between fact and opinion in the presentation of investment analysis and recommendations
• Ensure clients understand basic characteristics of relevant investments
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C: Record retention
• Develop and maintain appropriate records to support investment analysis, recommendations and actions, as well as
investment-related communications with clients and prospective clients
• Retain records - recommended for 7 years minimum if no other
regulatory standards
• Records are property of employer, not analyst, etc
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Section 6
Standard 6: Conflicts of Interest
A: Disclosure of conflicts
B: Priority of transactions
• Make full and fair disclosure of all matters that can reasonably
be expected to impair independence or objectivity or interfere
with duties to clients, prospective clients or employers
• Investment transactions of clients must have priority over own
transactions
• Ensure that such disclosures are prominently delivered in plain
language and effectively communicated
• Own transactions only after clients and employer had opportunity to transact
• Examples: direct or indirect beneficial ownership of shares
• Own transactions include those for direct family or entity in
which has a beneficiary interest
• Relationships with company management, etc
• Family members as beneficiaries: just another client
• Financial interest or relationships between analyst or their company with companies, brokers, etc
Compliance procedures:
• Analysts within brokerage company or investment division of a
bank where a negative research report on a current or potential
client may compromise actual or potential income
• IPOs: may create perception of competing with clients for limited opportunity, should declare any interests in IPOs and not
benefit from client’s participation (front-running: you know
that someone is going to buy a lot of shares so you buy before
price rises)
• Private placements: strict limitations for this regarding investment personnel, not often competing with clients but could be
seen as a favor designed to influence analyst future judgements,
clear incentive to recommend these deals to clients if self already a beneficiary
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• Blackout/ restricted periods: period prior to trade for clients
wherein those involved in investment process may not trade in
security for themselves (avoid front-running), in large firms
this could be a ban on private trading as the firm is nearly always trading in most securities on behalf of one or more clients
• Reporting requirements: regular disclosure by investment personnel of private holdings, duplicate copies of private security
transactions to be received by employer from personnel’s brokers, investment personnel required to receive preclearance for
all private transactions
C: Referral fees
• Must disclose to employer, clients and prospective clients any
compensation, consideration or benefit received or paid for recommendation of products or services
• Acceptable but: must be disclosed to all relevant parties
(whether paid in cash or not) so that can evaluate possible partiality in recommendations
• Personal investments: acceptable provided does not disadvantage clients, investment professional does not benefit personally from trades done for clients, compliance with all regulatory
requirements
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Section 7
Results & Summary
• Public loses confidence in investment industry - compromises
sustainability
• Need a well-functioning financial system - spurs economy
growth
• External controls e.g. Legislation and regulatory bodies
• Internal controls e.g. Company rules, compliance officer and
code of ethics
• CFA Institute Code of Ethics
• Investors get hurt - financially and otherwise
• Costs to the economy of short-term selfish application of financial resources
• Offenders may go to prison
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Summary:
4. Duties to Employers
1. Professionalism
• Loyalty
• Knowledge of the law
• Additional compensation arrangements
• Independence and objectivity
• Responsibilities of supervisors
• Misrepresentation
• Misconduct
5. Investment analysis, Recommendations and Action
• Diligence and reasonable basis
2. Integrity of Capital Markets
• Communication with clients and prospective clients
• Material non-public information
• Record retention
• Market manipulation
6. Conflicts of Interest
3. Duties to Clients
• Disclosure of conflicts
• Loyalty, prudence and care
• Priority of transactions
• Fair dealing
• Referral fees
• Suitability
• Performance presentation
• Preservation of confidentiality
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Chapter 2
The Investment Environment
Investment: the current
commitment of money or other
resources in the expectation of
reaping future benefits.
Section 1
Real Assets vs Financial Assets
Real Assets
• Determine the productive capacity and net
income of the economy
• Something that produces income in the
economy
• E.g.: land, buildings, machines, knowledge
used to produce goods and services
• Financial assets are claims to the income generated by real assets
• NOTE: patent = real asset
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Section 2
Financial Assets
Claims on real assets.
• E.g. Dividends not paid in poor times
Three types:
1. Fixed income or debt e.g. Bonds
Derivatives:
2. Common stock or equity e.g. Shares
• Value derives from prices of other securities, such as stocks
and bonds
3. Derivative securities - derived from other assets
• Used to transfer risk
Fixed income:
• Use them to gear your returns
• Payments fixed or determined by a formula
• Money market debt: short term (less than a year maturity),
highly marketable, usually low credit risk
• Capital market debt: long term bonds, can be safe or risky
Common stock:
• Equity or ownership in a corporation
• Payments to stockholders are not fixed, but depend on the success of the firm
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Section 3
Financial Markets and the Economy
The Information Role of Financial Markets:
Allocation of Risk:
• Financial markets play a central role in the allocation of capital
resources
• Investors can select securities consistent with their tastes for
risk
• If a corporation seems to have good prospects for future profitability, investors will bid up its stock price
• Capital therefore flows to companies with good prospects
• Also benefits the firms that need to raise capital to finance
their investments because each security is being sold for the
best possible price because investors are selecting security
types which best suit their risk-return preferences
• Some companies may be “hot” for a short period of time, attract a large flow of investor capital, and then fail after only a
few years
Separation of Ownership and Management:
• No-one knows with certainty what will happen - stock market
encourages allocation of capital to those firms who appear at
the time to have the best prospects
• Gives a firm stability
• Firm’s management should pursue strategies that enhance the
value of their shares - can lead to agency problems
Consumption Timing:
• Agency problems: managers, who are hired as agents of the
shareholders, may pursue their own interests instead
• Individuals may earn more than they currently wish to spend
• Ways to combat agency problems:
• They can “store” their wealth in financial assets
1. Compensation plans tie the income of managers to the success of the firm
• Investment in financial assets and postpone immediate consumption for future consumption
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2. Boards of directors can force out management teams that are
underperforming
3. Outsiders, such as security analysts, monitor the firm closely
and make the life of poor performers at the least uncomfortable
4. Bad performers are subject to the threat of takeover
Corporate Governance and Corporate Ethics:
• For markets to effectively serve their purpose, there must be an
acceptable level of transparency that allows investors to make
well-informed decisions
• Accounting scandals e.g. Enron, Rite Aid, HealthSouth
• Auditors - watchdogs of the firms
• Analyst scandals e.g. Arthur Andersen
• Sarbanes-Oxley Act in America to tighten the rules of corporate governance
• South Africa has the King III
• E.g. Law that says you must have a certain number of independent directors
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Section 4
The Investment Process
Investors make two types of decisions in constructing their portfolios:
1. Asset Allocation
• Choice among broad asset classes (top down)
2. Security Selection
• Choice of which securities to hold within asset class
• Security analysis to value securities and determine investment
attractiveness
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Section 5
Markets are Competitive
The Risk-Return Trade-off:
• Always risk associated with investments
• High return = high risk
Efficient markets:
• Financial markets process all relevant information about securities quickly and efficiently ie the security price usually reflects
all the information available to investors concerning its value
If markets are efficient and prices reflect all relevant information
- it is probably better to follow passive strategies instead of
spending resources in a futile attempt to outguess your competitiors
• Active Management: attempt to improve performance either by identifying misplaced securities or by timing the performance of broad assets classes - try beat the market
• Passive Management: holding highly diversified portfolios
without spending effort or other resources attempting to improve investment performance through security analysis - no
attempt to find undervalued securities - don’t fight the market
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Section 6
The Players
• Business firms - net borrowers
• Households - net savers
• Government - can be both borrowers and savers
• Investors trade previously issued securities among themselves
in the secondary markets
• Commercial Banking:
• Take deposits and make loans
Financial intermediaries:
• Pool and invest funds
• Bring lenders and borrows together
• Banks, investment companies, insurance companies and credit
unions
Universal Bank Activities:
• E.g. Standard Bank, ABSA, FNB, etc
• Investment Banking: e.g. Investec :
• Underwrite new stock and bond issues
• Sell newly issued securities to public in the primary market
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Section 7
Rise of Systematic Risk
Systematic Risk: a potential breakdown of the financial system in which problems in one market spill over and disrupt others
• One default may set off a chain of further defaults
• Potential contagion from institution to institution, and from
market to market
• When firms are fully leveraged, losses on their portfolios ca
force them to sell some of their assets to bring their leverage
back into line
• Wavs of selling from institutions that simultaneously need to
“de-leverage” can drive down assets prices and exacerbate portfolio losses - forcing additional sales and further price declines
in a downward spiral
Policies that limit this risk:
• Transparency - to allow traders and investors to assess the risk
of their counterparties
• Capital adequacy - to prevent trading participants from being
brought down by potential losses,
• Frequent settlement of gains or losses - to prevent losses from
accumulating beyond an institution’s ability to bear them
• Incentives - to discourage excessive risk taking
• Accurate and unbiased risk assessment
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Chapter 3
Asset Classes & Financial Instruments
Money market instruments: shortterm debt
Capital market instruments: bonds,
long-term debt, equity securities,
derivative securities
Section 1
Equity Securities
Common Stock as Ownership Shares
American Depository Receipts:
• Each share entitles its owner to one vote on any matters of corporate governance that are put to a vote at the corporation’s
annual meeting
• Way for a company to raise money from international investors
without listing on a foreign stock exchange
• Also entitles owner to a share in the financial benefits of ownership
• Its not a share trading on the foreign stock exchange - instead
a bank will give an investor exactly the same proceeds and
rights as if they did have a share
• Residual claim: to the profits of the company ie everyone else
gets paid first and then the company only pays dividends
• You have the same rights as a shareholder who owns the actual
share
• Limited liability: the most shareholders can lose in the event of
failure of the corporation is their original investment
Preferred Stock: perpetuity
• Fixed dividends
• Priority over common stock
• Tax treatment - treated like a dividend
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Section 2
Stock Market Indexes
Dow Jones Industrial Average:
Final value: $30 + $90 = $120
• Includes 30 large blue-chip corporations: tells you how their
prices changed
Percentage change in portfolio value = (120-125)/125 = -4%
• Computed since 1896
• Price-weighted average: all 30 shares are treated equally, as if
they have the same importance, but if the largest share went
up - this should be more important than if the smallest share
did so price-weighted
Index:
Initial index value: (25+100)/2 = 62.5
Final index value: (30 + 90)/2 = 60
Percentage change in index = (60-62.5)/62.5 = -4%
NOTE:
Standard & Poor’s Indexes:
Index: “score” for total stock exchange. Measurement of how the
shares in the stock exchange changed. E.g. Allshare 50 = score of
top 50 shares.
• S&P 500: broadly based index of 500 firms, market-valueweighted index
• Investors can base their portfolios on an index:
• Buy an index mutual fund
Example: Price-Weighted Average
• Buy exchange traded funds (ETFs)
Portfolio:
Initial value: $25 + $100 = $125
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Other Indexes:
Two company example:
Yesterday:
US Indexes:
Company A: R11 x 20m = R220m cap
• NYSE Composite
Company B: R5 x 100m = R500m cap
• NASDAQ Composite
Total cap: R720m = 100 (base value)
• Wilshire 5000
Today:
Foreign Indexes:
A: R15 x 20m = R300m cap
• Nikkei (Japan)
B: R4 x 100m = R400m cap
• FTSE (UK)
Total cap: R700m
• DAX (Germany)
• Hang Seng (China)
700/ 720 x 100 = 97 (new index)
• TSX (Canada)
FTSE/ JSE Allshare Index:
• Market cap weighted but only the free float
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Section 3
Debt, Preference Sahres & Ordinary Shares
DEBT
ORDINARY
SHARES
PREFS
MATURITY
Finite
Perpetual
Finite to
perpetual
SENIORITY
OF CLAIM
Contractual.
Priority over
ordinary shares
and prefs
Residual.
Subordinate to
debt and prefs
Residual.
Subordinate to
debt and have
priority over
ordinary
TAX FOR
COMPANY
Interest
deductible
Not deductible
Not deductible
TAX FOR
INVESTOR
Taxable
Taxable from 1
April 2012
Taxable from 1
April 2012
Right to vote
Limited voting
rights - can only
vote on issues
that affect your
rights
VOICE IN
MANAGEME
NT
No influence
except in
liquidation
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Chapter 4
How Securities are Traded
The first time a security trades is
when it is issued to the public.
Next, already-issued securities may
be traded among investors.
Section 1
How Firms Issue Securities
Primary market:
• Firms issue new securities through underwriter to the public
• Private placement: issue that is usually sold to one or a few institutional investors and is generally held to maturity, cheaper
than public offerings, not traded in secondary markets
• Investors get new securities and the firm gets funding
Markets for securities:
Secondary market:
• Investors trade previously issued securities among themselves
1. Institutional Security markets:
• Formal, organized exchanges
• Buyer and seller trading through broker
Stocks:
• IPO - Initial Public Offering: stocks issued by a formerly privately owned company that is going public for the first time
• Seasoned offering: offered by companies that already have
floated equity
• Only members can trade (stock brokers)
• Listing requirements
• E.g. LSE, JSE, NYSE, etc
2. Over-the-Counter (OTC) markets:
Bonds:
• Decentralized market
• Public offering: issue of bonds sold to the general investing public that can be traded on the secondary market
• Unlisted shares
• Network of dealers buy and sell shares
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• E.g. NASDAQ, BJM OTC
3. Direct trading between two parties
NOTE:
- broker market: broker brings the buyer to the seller
- dealer market: e.g. NASDAQ: the intermediary holds the share
themself, intermediary brings the stock to the buyer
Investment banking:
• Underwriting: investment bank helps the firm to issue and market new securities
• Prospectus: describes the issue and the prospects of the company - preliminary prospectus = red herring because it includes
a statement printed in red stating that the company is not attempting to sell the security before the registration is approved
• Road show: investment bankers and the company go and visit
the people who they want money form - try to convince them
to invest - investors also give an indication of what they think a
fair price is
• Firm commitment/ best effort agreement: investment bank purchases securities form the issuing company and then resells
them to the public
• Shelf registration: SEC Rule 415: allows firms to register securities and gradually sell them to the public for two years
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Section 2
How Securities are Traded
Types of Markets:
Types of Offers:
• Direct search markets
• Market order
- buyers and sellers seek each other out
- executed immediately
• Brokered markets
- Trader receives current market price
- brokers search out buyers and sellers
• Price-contingent order
• Dealer markets
- traders specify buying or selling price
- dealers have inventories of assets from which they buy and seek
NOTE: a large order may be filled at multiple prices
• Auction markets
- traders converge to one place to trade
Bid Vs Ask Prices:
BID PRICE:
• Bids are offers to buy
• Price that investors are willing to pay
PRICE BELOW
THE LIMIT
PRICE ABOVE
THE LIMIT
BUY
Limit-Buy Order
Stop-Buy Order
SELL
Stop-Loss Order
Limit-Sell Order
ASK PRICE:
• Asks are offers to sell
• Investors must buy at the ask/ offer price
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Limit order vs option:
• Limit order does not protect against sudden, large market
movements
• Option does, but more expensive
• Can get the same protection of a limit order using an option option costs you money (premium) whereas dont have to put in
money for a limit order
• As the market price moves the computer looks and sees if the
price is the same as the limit order - as soon as it goes above
the limit it stops the order
• But market might change in seconds and may not ‘touch’ every
price so limit order doesn’t protect you if the price goes up by
R5 suddenly and passes your limit
Trading mechanisms:
• Dealer markets
• Electronic communication networks (ECNs): true trading systems that can automatically execute orders
• Specialist markets: trading is managed by a specialist assigned
responsibility for that security: maintain a “fair and orderly market”
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Section 3
US Securities Markets
NASDAQ:
Electronic Communications Networks:
• Lists about 3200 firms
• ECNs: private computer networks that directly link buyers
with sellers for automated order execution
• Originally, NASDAQ was primarily a dealer market with a
price quotation system
• Today, NASDAQ’s Market Centre offers a sophisticated electronic trading platform with automatic trade execution
• Large orders may still be negotiated through brokers and dealers
New York Stock Exchange:
• Major ECNs include NASDAQ’s Market Centre, arced, Direct
Edge, BATS, and LavaFlow
• “flash trading”: computer programs look for even the smallest
misplacing opportunity and execute trades in tiny fractions of
a second
Bond Trading:
• Lists about 2800 firms
• Most bond trading takes place in the OTC market among
bond dealers
• Automatic electronic trading runs side-by-side with traditional
broker/ specialist system
• Market for many bond issues is “thin”
• SuperDot: electronic order-routing system
• DirectPlus: fully automated execution for small orders
• NYSE is expanding its bond-trading system
• NYSE bonds is the largest centralized bond market of any US
exchange
• Specialists: handle large orders and maintain orderly trading
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Section 4
Market Structure in Other Countries
London - predominately electronic trading
Euronext - market formed by combination of the Paris, Amsterdam and Brussels exchanges, then merged with NYSE
Tokyo Stock Exchange - switched to all-electronic trading
Globalization and consolidation of stock markets - NYSE
mergers and acquisitions: Archipelago (ECN), American Stock
Exchange, Euronext
- NASDAQ mergers and acquisitions: Instinet/ INET (ECN),
Boston Stock Exchange
- Chicago Mercantile exchange acquired: Chicago Board of
Trade, New York Mercantile Exchange
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Section 5
Trading Costs
1.Brokerage Commission:
• Fee paid to broker for making the transaction
• Explicit cost of trading
• Full service (research team) vs discount brokerage (“no frills”
service)
2. Spread
• Difference between the bid and asked prices
• Implicit cost of trading
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Section 6
Buying on Margin
• Buying part of the total purchase price of a position using a
loan from a broker
• Only available to you if your broker allows it
• Investors contributes the remaining portion
• Margin refers to the percentage or amount contributed by the
investor
Stock price falls to $70 per share
New position:
Stock $7000 - Borrowed $4000
(
(
(
Equity $3000
Margin % = $3000/ $7000 = 43%
• You profit when the stock appreciates
Example 2:
How far can the stock price fall before a margin call? Let maintenance margin = 30%
Example 1:
Share price: $10
Equity = 100P - $4000
60% - Initial margin
40% - Maintenance margin
100 shares purchased
Percentage margin = (100P - $4000)/ 100P
So (100P - $4000)/ 100P = 0.3
P = $57.14
Initial position:
Stock $10 000 - Borrowed $4000
(
(
(
Equity $6000
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Section 7
Short Sales
PURPOSE: to profit from a decline in the price of a stock or security
MECHANICS: selling a share you do not have
• Borrow stock through a dealer
• Sell it and deposits proceeds and margin in an account
• Closing out the position: buy the stock and return to the party
from which is was borrowed
Example 1:
Stock owed: 1000 shares
Stock falls to $70 per share
Assets(
(
(
(
$100 000 (sale proceeds)( (
Liabilities
$70 000 (buy shares)
$50 000 (initial margin)( (
(
(
(
(
(
(
Equity
(
(
(
(
(
(
$80 000
Profit = ending equity - beginning equity
= $80 000 - $50 000 = $30 000
1000 shares
50% - margin
(
(
= decline in share price x number of shares sold short
30% - maintenance margin
$100 - initial price
Example 2:
How much can the stock price rise before a margin call?
Sales proceeds: $100 000
Margin & equity: $50 000
[($100 000 + $50 000) - 1000P]/ 1000P = 30%
P = $115.38
40
Section 8
Regulation of Securities Markets
Major regulations:
Insider trading:
• Securities Act of 1933
• Officers, directors and major stockholders must report all transactions in firm’s stock
• Securities Act of 1934
• Securities Investor Protection Act of 1970
• Insider do exploit their knowledge
- Jaffe study
Self-regulation:
• Financial Industry Regulatory Authority
- Inside buyers > inside sellers = stock does well
- Inside sellers > inside buyers = stock does poorly
• CFA Institute standards of professional conduct
• Sarbanes-Oxley Act
- Public Company Accounting Oversight Board
- Independent financial experts to serve on audit committees of
boards of directors
- CEOs and CFOs personally certify firms’ financial reports
- Boards must have independent directors
41
Chapter 5
Macroeconomic & Industry Analysis
Valuation: Fundamental Analysis:
models a company’s value by
assessing its current and future
profitability. Purpose: to identify
mispriced stocks relative to some
measure of “true” value derived
from financial data.
Section 1
Introduction to Valuation
Fundamental Analysis:
Mispriced stocks:
• Fundamental value > market value : BUY
• Fundamental value < market value : SELL
There are various Models of Equity Valuation as well as
Valuation by Comparables. How then do you select a
method?
• Multiples incorporate current market conditions more so than
free cash flows - so time period is important
• Certain valuation methods suit certain industries better - gold
mines valued with reference to their assets - comparable
model. Banks use price to book ratio. Retail and industrial companies - free cash flow to firm
Forecasting Earnings: Basics
Top-Down Analysis:
• CA approach
Method selection:
• Data availability - if given accounting information, cant do
FCF
• Minority share vs majority share - minority share: cannot influence dividend policy because don’t have control, best return
you can expect is future dividends; if you can influence dividend policy: move away from dividend valuation to an earnings
valuation
1. Analyse and forecast macroeconomic variables - which
country is best? Which asset class is better ie debt or equity?
2. Analyse and forecast the industry - consider business cycles
and which industry is better in which business cycle
3. Within industries with best expected future, select the firms
with best future expectations relative to their current prices to
invest in
• Free cash flow models are theoretically better
43
Bottom-Up Analysis:
Logic of Top-Down:
1. Analyse and forecast company-specific issues
Safest way: diversify portfolio and aim to beat the market by a little.
2. Consider future of industry and macroeconomy as part of
company specific issues
3. Invest in companies with the best expected future earnings
relative to their current price
Which is better? IT DEPENDS.
Top-Down: don’t really expect share to go up or down that much
= markets are efficient
Bottom-Up: don’t expect market efficiency
• Economic Analysis: determines asset class allocation bonds, equities, property, cash, etc
• Industry Analysis: determines industry sector weightings
in equity portfolio - financials, industrial, resources, etc
Top-Down Analysis:
• Global economic forecast
• Company Analysis: determines company weightings per sector
• Domestic macroeconomic forecast
• Industry analysis
• Analyse company positioning within the selected industries
• Specific company valuations
• Selection and investment decision
44
Section 2
The Global Economy & the Domestic Macroeconomy
• Stock markets around the world responded in unison to the financial crisis of 2008
• Performance in countries and regions can be highly variable
• It is harder for businesses to succeed in a contracting economy
than in an expanding one
• Political risk: the global environment may present much
greater risks than normally found in US-based investments
• Exchange rate risk: changes the prices of imports and exports
• Stock prices rise with earnings
• P/E ratios are normally in the range of 12-25
• The first step in forecasting the performance of the broad market is to assess the status of the economy as a whole
• Key variables: GDP, unemployment rates, inflation, interest
rates (increase in rates will suppress the global economy),
budget deficit, consumer sentiment
• Indebtedness (not just of the consumer but of the government
sector and banking sectors as a whole)
45
Section 3
Demand & Supply Shocks
Demand shock:
Supply shock:
• An event that affects demand for goods and services in the
economy
• An event that influences production capacity or production
costs
• Positive: reductions in tax rates, increases in the money supply,
increases in government spending or increases in foreign export demand
• Examples: changes in the price of imported oil, freezes, floods
or droughts that might destroy large quantities of agricultural
crops, changes in the educational level of an economy’s workforce, or changes in the wage rates at which the labour force is
willing to work
• Usually characterized by aggregate output moving in the same
directions as interest rates and inflation
• E.g. Increase in government spending - increase GDP - increase interest rates by increasing the demand for borrowed
funds by the government as well as by businesses that might desire to borrow to finance new ventures - increase inflation rate
if demand for goods and services is raised to a level at or beyond the total productive capacity of the economy
• Usually characterized by aggregate output moving in the opposite direction of inflation and interest rates
46
Section 4
Federal Government Policy
Demand-side policies:
• Fiscal policy - government spending and tax
• Monetary policy - manipulation of the money supply
• Increasing the money supply lowers interest rates and stimulates the economy
• Less immediate effect than fiscal policy
• Tools of monetary policy include open market operations, discount rate and reserve requirements
Fiscal policy:
• Most direct way to stimulate or slow the economy
• Formulation of fiscal policy is often a slow political process
• To summaries the net effect of fiscal policy, look at the budget
surplus or deficit:
• Deficit stimulates the economy because: it increases the demand for goods (via spending) by more than it reduces the demand for goods (via taxes)
Supply-side policies:
• Goal: to create an environment in which workers and owners
of capital have the maximum incentive and ability to produce
and develop goods
• Supply-siders focus on how tax policy can be used to improve
incentives to work and invest
Monetary policy:
• Manipulation of the money supply to influence economic activity
47
Section 5
Business Cycles
The transition points across cycles are called peaks and troughs.
Defensive industries:
• A peak is the transition from the end of an expansion to the
start of a contraction
• Little sensitivity to the business cycle
• A trough occurs at the bottom of a recession just as the economy enters a recovery
Cyclical industries:
• Above-average sensitivity to the state of the economy
• Examples: producers of consumer durables e.g. Autos and capital goods ie goods used by other firms to produce their own
products
• High betas
• Examples: food producers and processors, pharmaceutical
firms and public utilities
• Low betas
Economic indicators:
• Leading indicators tend to rise and fall in advance of the economy
• Coincident indicators move with the market
• Lagging indicators change subsequent to market movements
• If you forecast an upturn you want to be in a cyclical industry
• If you forecast a slowdown you don’t want to be in a cyclical industry
48
Section 6
Industry Analysis
General principle: company will perform better if the whole
economy is doing well
Defining an industry:
• North American Industry Classification System (NAICS)
codes
• Firms with the same four-digit NAICS codes are commonly
taken to be in the same industry
Sensitivity to the Business Cycle:
• Three factors determine how sensitive a firm’s earnings are to
the business cycle
1. Sensitivity of sales: necessities vs discretionary goods. Items
that are not sensitive to income levels e.g. Tobacco and movies
vs items that are e.g. Machine tools, steel, autos
2. Operating leverage: the split between fixed and variable costs.
Firms with low operating leverage (less fixed assets) are less
sensitive to business conditions. Firms with high operating lev-
erage (more fixed assets) are more sensitive to the business cycle
3. Financial leverage: the use of borrowing. Interest is a fixed cost
that increases the sensitivity of profits to the business cycle
BUSINESS CYCLE PICTURE ****************
Sector Rotation:
• Portfolio is shifted into industries or sectors that should outperform according to the stage of the business cycle
• As the business cycle moves you put more emphasis on the sectors that are favored at that point in the business cycle - portfolio shift
• Peaks - natural resource extraction firms
• Contraction - defensive industries such as pharmaceuticals and
food
• Trough - capital goods industries
• Expansion - cyclical industries e.g. Consumer durables
49
SECTOR ROTATION PICTURE ************
NOTE: not an “absolute” rotation - more like a shift of emphasis
Industry Life Cycles:
4.Bargaining power of buyers: higher bargaining power = can
demand price concessions
5. Bargaining power of suppliers: if a supplier has monopolistic control over the product it can demand higher prices for
the good and squeeze profits
• Start-Up Stage - rapid and increasing sales growth
• Consolidation - stable sales growth
• Maturity - slowing sales growth
• Relative decline - minimal or negative sales growth
Which life cycle is most attractive? IT DEPENDS.
Industry structure and performance:
To evaluate the competitiveness of an industry: use the Five Determinants of Competition:
1. Threat of entry: harder to enter = less competitive
2. Rivalry between existing competitors: more firms already
in industry = more price competition
3. Pressure from substitute products: industry may face competition from firms in related industries
50
Chapter 6
Equity Valuation Models
Models of Equity Valuation:
• Balance sheet models
• Dividend Discount Models
• Price/ Earnings Ratios
• Free Cash Flow Models
Section 1
Valuation by Comparables
• Compare valuation ratios of firm to industry averages
• Ratios like price/ sales are useful for valuing start-ups that have
yet to generate positive earnings
Limitations of book value:
• Book values are based on historical cost, not actual market values
• It is possible, but uncommon, for market value to be less than
book value
• “floor” or minimum value is the liquidation value per share
• Tobin’s q is the ratio of market price to replacement cost - in
the long run this ratio will tend towards one (in theory)
52
Section 2
Intrinsic Value vs Market Price
The return on a stock is composed of dividends and capital gains
or losses.
Required return:
CAPM gives the required return, k:
Is a stock attractively priced today given your forecast of next
year’s price?
K = rf + B[E(rM) - rf]
The expected holding-period return:
Expected HPR = E(r) = (E(D1) + [E(P1) - P0])/ P0
If the stock is priced correctly, k should equal expected return.
K is the market capitalization rate.
The expected HPR is the sum of the expected dividend yield,
E(D1)/P0 and the expected rate of price appreciation = capital
gains yield [E(P1) - P0]/ P0.
The expected HPR may be more or less than the required return, based on the stock’s risk.
The intrinsic value (IV) is the “true” value, according to a model.
The market value (MV) is the consensus value of all market participants.
IV > MV : BUY
IV < MV : SELL or SHORT SELL
IV = MV : HOLD or FAIRLY PRICED
53
Section 3
Dividend Discount Models (DDM)
Vo = D1/ 1+k + D2/ (1+k)2 + D3/ (1+k)3 + …
The DDM says the stock price should equal the present value of
all expected future dividends into perpetuity
Example: constant growth DDM
A stock just paid an annual dividend of $3/ share. The dividend is
expected to grow at 8% indefinitely and the market capitalization rate is 14%.
V0 = $3(1+0.08)/0.14-0.08
Constant growth DDM:
= $54
V0 = D0(1+g)/k-g = D1/k-g
DDM Implications:
Example: preferred stock
No growth case.
Value a preferred stock paying a fixed dividend of $2 per share
when the discount rate is 8%.
V0 = $2/0.08 - 0
= $25
The constant growth rate DDM implies that a stock’s value will
be greater:
1. The larger its expected dividend per share
2. The lower the market capitalization rate, k
3. The higher the expected growth rate of dividends
The stock is expected to grow at the same rate as the dividends.
54
Estimating Dividend Growth rates:
g = ROE x b
P0 = $2/0.15 -0.06
= $22.22
Where b = retention ratio = 1 - payout ratio
PVGO = price per share - no growth value per share
Present value of Growth Opportunities:
= $22.22 - $5/0.15
The value of the firm equals:
= $11.11
The value of the assets already in place, the no-growth value
of the firm plus
The NPV of its future investment, the present value of
growth opportunities.
Life Cycles and Multi-stage Growth Models:
Expected dividends for Honda:
2010 - $0.50
P0 = E1/k + PVGO
2011 - $0.66
2012 - $0.83
Example:
Firm reinvests 60% of its earnings in projects with ROE of 10%,
capitalization rate is 15%. Expected year-end dividend is $2/
share, paid out of earnings of $5/share.
2013 - $1.00
Since the dividend payout ratio is 30% and ROE is 11%, the
“steady-state” growth rate is 7.7% since:
g = ROE x b = 0.11 x (1 - 0.30) = 7.7%
g = ROE x b = 0.1 x 0.6 = 6%
55
Honda’s beta is 0.95 and the risk-free rate is 3.5%. If the market
risk premium is 8% then k is:
K = 3.5% + 0.95 x 8% = 11.1%
So:
P2013 = D2014/ k -g = D2013(1+g)/ k -g = $1(1+ 0.077)/0.111-0.077
= $31.68
Finally:
V2009 = $0.50/1.111 + $0.66/1.1112 + $0.83/1.1113 +
($1+$31.68)/1.1114
In 2009, one share of Honda Motor Company Stock was worth
$23.04.
56
Section 4
Price-Earnings Ratio & Growth
The ratio of PVGO to E/k is the ratio of firm value due to
growth opportunities to value due to assets already in place (ie
the no-growth value of the firm)
Wall Street rule of thumb: the growth rate is roughly equal to
the P/E ratio. However, not true if you’re at extremes e.g. P/E =
25, not true that growth rate will be the same.
P0/E1 = 1/k(1 + PVGO/(E/k))
When PVGO = 0, Po = E1/k. The stock is valued like a nongrowing perpetuity.
P/E ratios and Stock Risk:
When risk is higher, k is higher, so P/E is lower.
P/E rises dramatically with PVGO ie high P/E = high growth opportunities.
P/E = 1 - b/ k - g
- higher growth rate = higher PE
- Higher required return = lower PE
P/E increases:
• As ROE increases
• As plowback increases, as long as ROE > k
Pitfalls in P/E Analysis:
Use of accounting earnings:
P0/E1 = (1 - b)/(k - ROE x b)
- forward looking because takes into account next period’s earnings
• Influenced by accounting rules e.g. Historical cost of depreciation
• Earnings management: practice of using flexibility in accounting tules to improve the apparent profitability of the firm
• Choices on GAAP
57
• Inflation
• Reported earnings fluctuate around the business cycle
Other Comparative Valuation Ratios:
• Price-to-book ratio: bank
• Price-to-cash-flow ratio
• Price-to-sales ratio: start up business
58
Section 5
Free Cash Flow Approach
Value the firm by discounting free cash flow at WACC.
Comparing the valuation models:
Free cash flow to firm: attempts to value total firm, including
parts of firm financed by long term debt.
In practice:
Free cash flow to equity: need to take the after tax cost of debt
off and add back any net increase in long term debt.
• Values from these models may differ
• Analysts are always forced to make simplifying assumptions
FCFF:
After-tax EBIT
+ Non-cash flow items
+ Depreciation
- Capital expenditures
- Increase in net working capital
59
Section 6
The Aggregate Stock Market
Explaining Past Behaviour:
• Stock market is a leading economic indicator
• Economic events and the anticipation of such events do have a
substantial effect on stock prices
• Two factors with greatest impact: interest rates and corporate
profits
Forecasting the Stock Market:
• Earnings multiplier approach used to forecast
60
Chapter 7
Financial Statement Analysis
Can be used to discover misplaced
securities.
Financial accounting data is widely
available but, accounting earnings
and economic earnings are not
always the same thing.
Section 1
The Major Financial Statements & Earnings
Income Statement:
Economic earnings:
• Profitability over time
• Sustainable cash flow that can be paid to stockholders without
impairing productive capacity of the firm
• Useful to distinguish four broad classes: cost of goods sold, general and administrative expenses, interest expense on the firm’s
debt and taxes
Accounting earnings:
• Affected by conventions regarding the valuation of assets
Balance Sheet:
• Financial condition at a point in time
• List of the firm’s assets and liabilities at that moment
Statement of Cash Flows:
• Tracks the cash implications of transactions
62
Section 2
Profitability Measures
ROE:
Financial Leverage and ROE:
• Measures profitability for contributors of equity capital
• ROE can differ from ROA because of leverage
After-tax profit/ book value of equity
• Leverage makes ROE more volatile
• Let t = tax rate and r = interest rate then:
ROA:
• Measures profitability for all contributors of capital
EBIT/ total assets
• If there is no debt or ROA = r, ROE will simply equal ROA(1-t)
Past vs Future ROE:
• ROE is a key determinant of earnings growth
• Past profitability does not guarantee future profitability
• If ROA > r, the firm earns more than it pays out to creditors
and ROE increases
• If ROA < r, ROE will decline as a function of the debt-toequity ratio
• Security values are based on future profits
• Expectations of future dividends determine today’s stock value
63
Section 3
Ratio Analysis
Decomposition of ROE: Du Pont Method
Margin and turnover are unaffected by leverage.
Where:
ROA reflects soundness of firm’s operations, regardless of how
they are financed.
(1) = tax burden
(2) = interest burden
(3) = margin
(4)= turnover
Tax burden is not affected by leverage.
Compound leverage factor = interest burden x leverage
(5) = leverage
Choosing a Benchmark:
NOTE: Pretax = EBIT - Interest Expense
Also: interest burden is similar to the times interest earned ratio:
Interest coverage = EBIT/ Interest Expense
• Compare the company’s ratios across time
• Compare ratios of firms in the same industry
• Cross-industry comparisons can be misleading
NOTE: SEE RATIOS FROM FINANCE 1
64
Section 4
Accounting Fraud
Categories:
• Recording revenue too soon or of questionable quality - ZZZ,
Enron
• Boosting income with one-time gains
• Shifting current expenses to later period
• Failing to record liabilities
• Shifting current revenue to later period
• Shifting future expenses to the current period as special charge
- provisions
Signs:
• Weak control environment
• Extreme competitive pressure
• Management known or suspected of having questionable character
• Wary of fast-growing companies, basket case companies or
newly listed companies
65
Chapter 8
EPS & Dilution
EPS x P/E ratio = value
Section 1
Earnings per Share
Diluted Earnings per Share:
• For complex capital structures - earnings to ordinary shareholders is adjusted to correct for potential dilution
Subtract preferred dividends because do not accrue to ordinary
shareholders.
• Dilution - other parties have ability to convert other financial
obligations/ instruments into ordinary shares, result: ordinary
shareholder’s claim on company’s earnings reduced (diluted)
Weighted average number of shares (WANOS):
• Potentially dilutive securities - options, warrants, convertible
debt
• Number of common shares in issue during the year
• Securities are only diluted if - exercising them reduces EPS
• Weighted by the percentage of the year they were in issue
• If not - securities are anti-dilutive ie should they be converted/
exercised they will increase the EPS for ordinary shareholders
Alternatives:
• Headline - JSE defined exclusions
• Anti-dilutive securities are ignored when calculating diluted
earnings per share (DEPS), as rational holders will not exercise
them
• Recurring headline - PSG defined exclusions
• Attributable - excludes minority interest in profits
• Diluted
67
Adjusted # of shares =
WANOS
+ shares from conversion of preferred shares
Where:
+ Shares from conversion of convertible debt
Adjusted income available to common shares =
+ Shares issuable from share options
earnings available to common shares
- shares bought back with option proceeds
+ dividends on convertible preferred shares
+ after-tax interest on convertible debt
Basic Rules: Diluted EPS
Rule 1:
If dilutive securities in issue throughout year:
• Treat as though converted to ordinary shares at beginning of
year
If issued in year:
Where:
• Treat as though immediately converted on issue
Adjusted earnings =
(Net income - Preferred dividends)
Rule 2:
+ convertible preferred dividends
+ Convertible debt interest x (1 - tax rate)
• Exclude discontinued operations, extraordinary items and accounting changes from earnings
• Only earnings from continued operations are relevant
68
Rule 3:
Treasury Stock Method: Options
• For diluted EPS:
• Assumes funds hypothetically received from conversion of options used to repurchase shares in the market at the average
market price
• Denominator = basic EPS denominator adjusted for equivalent
# of ordinary shares created on conversion of all outstanding
dilutive instruments
Rule 4:
• Therefore, method reduces total number of shares that are created by hypothetical option exercise
• Net shares created on exercise = shares created by option exercise - shares repurchased
• Stock options and warrants:
• Are dilutive only when: exercise price < market price
Example:
• No adjustment to net income in numerator
Ordinary shares in issue for the year = 12000
• If dilutive, increase the # of shares using Treasury Share
method
Preferred shares (10%, par R100) in issue for the year = 1500
Convertible bonds in issue for the year (8%, par R1000) = 120
Rule 5:
• If dilutive securities are present that will cause WANOS to
change, numerator must be adjusted as follows:
• If there are dilutive preferred shares, add preferred dividends
back to income from continuing operations
• If convertible bonds are dilutive, add after-tax interest expense
back to earnings
Options in issue for year (1 option = 10 shares @ R25 each) = 250
Tax rate = 30%
Bond conversion factor: 1 bond = 10 ordinary shares
Net income = R112650
Average market price of shares through the year = R33
Calculate both basic and diluted earnings.
69
1500 preferred shares in issue for the year, means preferred dividends of R15000
Net new total shares = 12000 + 2500 - 1894 = 12606
New earnings:
= basic earnings - pref divs
Next Step: test every potentially dilutive security for possible
dilutive effect
= 112650 - 15000
Convertible bonds - test for dilution:
= R97650
Number of new shares created on conversion:
# new shares created = 120 x 10 = 1200
New total shares = 12000 + 1200= 13200
Which is dilutive compared to 814 cents/share.
New earnings:
= basic earnings - pref divs + [conv debt int x (1 - tax rate)]
Both convertible bonds and options are dilutive, so both are
taken into account for DEPS calcs.
= 112650 - 15000 + [120 x 1000 x 8% x (1 - 0.3)]
= R104370
Options - test for dilution (Treasury Method):
Number of new shares created on exercising:
Shares bought back = (250 x 10 x R25)/ R33 = 1894
New earnings:
Options do not affect earnings, but convertible bonds do, so new
earnings = R104370.
Both convertible bonds and options affect # of shares used for
DEPS calcs.
70
New # of shares:
Ordinary shares (
(
(
(
12000
+ shares from conversion of convertible bonds( (
+1200
+ Shares issuable from share options( (
(
+2500
- shares bought back with share option proceeds(
-1894
= adjusted # of shares(
=13806
(
(
(
(
(
(
(
(
(
71
Chapter 9
High Frequency Trading
Section 1
HFT
What is it?
What information is used?
Current happenings in Europe:
Issue about STT in SA:
73
Chapter 10
Behaviour Finance & Technical Analysis
Conventional Finance: prices are
correct, resources are allocated
efficiently, consistent with EMH.
Behavioral Finance: what if
investors don’t behave rationally?
Section 1
The Behavioural Critique
Information processing:
Sample size neglect and representativeness:
Forecasting errors:
The notion of representativeness holds that people commonly
do not take into account the size of a sample, acting as if a small
sample is just as representative of a population as a large one.
People give too much weight to recent experience compared to
prior beliefs when making forecasts - memory bias.
Tend to make forecasts that are too extreme given the uncertainty inherent in their information.
Overconfidence:
People tend to overestimate the precision of their beliefs or forecasts, and they tend to overestimate their abilities.
Conservatism:
A conservatism bias means that investors are too slow in updating their beliefs in response to new evidence. Such a bias would
give rise to momentum in stock market returns.
They may therefore infer a pattern too quickly based on a small
sample and extrapolate apparent trends too far into the future.
Behavioral biases:
Framing:
Decisions seem to be affected by how choices are framed.
Individuals can act risk averse in terms of gains but risk seeking
in terms of losses.
Mental accounting:
Mental accounting is a specific form of accounting in which people segregate certain decisions.
75
E.g. An investor may take a lot of risk with one investment account, but establish a conservative position with another account that is dedicated to her child’s education.
The house money effect refers to gamblers’ greater willingness to
accept new bets if they currently are ahead.
Limits to Arbitrage:
Fundamental risk:
The fundamental risk incurred in exploiting apparent profit opportunities presumably will limit the activity of traders.
Regret avoidance:
E.g. Buying an underpriced security - risk is that the price will
never converge to the intrinsic value.
It has been found that individuals who make decisions that turn
out badly have more regret when that decision was more unconventional.
“Markets can remain irrational longer than you can remain solvent.”
E.g. Buying a blue-chip share which fails is not as painful as buying an unknown share that fails. Blue-chip failure can be attributed to bad luck whereas unknown failure can be put to bad decision making.
Prospect theory:
Prospect theory modifies the analytic description of rational
risk-averse investors.
Intrinsic value and market value may take too long to converge.
Implementation costs:
Short-selling a security entails costs, short0-sellers may have to
return the borrowed security on little notice, rendering the horizon of the short sale uncertain, other investors face limits on
their abilities to short securities.
Model risk:
Conventional view: Utility depends on level of wealth.
Behavioral view: Utility depends on changes in current wealth.
There is a risk that an apparent profit opportunity is more apparent than real.
76
Section 2
Technical Analysis & Behavioural Finance
Technical Analysis and Behavioral Finance:
• Technical analysis attempts to exploit recurring and predictable patterns in stock prices
• Prices adjust gradually to a new equilibrium
• Market values and intrinsic values converge slowly
• Disposition effect: the tendency of investors to hold on to losing investments
• Demand for shares depends on price history
• Can lead to momentum in stock prices
Trends and Corrections:
Dow Theory:
1. Primary trend: long-term movement in prices, lasting from several months to several years
2. Secondary or intermediate trend: short-term deviations of
prices from the underlying trend line and are eliminated by corrections
3. Tertiary or minor trends: daily fluctuations of little importance
Moving Averages:
• The moving average is the average level of prices over a given
interval of time
• Bullish signal: market price though the moving average line
from below - time to buy
• Bearish signal: when prices fall below the moving average, it is
time to sell
Breadth:
• Often measured as the spread between the number of stocks
that advance and decline in price
• A measure of the extent to which movement in a market index
is reflected widely in the price movements of all the stocks in
the market
77
Sentiment Indicators:
Warning:
Trin Statistic:
• It is possible to perceive patterns that really don’t exist
Common patterns:
Ratios above 1.0 are bearish
Confidence index:
• The ratio of the average yield on 10 top-rated corporate bonds
divided by the average yield on 10 intermediate-grade corporate bonds
• Higher values are bullish
Head-and-Shoulders
• Three peaks close together but rising off a common base in a
share price graph, with the middle one being the “head” and
the other two either side of it the “shoulders”
• Normally considered to be a very bearish (negative) pattern ie
leading to a sharp decline in share price
• Theory: if the share price persistently increases before falling
back, but by the third time is unable to reach previous high, investors are likely losing confidence in the share
Put/ call ratio:
Cup-and-Handle
• Calls are the right to buy - a way to bet on rising prices
• Puts are the right to sell - a way to bet on falling prices
• A rising ratio may signal investor pessimism and a coming market decline
• Contrarian investors see a rising ratio as a buying opportunity
• Consists of a “cup” in shape of U, followed by a period of relative price stability (the “handle”), which may have slight downward drift
• This is a bullish pattern, considered to be the prelude to an important price breakout
• Theory: as the share price rises from the low levels inside the
“cup”, it starts to test old highs. People who had previously
78
bought at these highs sell their shares so as to break-even (the
lows reached in the cup being fresh in their minds). This results in the share trading sideways for a while (increased supply
temporarily halting the upward trend). Once all the excess supply is in the market (the above group having sold), the share is
set to resume its upward trend (albeit at an accelerated pace the breakout).
The Double Bottom:
• Resembles a “W”
• Considered to be very bullish, preceding a rise in share price
• Idea is to buy when the price passes the highest point in the
“handle” (the middle peak) on its way out of the double bottom
• Theory: after two declines in the share price, all the uncertain
investors have sold, and only the long-term investors are not
still in the share. This means few potential sources of supply remain, resulting in a higher share price going forward.
79
Chapter 11
Risk and Return
Section 1
Background to Portfolio Theory
Markowitz’s Theory:
Introduced diversification, the idea that risks are correlated in
some way, a distinction between efficient and inefficient portfolios, the idea of risk-return trade offs on a portfolio and expected
return and variance as criteria for portfolio selection
Assumed only positive investments and said the investor will
choose a portfolio from an efficient frontier
Roy:
Assumed positive and negative investments and said the investor
will choose a specific portfolio
Tobin:
Separation Theorem:
First select a portfolio of risky assets - subject to market risk.
Decide on mix of this and the risk-free asset
81
Section 2
Introduction to Risk and Return
Let:
Given:
R = nominal rate
t = tax rate
r = real rate
R = nominal rate of interest
i = inflation rate
After-tax rate is R(1 - t)
Real after-tax rate is R(1 - t) - i
Excess returns and risk premiums:
Also if:
E(i) = current expectation of the inflation rate that will prevail
over the coming period
Excess return = the difference between the actual rate of return
on a risky asset and the actual risk-free rate.
Sharpe ratio:
Measures the excess returns per unit of risk.
82
Chapter 12
Risky Aversion and Capital Allocation to
Risky Assets
Section 1
Risk Aversion and Capital Allocation to Risky Assets
Speculator: assumes considerable risk in order to obtain commensurate gain (positive risk premium)
Gamble: bet or wager on an uncertain outcome
Risk averse investors are willing to consider only risk-free prospects or speculative prospects with positive risk premiums
Risk-neutral investors look only at expected returns when making an investment decision. (e.g. Receive R100 @ 100% certainty
vs. 50% chance of R200). Risk-neutral investors wouldn’t mind
either way.
Evidence of risk aversion:
Insurance – protects you against the risk of a loss of an asset
Preference of Bonds – people buy bonds with lower return rather
than high risk equity with higher return.
Are people completely risk loving or risk averse?
Insurance
The Basic assumption remains…
Portfolios with higher expected returns and lower risk are considered more attractive and will therefore be ranked higher in terms
of preference
“Most investors devoting large sums of money to a portfolio are
risk averse”
If faced with portfolios of equal return, the risk averse investor will choose the portfolio with the lower risk…conversely…..
84
The$U&lity$Func&on$–$Differences$in$investor$risk$
aversion$
High%U'lity%
• The$risk3return$trade3
off$is$one$in$which$
greater$risk$is$taken$if$
greater$returns$can$be$
expected,$resul&ng$in$
a$posi&ve$slope.$
• The$highest$
indifference$curve$(the$
one$in$the$most$
Lower%U'lity%
northwestern$
posi&on)$offers$the$
greatest$u&lity.$$
Moderately%%Risk%
Averse%
Expected%Return%
Expected%Return%
The$U&lity$Func&on$
More%Risk%
Averse%
Less%%Risk%Averse%
Risk%Loving%
$
Same%measure%of%
u'lity%
Standard%Devia'on%
(Risk)%
Standard%Devia7on%(Risk)%
Note:%
• The$risk$lover$is$
prepared$to$accept$
lower$expected$
returns$as$risk$
increases.$
• By$contrast,$the$Risk$
Averse$investor$$
demand$significantly$
higher$expected$
returns$per$unit$of$
risk.$
$
The"U$lity"Func$on"
Portfolio that is best for us will be the one that is tangent to the
highest utility
• To"rank"your"investment"por=olios"you"need"to"assign"them"a"level""
or"measure"of"u$lity"which"is"based"on"expected"return"and"
variance.""This"is"defined"by"…"
Where:&
U""""""""""!
="U$lity"(Certainty"Equivalent"Rate"of"Return)"
E"("r")!="expected"Return"on"the"asset"or"por=olio"
A!="coefficient"of"risk"aversion"
σ2"="variance"of"returns"
Note"that"rates"of"return"are"expressed"as"
decimals"and"not"integers"in"this"equa$on."
85
As can be seen from the equation, utility is enhanced by higher
expected returns and lower risk.
At the risk free level utility will be equal to the E(r) ….1/2Aδ2 will
be equal to zero since risk (δ ) is equal to zero.
Risk averse investors will have a higher value “A”, they tend to penalise risky investments severely.
The utility score of a risky portfolio represents the certainty
equivalent rate of return that the risk free asset must meet or
beat if it is to be chosen. A portfolio will only be desired if the
CERR is greater than that of a risk free alternative.
!="($)−​#/$ )*$(
The&U3lity&Func3on&
Example(1:(
Assume&an&investor&with&the&following&u3lity&func3on:&&
U(=(E(r)(0(3/2(σ2).(
He&has&a&choice&between&2&assets&with&E&(&r)&and&σ&as&follows:&
&
P('n(P(
Shoprite(C(
&
E((r()(
10%&
15%&
&
σ(
10%&
10%&
&
&
Which&asset&(share)&should&he&invest&in?&
Solution:
Objective is to Maximize the investor’s utility, given their level of
risk aversion (generally referred to as the co-efficient of risk aversion)
!="($)−​#/$ )*$(
An investor that is risk neutral will have a risk index of zero. For
them the portfolios CERR= E(r); risk is irrelevant.
A risk lover will have a risk index <0. These investors are happy
with “fair game” and “gambles”
The"U9lity"Func9on"
Example(2:(
A"por&olio"has"an"expected"rate"of"return"of"0.15"and"a"standard"
devia9on"of"0.15."The"risk<free"rate"is"6"percent."An"investor"has"the"
following"u9lity"func9on:"U"="E(r)"<"(A/2)σ2.""Investor""X"has""coefficient"
of"risk"aversion"of"4""and"investor"Y"has"a"coefficient"of"risk"aversion"of"
8."Which"investor"is"indifferent"between"the"risky"por&olio"and"the"risk"
free"asset."
"
86
Solution:
Objective is to find the coefficient of risk aversion (A) at which
the U given by U = E(r) - (A/2)σ2 is equal ton the utility of the
risk free asset.
We know that…U(Rf) = 6%.
We can substitute the coefficient of risk aversion for each investor into the formula U = E(r) - (A/2)σ2 or..
greater than any portfolio within that quadrant. Similarly, portfolio “A” has a risk (SD) is equal to or lower than any portfolio that
is located within quadrant IV.
The criteria on which the choice of portfolio “A” is made is referred to as the mean-standard deviation or the mean-variance
criterion. Under this criteria portfolio “A” will be superior to
portfolio “B” if E(rA) >/= E(rB) and δA </= δB .
We can solve for “A” given a Utility (U) of 6%
The$Trade(off$Between$Risk$and$Returns$$
Mean-variance criterion. "
E"(r")"
II"
I"
E"(rp")"
For this criterion to hold, if expected returns are equal [E(rA) =
E(rB)], then the SD of portfolio “A” must be less than that of
portfolio “B” [δA < δB ]. Alternatively, if the return on portfolio
“A” id greater than that of portfolio “B’ [E(rA) > E(rB)], it is sufficient for the risk (SD) of the 2 portfolios to be equal [δA = δB ]
for the mean-variance criterion to hold.
Por,olio"“A”"
III"
IV"
σp"
Por,olio"“B”"
Anything in quadrant 4 is worse than anything in quadrant 1
σ"
For the investor the portfolio represented by ”A” has an expected
return of E(rp) and a SD of δp. The risk averse investor will prefer this portfolio to any other portfolio in quadrant IV because
portfolio “A” has an expected return that is at least equal to or
87
criterion. Under this criteria portfolio “A” will be superior to
portfolio “B” if E(rA) >/= E(rB) and δA </= δB .
The$Trade(off$Between$Risk$and$Returns$$
Mean-variance criterion conditions:. "
Under$ this$ criteria$ por7olio$ “A”$ will$ be$
superior$ to$ por7olio$ “B”$ if$ E(rA)" >/=" E(rB)"
and$δA"</="δB".""
$
$
If$expected$returns$are$equal$[E(rA)"="E(rB)],$
then$ $the$SD$of$por7olio$“A”$must$be$less$
than$that$of$por7olio$“B”$[δA"<"δB"].""
I"
II"
III"
IV"
$
AlternaEvely,$ if$ the$ return$ on$ por7olio$ “A”$ is$ greater$ than$ that$ of$
por7olio$ “B’$ [E(rA)" >" E(rB)]," it$ is$ sufficient$ for$ the$ risk$ (SD)$ of$ the$ 2$
por7olios$ to$ be$ equal$ [δA" =" δB" ]" for$ the$ mean(variance$ criterion$ to$
hold.$
For this criterion to hold, if expected returns are equal [E(rA) =
E(rB)], then the SD of portfolio “A” must be less than that of
portfolio “B” [δA < δB ]. Alternatively, if the return on portfolio
“A” id greater than that of portfolio “B’ [E(rA) > E(rB)], it is sufficient for the risk (SD) of the 2 portfolios to be equal [δA = δB ]
for the mean-variance criterion to hold.
What about quadrants II and III?"
NB: for exam purposes. Be about to explain the mean variance.
The$Trade(off$Between$Risk$and$Returns$$
Mean-variance criterion. "
E"(r")"
Let’s turn our focus back to figure 6.1. For the investor the portfolio represented by ”A” has an expected return of E(rp) and a SD
of δp. The risk averse investor will prefer this portfolio to any
other portfolio in quadrant IV because portfolio “A” has an expected return that is at least equal to or greater than any portfolio within that quadrant. Similarly, portfolio “A” has a risk (SD) is
equal to or lower than any portfolio that is located within quadrant IV.
E(r) >
δ
I"
II"
Por,olio"“E”"
A;C;D;E;F$
Dominates$B$
Por,olio"“C”"
D;A;C$$($
Indifferent$
Por,olio"“F”"
E"(rp")"
Por,olio"“A”"
E(r) <
δ
Por,olio"“D”"
Por,olio"“B”"
F;E$
Dominates$
D;A;C$;B$
IV"
III"
σp"
σ"
The criteria on which the choice of portfolio “A” is made is referred to as the mean-standard deviation or the mean-variance
88
When it comes to quadrant II and III we need to consider the
trade off between risk and return. It is possible for the investor
to identify other portfolios that are equally preferred to portfolio
“A”. However this will call for some trade-off between risk and return. One such portfolio is portfolio “C”. The investor will be indifferent between portfolio C ,A and D, hence the indifference
curve
Capital'Alloca+on'
Across&Risky&and&Risk.Free&
Por2olios&&
• Control'risk''
– Asset'alloca+on'choice'
• Frac+on'of'the'por4olio'invested'in'
Treasury'bills'or'other'safe'money'
market'securi+es'
Risk free
asset
Any portfolio to the left of the indifference curve is preferred to
portfolios identified by the indifference curve because they have
a higher return and lower risk (quadrant I) or a significantly
higher return for marginal increase in risk (quadrant IV).
Conditions that apply to quadrants 2 and 3:
Would only accept investments in quadrant 2 if the increase in
Expected Return is greater than the increase in standard deviation
Would only accept investments in quadrant 3 if the decrease in
Expected reurn is less than the decrease in standard deviation
Risky asset
Level of
Risk
You can control risk by simply re-allocating assets from risky to
risk free assets.
Incr. risk in risk free - risk goes down.
Portfolios of one risky asset and one risk-free asset:
It’s possible to split investment funds between safe and risky assets.
Risk free asset: proxy; T-bills
Risky asset: stock (or a portfolio)
89
Construc.on$of$a$Por(olio$of$1$Risky$Asset$$
and$1$Risk?Free$Asset$
The$por(olio$consis.ng$of$risk$free$and$risky$assets$in$known$as$a$complete(
por*olio.(To$construct$the$complete$por(olio$we$will$need$the$following:$
Two versions of the formula. Can use either, depending on the
information given and which would be best suited.
Two sources of income for the portfolio. Return from risky asset
and the return from risk-free asset.
FOR(THE(RISK(FREE(ASSET:(
The$rate$of$return,$known$as$the$risk$free$rate$of$return$$?$
r f(
By$defini.on,$there$is$no$risk$aDached$to$the$risk$free$rate$of$return$therefore$
its$standard$devia.on$is$zero$$
σ(rf(=(0%(
FOR(THE(RISKY(ASSET:(
The$expected$return.$There$is$no$certainty$aDached$to$this$rate$of$return,$
hence$expected$$
E(rp)((
The$risk$aDached$to$the$expected$return$on$the$risky$asset$$
σp((
The$percentage$(of$funds$invested$)$in$the$risky$asset$–$
y"
By$defini.on$therefore$the$percentage$(of$funds$invested$)$in$the$risk$free$$
asset$is$$$
(1%y)""
Por$olios(of(One(Risky(Asset(and(a(Risk4Free(
Asset(
Example:)
Assump-ons:)
• rf)=)7%)
• E(rp))=)15%)
• σp)=)22%)
Expected)return)and)standard)
devia-on)for)the)complete)por>olio:)
E(rc))=))rf)+)y[E(rp)))–)
rf])
)
E(rc))=))7)+)y[15C7])
)
E(rc))=))7)+)y8)
σc)=)y)σ)p)
(
σc)=)y)22)
(
σc)=)22y)
Interpreta-on:)
We#can#now#define#the#expected#return#and#risk#
for#the#complete#por1olio#
In#terms#of#expecta/on#of#the#por1olio’s#rate#of#return…#
• The(basic(rate(earned(on(this(por$olio(is(the(risk(free(rate(of(7%.(
• The(por$olio(is(expected(to(earn(a(maximum(risk(premium(of(8%,)if(100%)is(invested(
in(the(risky(asset((i.e.)y=1)(or(less)than)8%)depending(on(the(value)of)y)
• The(total(expected(return(for(the(completed(por$olio(for(the(given(set(of(numbers(is(
depends(on(the(amount)invested)in)the)risk)asset)(y))which(is(determined(by(the(
investors’)level)of)risk)aversion)(will(touch(on(this(later)(
E(rc)"=""yE(rp&)&+&(1*y)rf&
&
E(rc)"=""rf&+&y[E(rp&)&–&rf]"
"
Risk premium - [E(rp ) – rf]
The#complete#por1olio’s#risk#is#given#by….#
σc&=&y&σ&p&
90
Asset%Alloca*on%
•
Investment(Opportunity(Set(
•
•
E'(r')'
Capital'Alloca)on'Line''
(CAL)'
•
P
E'(rp')'='
15%'
rf'='7%'
M
Slope'='
8/22'
E'(rp')'B'rf'='8%'
σc'='y'σ'p'
σp'='22%'
σ
E"(r")"
Slope(of(the(CAL(is(o>en(
referred(to(as(the(price(of(
risk(
15%"
The(slope(of(the(CAL(
indicates(that(there((is(
increase(an(expected(
return(of(the(complete(
porDolio(per(unit(of(
addiEonal(risk(
•
Por,olio'P''is(100%(
invested(in(the(risky(assets(
(y=1)'
•
Por,olio'F'is(100%(
invested(in(the(risk(free(
asset((y=0)'
•
Por,olio'M'consists(of(a(
mixture(of(risky(and(risk(
free(assets((0<y<1)'
F
E(rc)'=''rf&+&y[E(rp&)&–&rf]'
Slope&=([E(rp&)&–&rf]/&σp&
0.36&
&
This(is(the(Sharpe'Ra)o'
Assump&on:*y=1*
Return:*
E(rc)"=""rf*+*y[E(rp*)*–*rf]*
E(rc)!=%0.07%+%1%(0.15%4%0.07)%%=%0.15%
*
Risk:*
σc*="yσp*
σc!=%(1.0)%(0.22)%=%0.22%
(CAL)"
P
M
11%"
rf"="7%"
Y"="1"
Y"="0.5"
F
σp"="%" σp"="11%"
Y"="0"
σp"="22%"
Assump&on*y=0*
Return:*
E(rc)"=""rf*+*y[E(rp*)*–*rf]*
E(rc)!=%0.07%+%0%(0.15%x%0.07)%%=%0.07%
σ
Risk:*
σc*="yσp*
σc!=%(0.0)%(0.22)%=%0%
Assump&on*y=*0.5*
Return:*
E(rc)"=""rf*+*y[E(rp*)*–*rf]*
E(rc)!=%0.07%+%0.5%(0.15%4%0.07)%%=%0.11%
*
Risk:*
σc*="yσp*
σc!=%(0.5)%(0.22)%=%0.11%
91
E(rc)&=&&rf&+&y[E(rp&)&–&rf]&
Asset%Alloca*on%–%Borrowing%@%rf%
AssumpBon:&
Borrow%at%the%RiskCFree%Rate%and%invest%in%the%%Risky%
asset,%essen*ally,%Lending at 7% and Borrowing at
7%
E&(r&)&
rf&=&7%&
Return:&
E(rc)&=&&rf&+&y[E(rp&)&–&rf]&
&
E(rc)!=%0.07%+%(1.5%x%0.08)%=%0.19%
Risk:&
σc&=&yσp&
P
&
σc!=%(1.5)%(.22)%=%0.33%
E&(rp&)&/&rf&=&
8%&
Slope&=&
8/22&
Using&50%&Leverage:&
Asset%Alloca5on%–%Borrowing%@%>%rf%
AssumpAon:&
Borrow%at%the%Risk.Free%Rate%and%invest%in%the%%Risky%asset,%essen5ally,%Lending
at 7% and Borrowing at 9%
E&(r&)&
(CAL)&
(CAL)&
E&(rp&)&=&
15%&
σc&=&y&σ&p&
%
Sharpe&ra9o&
Slope&=&[E(rp&)&–&rf]/&σ&p&
F
P&
E&(rp&)&=&15%&
rB&=&9%&
rf&=&7%&
S&(y≤&1)&=&0.36&
σ
σp&=&22%&
&
Slope&=&0.36&
Sharpe'ra(o'
Slope'=&[E(rp')'–'rf]/'σ'p'
'
F&
Slope'=&(0.15'–'0.09)/'0.22'
'
Slope'=&0.27'
&
Slope&=&[(0.15)&–&0.07]/&0.22&
S&(y&<&&1)&=&0.27&
σp&=&22%&
σ
E(rc)&=&&rf&+&y[E(rp&)&–&rf]&
Asset%Alloca*on%–%Borrowing%@%rf%
AssumpBon:&
Borrow%at%the%Risk<Free%Rate%and%invest%in%the%%Risky%
asset,%essen*ally,%Lending at 7% and Borrowing at
7%
E&(r&)&
(CAL)&
P
E&(rp&)&=&
15%&
rf&=&7%&
Slope&=&
8/22&
E&(rp&)&/&rf&=&
8%&
σc&=&y&σ&p&
Using&50%&Leverage:&
Alterna:ve%
Return:&
E(rc)&=&&Y[E(&Rp)]&+&(16y)&(Rf)&
&
E(rc)%=%1.5%(0.15)%+%(1<1.5)%(0.07)%
%
E(rc)%=%22.5%–%3.5%
%
E(rc)%=%0.19%
F
σp&=&22%&
σ
92
E(rc)*=**rf&+&y[E(rp&)&–&rf]*
*
Asset%Alloca*on%–%Borrowing%@%>%rf%
Assump&ons:*
• rf%=%7%%
•
E(rp)%=%15%%
•
σp%=%22%%
•
Have%R100%to%invest%
•
Borrow%another%
R100%@%9%%to%invest%
in%Risky%assets%
What*is*the…**
• E%(rc)%%
• σc%
σc*=*y*σ*p*
E(rc)*=**rBf&+&y[E(rp&)&–&rBf]*
Investment(Opportunity(Set(
Using&50%&Leverage:&
E&(r&)&
%
E(rc)%=%%9%++1+(15%/9%)+++0.5(+15%/9%)+
+
E(rc)%=%%9%+++6%+++3%+
+
Lower%than%borrowing%@%
E(rc)%=%%18%+
rf%rate….why?%
+
+
Compare+the+return+to+that+under+the+
scenario+of++borrowing+at+the+risk+free+
rate…refer+to+graph+
&
&
σc*=*y*σ*p*
%
σc%=%1.5%(0.22)% Why%has%this%not%change?%
%
σc%=%0.33%
&
&
(CAL)&
P
E&(rp&)&=&
15%&
rB&=&9%&
rf&=&7%&
S&(y&<&&1)&=&0.27&
Return:(
E(rc)&=&&rf(+(y[E(rp()(–(rf]&&
(
E(rc)!=(0.07(+((1.5(x(.06)(
(
(0.15)(=(.16(
Risk:(
σc(=&yσp(
S&(y≤&1)&=&0.36&
F
(
σc!=((1.5)((.22)(=(.33(
σp&=&22%&
σ
Assump@on:&
Borrow(at(the(RiskBFree(Rate(and(invest(in(the((Risky(
asset,(essenFally,(Lending at 7% and Borrowing at
7%
(
Sharpe(ra9o(
Slope(=&[E(rp()(–(rf]/(σ(p(
(
Slope(=&[(0.15)(–(0.07]/(0.22(
(
Slope(=&0.36(
*
Effect of borrowing at a rate higher than the risk free
rate, is a decreased expected return
Higher levels of risk aversion lead to a larger proportion of your
investment in the risk free asset (1.e. lower “y” values)
Lower levels of risk aversion lead to larger a proportion of your
investment in the portfolio of risky assets (i.e. higher “y” values)
93
Risk Tolerance and Asset Allocation:
Example:
Expected return of the complete portfolio is given by:
rf = 7%
E(rp) = 15%
σp = 22% (risk in the portfolio)
Variance is:
Investor X has coefficient of risk aversion of 6
Investor Y has a coefficient of risk aversion of 2
Utility:
Max U:
Calculate the E(rc); σc and the reward-to- risk ration for each
investor.
Allocation to risky asset
Use formula: Y* = (E(rp) – rf)/ Aσ2p
Investor X: Y* = 0.28
Investor Y: Y* = 1.65
Expected return:
Use Formula: E(rc) = rf + y[E(rp ) – rf]
Investor X: 0.0924
94
E(rc)&=&&rf&+&y[E(rp&)&–&rf]&
Investor Y: 0.202
σc&=&y&σ&p&
Asset%Alloca*on%–%Borrowing%@%rf%
Standard Deviation:
E&(r&)&
(CAL)&
Investor&&y&
E&(rc&)&=&20.2%&
Use formula: σc = yσp
Investor X: 0.0616
Investor Y: 0.363
Note:
If the borrowing rate = risk free rate, then the capital allocation
line will be straight.
Investor&&X&
E&(rc&)&=&9.24%&
rf&=&7%&
Risk-to-Reward Ratio:
Slope = 0.36
P
E&(rp&)&=&15%&
F&
σp&=&6.16%&
σp&=&22%&
σp&=&36.3%&
σ
Optimal portfolio in this particular example for the respective
investors. Investor X is more risk averse than investor Y.
Capital'Market'Line'
E&(r&)&
(CML)&
Investor&&Z&
E&(rc&)&=&20.2%&
Evaluating the effect of a different coefficient of variance and its
impact on expected return and risk of the portfolio.
Shows low risk is in line with a lower return and a high risk with
a higher expected return.
P
E&(rp&)&=&15%&
Investor&&X&
PorDolio&of&assets&
E&(rc&)&=&9.24%&
rf&=&7%&
F&
Reward-to-risk will be the same for both.
σp&=&6.16%&
σp&=&22%&
σp&=&36.3%&
σ
95
Chapter 13
Optimal Risky Portfolios
Section 1
Optimal Risky Portfolios
Assumptions:
We are looking at a short term horizon
Reduces the need to be concerned with skewness that is typical
of investments over a long term horizon
Normal distribution returns of a short term investment horizon
therefore allows us to assume holding period return that are sufficiently accurate.
Now our only concern is the mean and variance of the portfolio
Systema*c-risk:-cannot-be-diversified-out-of-a-por>olio.-
Diversifica*onPor$olio'Risk'as'a'Func0on'of'the'Number'of'Stocks'in'the'Por$olio'
Risk (σ )
Stocks- may- have- different- risk–
return-characteris*cs.-By-includingshares- with- different- risk–returncharacteris*cs- - in- a- por>olio- theinvestor- can- effec*vely- managehis-overall-por>olio-risk.-
Diversifiable Risk
a.k.a. Unique risk;
Company-specific risk;
Non-systematic risk
Unique risk
Non-diversifiable risk
a.k.a. Market risk ;
Systematic risk
Note:'Systema*c-Risk-is-defined-by-the-variability-in-thereturn-on-all-risky-assets-cause-by-macroeconomic-variablesuch-as-growth-in-money-supply,-changes-in-interest-ratesetc.-
Market risk
Number of
companies
(n)
Chalkboard:
In the above diagram the risk return characteristics of the stock
represented by the blue and red lines are “opposite” to each
other i.e. they are negatively correlated, a concept which we may
revisit a little later. If these 2 stocks made up the portfolio the
risk would be represented by the blue line instead of the red or
black line which would have been the case if the investor had
only one of these 2 stocks in his portfolio. The risk-return charac97
teristics represented by the blue line(stock) is more stable than
that represented any of the other lines (stock).
This is the impact of diversification; it stabilises the risk-return
characteristics of a portfolio.
However, it goes beyond just stabilising the risk-return characteristics; it also reduces the risk in the overall portfolio. As can be
seen in the diagram on the left, the more stock of different riskreturn characteristics your add to your portfolio, the lower the
portfolio’s overall risk will be.
But….not all risk is can be diversified, Market risk or systematic
risk i.e. that which impacts the stock of all companies, cannot be
diversified. An example of such risk is a slow down in the economy, all companies will be negatively impacted by this.
It is only company specific risk, otherwise known as unique risk
or non-systematic risk that can be diversified..
Two$Security-Por/olio:-Return!"="#$%$+#&%&!
Linear-relaPonship-
rp-=-Por/olio-ReturnwD-=-Bond-WeightrD-=-Bond-ReturnwE-=-Equity-WeightrE-=-Equity-Return-
E(R)
13%
Equity
8%
E--(r-p)-=-wD!E!(rD!)!+-wE!E!(rE!)!
The-por/olio-return-is-equal-tothe-sum-of-the-returns-derivedfrom-the-individual-stocks-thatmake-up-the-por/olio-i.e-WDrDand-WErE-.-!
Similarly,!the!Expected!
Returns…..!
An-investor-may-invest-all-hismoney-in-debt-funds-at-a-return-of8%;-he-may-invest-all-his-money-inequity-at-a-return-of-13%,-or-hecan-invest-in-a-combinaPon-ofdebt-and-equity-and-end-up-with-apor/olio-return-that-will-be-theweighted-average-of-the-returnsbetween-debt-and-equity.--
Debt Fund
(Corp. Bond)
-0.5
0
1.0
1.5
1.0
0
Short Equity
2.0
W(Equity)
-1.0
W(Bonds) =1-W(Equity)
Short Debt
We will now consider the construction of portfolio of 2 risky assets. One of these assets is a long term bond, denoted in the slide
by the letter “D”, the other is Equity, denoted on the slide by the
letter “E”.
The weight of D and E in the portfolio will be denoted by WD
and WE as indicated on the slide, and the return from D and E
will be denoted by rD and rE, respectively. The return of the
portfolio is denoted by rP. The equation for the portfolio return
simply states that the portfolio return is equal to the sum of the
returns derived from the individual stocks that make up the portfolio i.e WDrD and WErE .
Similarly, the Expected returns of the portfolio will be defined by
the summation of expected returns of the bonds and the expected returns of the equity.
98
Two-security Portfolio: Risk
You will note that there is a linear relationship between the returns of the individual stocks and the portfolio.
As indicated in the graph to the right of the slide, an investor
may invest all his money in debt funds at a return of 8%; he may
invest all his money in equity at a return of 13%, or he can invest
in a combination of debt and equity stock and end up with a portfolio between the 2 extremes (debt and equity) at a return that
will be the weighted average of the returns between debts and equity based on their weights in the portfolio.
We have a fixed budget – can either go 100% debt and 100% equity or anything in between. But we can also borrow. If we invest
200% in our equity fund, it means we will go short debt and long
equity, thus borrowing at the rate of debt.. Can borrow and lend
in this portfolio. Short equity - borrowing at the rate of equity
and if you short debt you are borrowing at the rate of debt.
Covariance and correlation:
Portfolio risk depends on the correlation between the (expected)
returns of the assets in the portfolio
Covariance and the correlation coefficient provide a measure of
the way returns of two assets vary
Correlation coefficient of +1:
σP = WDσD + WE σE
Correlation coefficient of -1:
σP = WDσD - WE σE
Examples:
In these examples we want to see what the impact of changing
the correlation coefficient is on the portfolios risk. Therefore we
assume that the assets have the same expected return and standard deviation.
Assume we have expected returns and standard deviations for
debt and equity as follows:
E(rD) = 0.20
E(rE) = 0.20
E(σD)=0.10
E(σE)=0.10
99
Assume further that you have a portfolio of debt and equity
which is equally weighted.
The$Trade(off$Between$Risk$and$Returns$–$
About$por8olio$dominance$$
Mean-variance criterion. "
Examples (cont):
Calculate the standard deviation for the portfolio of assets given
the following correlation coefficients.
E"(r")"
E(r) >
I"
II"
Por,olio"“E”"
E"(rp")"
Por,olio"“C”"
D;A;C$$($
Indifferent$
Por,olio"“A”"
E(r) <
δ
Por,olio"“D”"
Por,olio"“B”"
rD,E = 0.50
rD,E = -0.50
A;C;D;E;F$
Dominates$B$
Por,olio"“F”"
rD,E = 1.00
rD,E = 0.00
δ
F;E$
Dominates$
D;A;C$;B$
IV"
III"
σp"
σ"
rD,E = -1.00
Before we continue, we need to just re-cap the concept of portfolio dominance. If you recall, we discussed in a previous lecture
the significance of the position of portfolio P on the risk – return plane. We said that any portfolio that lies within quadrant
four i.e. to the south-east of portfolio P will be inferior to portfolio P. This is because the risk of such portfolio will be higher
than that of portfolio P and the return will be lower. There for
portfolio P in this example is said to dominate portfolio A.
The conditions under which portfolio P will dominate Portfolio
A is when the rP ≥ rA and the δP≤ δA ; Dominance requires at
least one > condition to hold.
100
If we consider portfolio E relative to Portfolio P, we will see that
we have an increase in risk, but that the increase in risk is less
than proportional to the increase in return. Similarly, if we compare portfolio D with portfolio P, we can see that there is a decrease in return, but the decrease in return is less than proportional to the decrease in risk. As we have discovered in a previous lecture, with portfolios P, D and E being on the investor’s indifference curve, the investor will be equally satisfied any of
these portfolios. We can therefore say that portfolio E is equivalent to portfolio P and portfolio P is equivalent to portfolio D,
with portfolio D being equivalent to portfolio E. This is equivalent to saying that none of the portfolios on the IC dominates
each other.
Por$olio'Expected'Return'as'a'Func5on'of'
Standard'Devia5on''
E
Expected
Return
ρ"=##)1#
ρ"=##0#
rf
C#
ρ"=##0.30#
B#
C1#
B1#
A#
ρ"=#1#
A1#
D
Standard
Deviation
Any portfolio to the right of the IC will be dominated by a portfolio on the IC. Any portfolio on the IC will be dominated by a
portfolio to the left of the IC.
We have seen that…
As you will see in the next slide, this principle is important in understanding the logic behind the shape of the efficient frontier of
risky assets.
The smaller the correlation, the greater the risk reduction potential and the greater the impact of diversification.
We also saw that…
If ρ = +1.0, no risk reduction is possible
101
Asset%alloca*on%including%a%risk%free%asset%
Capital Allocation Line
Expected
Return
P8
Indifference
Curve
P7
P6
P5
P4
P3
rf
P2
P1
!"="​$"(&")−&(/*" !
Standard
Deviation
Any asset to the north-west of portfolio 1, dominates portfolio 1.
P3 – P8 = The efficient frontier
Based on the mean-variance criterion and the concept of portfolio dominance we can see that portfolio P1 is not dominant, portfolios P2 to P 6 can offer higher returns at lower risk. As a result,
portfolio P does not form part of the efficient frontier.
We can extend this argument to portfolios P2 and P3 as well
(portfolio P3 just about dominated by portfolio P4). Portfolios
P2 and P3 therefore do not form part of the efficient frontier.
However, portfolios P4 to P8 meet all the criteria of dominant
portfolios; there is no other possible portfolio in quadrant 1 or
north-west of any of these portfolios. For all these portfolios the
criteria form dominance referred to earlier is met. In the sense
that there is no portfolio that dominates the portfolios on the
frontier beyond portfolio P3, this part of the frontier can be referred to as an efficient frontier.
We now want to add a risk free asset to the portfolio. The risk
free rate is indicated by rf . The risk free portfolio is indicted on
the y-axis. Why do you think this so? Because by definition the
risk free asset has zero standard deviation.
An investor can decide to allocate his investment between risk
free assets and risk assets, remember risk assets is represented by
efficient frontier. Lets say that he chooses to allocate his funds
between the risk free asset and portfolio P3 of the risky assets.
This would imply that his portfolio will be allocated anywhere
between rf and portfolio P3. the more risk averse the investor
the more of the risk free asset he will include in his portfolio.
His portfolio will lie closer to the y-axis indicating a greater proportion of risk free assets in it. Conversely, a less risk averse investor will include more of the risky asset in his portfolio; his
portfolio would lie closer to portfolio P3. You should be familiar
with this from one of our earlier lectures.
But this is not the most optimal decision by the investor. Lets assume he decides to invest in the risk free asset and portfolio P4.
Any portfolio on the line between rf and portfolio P4 will domi-
102
nate portfolios represented by portfolio P3 and the risk free asset. This should be clear from our earlier discussion.
One could follow the same argument for at portfolio that consist
of the risk free asset and portfolio P5 or the risky assets, but not
for portfolio P7 of the risky assets…Why? Well that is quite simple. Any portfolio that is on the line between portfolio P5 and P7
of the risky portfolios is dominated by a portfolio on the efficient frontier.
The lines rf – P3, rf – P4, and rf – P4 are all Capital Allocation
lines. We can continue to move our CAL to the left until we
reach our optimal risky portfolio (P6) as defined by the tangent
between the CAL and the efficient frontier. There is no risky
portfolio that offers a better reward-volatility (risk) ratio than
this. Recall that the reward to volatility ratio is the Sharpe Ratio.
As we move our CAL to the left our sharp ratio increases (slope
of the CAL).
Finally, we can introduce the level of risk aversion of the investor.
In a previous lecture we were introduced to the calculation of
the proportion of assets that should be allocated to risky assets
give the level of risk aversion “A”. This will give us the complete
optimal portfolio which is diagrammatically this is represented
by the tangent between the CAL and the investors highest indifference curve.
Note: The objective of an investor interested in the minimum
variance portfolio is to minimize the risk of the portfolio.
Whereas the objective of an investor interested in the optimal
risky portfolio is to obtain the best risk and return trade-off available.
Efficient(Fron,er(and(3(Possible(“op,mal”(
Por8olios(
Capital Allocation Line
Expected
Return
P8
Indifference
Curve
P7
Optimal Risky Portfolio
P6
Optimal
Complete
Portfolio
P5
P4
Global Min-Variance Portfolio
rf
Standard
Deviation
Based on the mean-variance criterion and the concept of portfolio dominance we can see that portfolio P1 is not dominant, portfolios P2 to P 6 can offer higher returns at lower risk. As a result,
portfolio P does not form part of the efficient frontier.
We can extend this argument to portfolios P2 and P3 as well
(portfolio P3 just about dominated by portfolio P4). Portfolios
P2 and P3 therefore do not form part of the efficient frontier.
103
However, portfolios P4 to P8 meet all the criteria of dominant
portfolios; there is no other possible portfolio in quadrant 1 or
north-west of any of these portfolios. For all these portfolios the
criteria form dominance referred to earlier is met. In the sense
that there is no portfolio that dominates the portfolios on the
frontier beyond portfolio P3, this part of the frontier can be referred to as an efficient frontier.
We now want to add a risk free asset to the portfolio. The risk
free rate is indicated by rf . The risk free portfolio is indicted on
the y-axis. Why do you think this so? Because by definition the
risk free asset has zero standard deviation.
An investor can decide to allocate his investment between risk
free assets and risk assets, remember risk assets is represented by
efficient frontier. Lets say that he chooses to allocate his funds
between the risk free asset and portfolio P3 of the risky assets.
This would imply that his portfolio will be allocated anywhere
between rf and portfolio P3. the more risk averse the investor
the more of the risk free asset he will include in his portfolio.
His portfolio will lie closer to the y-axis indicating a greater proportion of risk free assets in it. Conversely, a less risk averse investor will include more of the risky asset in his portfolio; his
portfolio would lie closer to portfolio P3. You should be familiar
with this from one of our earlier lectures.
But this is not the most optimal decision by the investor. Lets assume he decides to invest in the risk free asset and portfolio P4.
Any portfolio on the line between rf and portfolio P4 will dominate portfolios represented by portfolio P3 and the risk free asset. This should be clear from our earlier discussion.
One could follow the same argument for at portfolio that consist
of the risk free asset and portfolio P5 or the risky assets, but not
for portfolio P7 of the risky assets…Why? Well that is quite simple. Any portfolio that is on the line between portfolio P5 and P7
of the risky portfolios is dominated by a portfolio on the efficient frontier.
The lines rf – P3, rf – P4, and rf – P4 are all Capital Allocation
lines. We can continue to move our CAL to the left until we
reach our optimal risky portfolio (P6) as defined by the tangent
between the CAL and the efficient frontier. There is no risky
portfolio that offers a better reward-volatility (risk) ratio than
this. Recall that the reward to volatility ratio is the Sharpe Ratio.
As we move our CAL to the left our sharp ratio increases (slope
of the CAL).
Finally, we can introduce the level of risk aversion of the investor.
In a previous lecture we were introduced to the calculation of
the proportion of assets that should be allocated to risky assets
104
give the level of risk aversion “A”. This will give us the complete
optimal portfolio which is diagrammatically this is represented
by the tangent between the CAL and the investors highest indifference curve.
Note: The objective of an investor interested in the minimum
variance portfolio is to minimize the risk of the portfolio.
Whereas the objective of an investor interested in the optimal
risky portfolio is to obtain the best risk and return trade-off available.
Objec-ve:#
Maximise:''
'Sp'='E(rp);'rf'/'δp'
Op#mal'Risky'Por0olio'
Step 1:
Wmin(D) =
E(RD)δ2E – E(RE)Cov(rD,rE)
[R#=#Risk#Premium]#
E(RE)δ2D + E(RD)δ2E – [E(RD) + E(RE)]Cov(rD, rE)
Step 2:
Wmin(E) = 1- Wmin(D)
CAL
E(R)
IC
Step 3:
ORP
OCP
E''(r'p)'='wD#E#(rD#)#+'wE#E#(rE#)#
Objec-ve:#
Minimise&Risk&
Global&Min*Variance&Por0olio&
G M-V P
r
f
Step 4:
δ2P = W2Dδ2D + W2Eδ2E + 2WDWECov(rD,rE)
SD
Step 1:
δ2E – Cov(rD,rE)
Wmin(D) =
CAL
E(R)
δ2D + δ2E – 2Cov(rD, rE)
IC
ORP
Step 2:
Wmin(E) = 1- Wmin(D)
OCP
G M-V P
r
f
Step 3:
SD
E&&(r&p)&=&wD#E#(rD#)#+&wE#E#(rE#)#
Step 4:
δ2P = W2Dδ2D + W2Eδ2E + 2WDWECov(rD,rE)
Cov(rD,rE) =
!DE!D!E
105
Determine the allocation of investment funds in the complete
portfolio for an investor with a coefficient of risk aversion of 5,
given an opportunity to invest in risk free treasury bills with an
interest rate of 6%
Op#mal'Complete'Por.olio''
Step 1:
CAL
E(R)
Percentage of Capital in Risky Portfolio
E(rp)'–'rf'
Y= '
''''A'δ2p''
IC
ORP
OCP
Repeat the above for an investor who has a coefficient of risk
aversion of 3
G M-V P
r
f
Step 2:
Percentage of Capital in Risky Portfolio
Risk Free Portfolio = 1-Y
Step 3:
Split of Securities
SD
25%
45%
Split Risky Portfolio into its percentage Debt and
Equity given Capital Allocation between Risky
and Risk free Portfolio
Solu'on!
30%
Risk free portfolio
Debt (Bonds)
Equity
Risky Portfolio
Assume the following characteristics of an optimal risky portfolio:
E (rp) = 15.12%
σp = 15.37%
Objec've:!
Allocate'investments'
consistent'with'level'of'
risk'aversion'
Op#mal'Complete'Por.olio'!
Step 1:
Percentage of Capital in Risky Portfolio
Step 2:
E(rp)'–'rf'
Y= '
''''A'δ2p''
Percentage of Capital in Risky Portfolio
Risk Free Portfolio = 1-Y
= -28.81%
Answer'='Y'='77.29%'
Step 3: Split Risky Portfolio
Bonds: 48.19%
Equity: 29.10%
The split between Bonds and equity in this portfolio is:
Bonds - 62.35%
Equity – 37.65%
Required:
106
portfolio would be made up of all the risky assets in proportion
to their market values, but it would not contain any risk-free asset.
Indifference Curve
(A=3)
Expected
Return
Optimal
Complete
Portfolio
CAL/ CML (if all risky assets i.e. the market)
Indifference Curve
(A=5)
Optimal
Complete
Portfolio
Optimal Risky Portfolio
Global Min-Variance Portfolio
rf
Standard
Deviation
Lending Portfolio
Borrowing Portfolio
Based on our earlier discussion, the capital market line (CML)
provides the best risk and return tradeoff for an investor. Along
the CML, a potential portfolio for the investor will be the market portfolio. However, it is not necessary for the investor to
pick that portfolio because he/she can pick any portfolio along
the CML. This investment decision will be heavily influenced by
the investor’s risk preference. The following graph illustrates
how the risk preference of an investor affects his/her investment
decisions:
2. If the investor’s risk preference is represented by indifference
curve 2, he/she will pick portfolio L, which has a lower level of
risk than the market portfolio. The only way to reduce the risk
of the portfolio is to include the risk-free security in it. In essence, the investor is “lending” part of his/her money at the riskfree rate. As a result, portfolio L is also known as a lending portfolio. From the graph above, we know that any portfolio along
the CML that is below the market portfolio can be termed a
lending portfolio.
3. If the investor’s risk preference is represented by indifference
curve 3, he/she will pick portfolio B, which has a higher level of
risk than the market portfolio. The only way to increase the risk
of the portfolio is to borrow at the risk-free rate to increase the
investment base (i.e. margin trading). As a result, portfolio B is
also known as a borrowing portfolio. From the graph, we know
that any portfolio along the CML that is above the market portfolio can be termed a borrowing portfolio.
1. If the investor’s risk preference is represented by indifference
curve 1, he/she will pick portfolio M, which is the market portfolio. In other words, the investor’s Business 442: Investments
Chapter 5-21 Dr. Siaw-Peng Wan
107
Identify the risk-return combinations
available from the set of risky
assets.
CAL/ CML (if all assets i.e. the market)
Expected
Return
Indifference
Curve
Optimal Risky Portfolio
Identify the Optimal portfolio of risky
assets by finding the portfolio
weights that resulted in the steepest
CAL.
Individual Assets
Optimal
Complete
Portfolio
free asset against a single optimal risky portfolio (i.e. the market
portfolio). From the graph, we know that in the situation where
a risk-free asset is not available, the investor can only reduce the
risk of the portfolio by shifting his/her funds towards “safer”
risky assets, which will move the investor downward along the
efficient frontier.
Global Min-Variance Portfolio
rf
Chose an appropriate complete
portfolio by mixing the risk-free asset
with the optimal risky portfolio
Standard
Deviation
• Efficient Frontier of risky assets
• Common risky portfolio for all clients
• Risk aversion factored in when a
decision is made between proportion
of risk free and risky portfolio
(separation property)
• The best risk Portfolio is the same for
all clients regardless of risk aversion
• Different managers may have
different “optimal portfolios
• Optimal Risky portfolios may be
affected by constraints e.g. client
preferences
Optimal Risk Portfolio does not take into account risk aversion
whereas your optimal complete portfolio does
According to James Tobin’s Separation Theorem, an investor’s decision making process is actually made up of two separate decisions:
i. To be on the CML, the investor initially decides to invest in
the market portfolio (i.e. portfolio M). This is the investment decision.
ii. Based on the investor’s risk preference, he/she makes a separate financing decision on whether to lend or borrow at the riskfree rate to get to the desired point.
In the above three scenarios, we have seen how an investor can
adjust his/her return or risk by borrowing or lending with a risk-
However, when the investor has access to the risk-free asset, he/
she can reduce the risk of the portfolio by investing or lending at
the risk-free rate, which will move him/ the portfolio at the expense of a lower return.
If you look at the graph carefully, you will realize that the portfolios on the CML provide a higher return than the portfolios on
the efficient frontier for the same level of risk. In other words,
the availability of the risk-free asset helps an investor reduce the
risk of his/her portfolio but also helps preserve most of the return of the portfolio.
Similarly, we know that in the absence of the risk-free asset, the
investor can improve the return of the portfolio by shifting his/
her funds towards “riskier” risky assets, which will move the investor upward along the efficient frontier.
However, with the presence of the risk-free asset, the investor
can improve the return of the portfolio by borrowing at the riskfree rate, which will move the investor along the CML. We know
both strategies will improve the return of the portfolio at the expense of a higher level of risk.
108
If you look at the graph carefully, you will realize that the portfolios on the CML experienced a smaller increase in risk level than
the portfolios on the efficient frontier for the same level of increase in return.
In other words, the availability of the risk-free asset helps an investor improve the return of his/her portfolio without taking on
as much risk. her downward along the CML. We know both
strategies will reduce the risk of
Allocate funds between the risky portfolio and the risk-free asset
• Calculate the fraction of the complete portfolio allocated to
the risky portfolio and the risk-free portfolio.
• Calculate the share of the portfolio in each risky asset and the
T.bill (risk-free)
Conclusion
Summary:
Specify the return characteristics of all securities:
• expected returns,
• variances, and co-variances
• correlation coefficients
Establish the risky portfolio, P.
• Calculate the optimal risky portfolio, P
• Calculate the properties of portfolio P using the weights determined in the pervious step.
In this chapter, we have looked at three different portfolios and
we learned three different lessons from these portfolios:
1. With a portfolio that contains a risky asset and a risk-free asset, we derived the basic relationship between risk and return
through the capital allocation line. We found that the return of a
portfolio can be broken down into two components: a guaranteed return and a compensation for taking risk.
2. With a portfolio that contains multiple risky assets, we derived
the efficient frontier that helped explain the concept of diversification. In other words, it is possible for an investor to put together a portfolio that has a risk level lower than the individual
risk levels of the assets it contained.
3. With a portfolio that contains multiple risky assets and a riskfree asset, we showed that it is possible for an investor to improve his/her risk or return situation by lending or borrowing at
the risk-free rate.
109
Chapter 14
Index Models
Section 1
Index Models
n = 50 estimates of variances;
(n2- n)/2 = 1,225 estimates of covariances;
1,325 estimates.
!
Markowitz!Por,olio!Selec1on!
!
CAL/ CML (if all risky assets i.e. the market)
Expected
Return
Indifference Curve
Optimal Risky Portfolio
Individual Assets
Optimal
Complete
Portfolio
Global Min-Variance Portfolio
N=50
1325 estimates
rf
Standard
Deviation
Examples)of)single)factors:)
• Business)cycles)
• Interest)rates)
)
The problem with this approach
This approach calls for a high level of data inputs. Consider the
following:
• For a 50 security portfolio, the Markowitz model requires the
following parameter estimates:
n = 50 estimates of expected returns;
The process is time consuming and error prone. An investment
analyst will find it very difficult to manage such a huge number
of inputs. Index Models are designed to overcome this problem,
but at a cost of accuracy. This will become evident later.
• For a 50 security portfolio, the single-index model requires the
following parameter estimates:
n = 50 estimates of expected excess returns, E(R);
n = 50 estimates of sensitivity coefficients, i;
n = 50 estimates of the firm-specific variances, 2(ei);
1 estimate for the variance of the common macroeconomic factor, 2M;
or (3n + 1 = 151) estimates.
In addition, the single-index model provides further insight by
recognizing that different firms have different sensitivities to
macroeconomic events. The model also summarizes the distinction between macroeconomic and firm-specific risk factors.
111
ri = E (ri) + m + ei
Building(a(Single(Factor(Model(
Return
Firm-specific (unique)
ri(=(E((ri)((+( ei(
“e” has a:
Mean = 0
SD = δ
Expected
return
Unexpected
return
ri(=(E((ri)((+(((((((+(ei(
m
Uncorrelated
“m” has a:
Mean = 0
SD = δ
Expected
return
Unexpected
return
Return emanating from the inclusion
of a common macro-economic
factor
Whilst(“m”(is(
common(to(all(
securi=es,(ei(will(be(
unique(to(firm(“i”,(
therefore,(by(
defini=on(m(and(ei(
are(not(correlated(
with(each(other;(
any(shock(resul=ng(
from(the(
macroeconomic(
factor(will(not(
impact(the(
unexpected(frimI
specific(return.(
Whilst “m” is common to all securities, ei will be unique to firm
“i”, therefore, by definition m and ei are not correlated with each
other, therefore any shock resulting from the macroeconomic factor will not impact the unexpected frim-specific return.
Actual return on security I = Expected return of security i + Surprise return on the security
Macro economic surprise return tends to zero as time passes.
ei = Firm specific surprise return.
Return:
Let us consider the rate of return on a single security..
ri = E(ri) + ei
The return for this security can be broken down into two portions, an expected return [E (ri )] and an unexpected return [ei ].
The unexpected return is assume to have a zero mean.
We can include a common macroeconomic factor into the
model. We call this factor “m”. Note that this factor will have an
impact on all securities. Note further that the return associated
with this macroeconomic factors will be unexpected as well. We
can therefore re-write the return on the portfolio as;
112
Building(a(Single(Factor(Model(
Risk
Because returns m and ei are
uncorrelated we can write the
variance as…
Variance
Because “m” is a common market factor, which by definition affects all securities, it will be correlated with all securities. This is
not the case with “e”; “e” is frim specific and will by definition be
uncorrelated across firms (or securities).
δi2$=$$δ2m$$+$δ2ei$
Systematic factors
(market factors)
Unsystematic factors
(firm specific factors)
Covariance
Cov(ri,$rj)$=$Covm$+$ei,$m+ej$
Assume(1(macro(factor.(No(
correla9on(between(firms.(Simply(
variance(of(security(to(the(market,(
rather(than(to(the(stocks(and(the(
market(as(with(morkovits.(
e is uncorrelated across
firms
Cov(ri,$rj)$=$Covm,m$=$δ2m$
Significantly reduced number of estimates because we are
now relating the variance of each security to the market
and not to each other as in the Markowitz model
Risk:
Variance:
We measure risk by looking at the variance of the return of the
stock. Within the portfolio the variance will be driven by the
market δ2m and the firm δ2i. This defines the systematic factors
(affects all securities) that impact stock returns and unsystematic
(or firm specific) factors that impact the stock returns , respectively. As we have seen under the section on returns m and ei are
uncorrelated. The variance is written as:
Covariance:
Let’s assume we have 2 securities, “i” and “j”. The covariance will
come from the uncertain parts of the returns. We can therefore
concentrate on m and e for security i and j. we can therefore
write the covariance as:
Cov(ri, rj) = Covm + ei, m+ej
We said that e was uncorrelated across firms, we can therefore
drop e from the expression which would then make the covariance equal to…
Cov(ri, rj) = Covm,m = δ2m
This reduces variance of securities i and j to the variance of the
common macro-economic factor (m).
δ2 = δ2m + δ2ei
113
m = a common macroeconomic factor that affects all security returns. The S&P 500 is often used as a proxy for m.
Building(a(Single(Factor(Model(
Beta = sensivity of security to the market
Systema8c#Risk#<#β2iδ2m#
##
Firm#Specific#Risk##<#δ2ei###
Return
ri#=#E#(ri)#+#βm#+#ei#
Variance
We can ignore ei
and ej we assume
them to be
uncorrelated.
δ2i#=#β2iδ2m#+δ2ei###
Covariance
Cov(ri,#rj)#=#Cov#(βi#m+ei,#βj#m+ej)#
ei = firm-specific surprises
Variance:
We should factor the sensitivity coefficient (β) into the equation
for the variance as well. The variance will be as follows:
δ2i = β2iδ2m δ2ei
Cov(ri,#rj)#=#βi#βj#δ2m###########################
[δax,by = ab δxy]
8-8
Return
β2δ2m will define the systematic (market) risk to which security
returns is exposed. δ2ei will define the unsystematic (firm specific) risk.
We can now re-write the single factor model by including beta.
The will be as follows:
ri = E (ri) + βm + ei
Covariance:
We should factor the sensitivity coefficient (β) into the equation
for the covariance as well. The covariance will be as follows:
This expression defines the single factor model where…
βi = response of an individual security’s return to the common
factor, m. Beta measures systematic risk.
Cov(ri, rj) = Cov (βi m+ei, βj m+ej)
We can ignore ei and ej because, as before, we assume them to be
uncorrelated. Remembering that [δax,by = ab δxy], we can rewrite the covariance as follows:
Cov(ri, rj) = βi βj δ2m
114
From Single Factor Model to the Single Index Model:
• To make the single factor model operative we relate a broad
market index such a the S&P500 or the JSE all share index as a
proxy for the macroeconomic factor.
• We will end up with an equation for the index model that is
similar to the single factor model…the reason we refer to as the
single index model that it uses the market index as a proxy for
the common macroeconomic factor.
• We have an abundance of historical information from which to
estimate market risk and market risk premium for the single
index model.
• The market risk premium will be denoted as: RM = rM – rf and
the associated risk is denoted as σM
• We can now regress the excess returns of the security (Ri = ri –
rf) on the excess returns of the market (RM). This will tell us
how sensitive the return of the security in relative to that of
the market.
To get the single index model we regress excess market returns
on excess index returns.
Now we want to estimate the single index model at time period
“t”. This will be stated as:
Ri (t) = αi +βiRM (t) +ei (t)
α i represents the intercept or the security return when the market return is zero. If market return is zero then βiRM (t) equal
to zero and the return on the security will be equal to α.
βi gives us the slope of the security, it is a reflection of the sensitivity of the security to movements in the market (sensitivity coefficient); is a measure of systematic risk.
The Single Index Model:
The error term (ei ) averages out to zero over the long term. This
is also referred to as the firm specific unexpected (surprise) return. Because ei is zero we can re-write the equation as:
E(ri) – rf = Ri and the E(rM) – rf = RM [note: we are using capital “M” to denote the market and not small “m” which denoted a
single economic factor; R= excess return]
Ri (t) = αi +βiRM (t)
or
115
E(Ri) = αi +βi E (RM)
βi E (RM) tells us that security’s risk premium is due to the risk
premium of the index. The extent to which the security “mimics” the index risk premium is given by the sensitivity coefficient
(βi ). This is referred to as the systematic risk premium.
The second part of the risk premium is given by α. This is a nonmarket risk premium. This is what investment managers will
look for…companies whose stock exhibit positive α. Generally is
a security is under-priced you will see a positive α for it. Note
that if one company gains +ve α another loses. The sum total of α
in the market is zero i.e it is competed away.
Single"index"model"
Return!
1" The"stock’s"expected"return"if"the"market"is"neutral"ie."When"
excess"return"(Rm;Rf)"="zero"
α"
2" The"component"of"return"due"to"movements"in"the"overall"
market"(β)"i.e."the"securiCes"responsiveness"to"market"
movements""
βi(rM;rf)"
3" The"unexpected"component"of"return"due"to"unexpected"events"
that"are"relevant"only"to"this"security"(firm"specific)"
ei"
!(#$)="&$+"'$!(#()!
Summary"of"result"of"the"single"index"model"
1" The"stock’s"expected"return"if"the"market"is"neutral"ie."When"
excess"return"(Rm;Rf)"="zero"
"
α"
2" The"component"of"return"due"to"movements"in"the"overall"
market"(β)"i.e."the"securiCes"responsiveness"to"market"
movements""
"
βi(rM;rf)"
3" The"unexpected"component"of"return"due"to"unexpected"events"
that"are"relevant"only"to"this"security"(firm"specific)"
"
ei"
4" The"variance"aIributable"to"the"uncertainty"of"the"common"
macroeconomic"factor""
"
β2iσ2M"
5" The"variance"aIributable"to"firm;specific"uncertainty"
σ2(ei)"
"
Risk Premium
116
For'a'single'security:'δ2i'='δ2'(ei)'
Index Model and Diversification:
Advantage&of&the&index&model&–&
diversifica1on&gets&rid&of&the&
unsystema1c&risk&
δ2p = β2pδ2M δ2 (ep)
Index&Model&and&diversifica1on&
The variance determined by the market is represented by
β2pδ2M and the variance represented by company specific components is represented by δ2 (ep).
Por$olio'Variance'
δ2p'='β2pδ2M'+δ2'(ep)'
It does not matter how much effort we put into diversification,
the variance that is determined by the market will persist, simple
because those factors that determine the variance will impact on
all securities within the portfolio.
However, the factors that impact on the variance attributable to
that which is company specific as represented by δ2 (ep), can be
diversified because these factors do not affect all securities; the
firm specific components (ei) are independent. The more stocks
we add to the portfolio the greater the change that the firm specific components will cancel each other out.
Because of this property of independence we can state that the
variance of an equally weighted portfolio is equivalent to the average of the variance of the portfolio x number of securities in the
portfolio….
Which would imply that as the number of securities in the portfolio increase the variance of the portfolio would decrease. If “n”
increases the average variance decreases to a negligible amount
Risk (σ )
δ2'(ep)'='δ2'(e)'/'n'
δ2'(ep)'
β2pδ2M'
Market risk
Number of
companies
(n)
8-17
What we would like to do is break this excess expected rate of
return down into its components to see which parts would be
subject to variance (risk).
If we assume that each security in our portfolio is equally represented then the weight of each security (Wi) must be equal to 1/n
and the sum of 1/n securities must be equal to the number of securities in the portfolio. We can therefore write the excess return as…
Rp= Σ WiRi or Rp= 1/n Σ Ri
(since wi = 1/n)
Note that we have changed the subscript in reference to the excess return from “p” to “i” and we are stating that the portfolio
excess returns equate to the sum of all equally weighted excess
117
returns of each security within the portfolio. Because we made
reference to the excess returns of security “i” in the equation
above we can now substitute Ri with αi + βi RM + ei in the above
equation since we have stated above that Ri=αi + βi RM + ei. We
therefore say that…
Rp= 1/n Σ Ri = 1/n Σ (αi + βi RM + ei)
Alpha*and*Security*Analysis*
Macro8Economic'Analysis:''
Es9mate*of…*
• risk*premium**E(RM)'and**
• risk*of*the*market*index*–*β2iδ2M'
**
We can break this down further as follows:
Rp = 1/nΣi α + (1/nΣβi)RM + 1/nΣei
Now we can clearly see that the excess return consists of a company specific return [1/nΣi α ] a market return [(1/nΣβi)RM ] and
a company specific “surprise” return [1/nΣei], this last element of
return is driven by events that are relevant to a specific security
(i).
The variance of returns therefore come from [(1/nΣβi)RM ] and
[1/nΣei], (market and the company). For a single security we represented the variance by….
δ2 = δ2m δ2(ei)
For the portfolio, by….
δ2p = β2pδ2M δ2 (ep)
The variance determined by the market is represented by
β2pδ2M and the variance represented by company specific components is represented by δ2 (ep).
Sta@s@cal'analysis:'Passive'
por$olio'analysis'
Es9mate*of*…*
• Beta*coefficients*(β)'and*
• Residual*variances*(δ2(ei)'of*all*
securi9es*
iste
Cons
ncy'
2 sources of premium
Por$olio'Manager:''
Establishes*a*market*driven*
expected*return*which*is*use*as*a*
benchmark*for*performance*
measurement*8*βiE(RM)'
Security'analysis:'Ac@ve'por$olio'
analysis'
Es9mate*of*…*
• Security*α*–*This*represents*
incremental'risk'premium'over8
and8above*the*market*risk*
premium*which*is*βiE(RM)'
This separation of macroeconomic analysis of market risk premium from statistical analysis of the alpha has some significant
advantages:
It allows us to separate the risk premium attributable to the market from that attributable to the security.
The risk premium attributable to the market can now be set as a
benchmark against which performance could be measured.
If alpha is positive it provides a risk premium over and above
that of the market…the security will be a good buy and should in
theory be “over-weighted” in the portfolio. This security is referred to as under-priced.
118
The opposite is also true, a security with an alpha that is negative
is generally a bad buy and therefore its portfolio weight should
be reduced. This security is referred to as over-priced.
• When%introducing%the%index%por0olio%we%assume%that%this%also%
contributes%as%risk%premium%in%propor8on%to%its%risk%(E(RM)&/&
σ2M),&leaving%the%por0olio%variance%undisturbed.%
Optimal Risky Portfolio:
• As%a%result,%the%assump8on%of%a%beta%of%1%for%the%overall%
por0olio%s8ll%holds.%
The investor has a choice to invest in the market only, in which
case he invest in the index portfolio, or he can invest in the security that gives him additional alpha as well as the market (i.e. index portfolio).
In exchange for additional alpha, the investor sacrifices some of
the benefit of diversification which is assumed in the index (market) portfolio. Adding an additional asset to the portfolio introduces company specific risk.
We start off with an assumption that the active portfolio has a
beta of 1.
When introducing an additional asset to the portfolio we assume
that the asset contributes a return (alpha) in proportion to its
risk (αA / σ2A), leaving the portfolio variance undisturbed.
• The%weight%in%the%ac8ve%por0olio%given%these%assump8ons%is%
given%by….%
!!"=#​$"/&$" /​(()*)/&$*
&
Or better stated…we can view this as the ratio of investment in
the active portfolio relative to the investment in the passive (market or index) portfolio.
• For$any$level$of$σ2A$the$correla2on$between$the$ac2ve$and$
passive$por9olio$is$greater$when$beta$of$the$ac2ve$por9olio$is$
higher.$If$so,$the$diversifica2on$benefit$in$the$passive$por9olio$
is$diminished$and$as$a$result$its$weight$in$the$ac2ve$por9olio$is$
reduced.$This$can$be$easily$seen$from$….$
$!∗"=#​$$"/%+(%−'")!$" $
!
119
Information ratio:
!∗"=#​$$"/%+(%−'")!$" !
Having decided on the weight of the active portfolio within the
overall risky portfolio, how do we ascertain whether we have
made the right decision, otherwise stated…how do we know
what value our security analysis has added to our portfolio?
• ….when!the!βA#is#>1,!the!weight!of!the!ac*ve#por0olio#increases#/!
the!weight!in!the!passive#por0olio#therefore#decreases#
• ….when!the!βA#is#<1,!the!weight!of!the!ac*ve#por0olio#decreases#/!!
the!weight!in!the!passive#por0olio#therefore#decreases#
• Note!that!(∗)#is!invested!in!the!ac5ve!por8olio!and!18#(∗)##is!
invested!in!the!passive!por8olio!
#
To answer that we look at the following equation…..
The information ratio is ((alpha)(A)/(sigma)(eA))2 portion of the
equation. It tells us how much extra return we can obtain from
security analysis relative to the risk incurred when we over or
under-weight securities relative to the passive market index.
To#re&iterate….#
• !!"=#​$"/&$" /​(()*)/&$* #
Regardless#of#beta,#when#w#0#
decreases,#so#does#w*.#
2
1
Other#things#held#equal#,w#0#is#smaller#the#
greater#the#residual#variance#of#a#
candidate#asset#for#inclusion#in#the#
por=olio.#
• !∗"=#​,!"/(+((−.")!!" ##
3
4
We can also refer to this as the information ratio of the active
portfolio; this is what we need to maximise in order to maximise
the overall Sharpe ratio.
Increased#firm&specific#risk#reduces#the#
extent#to#which#an#acAve#investor#will#
be#willing#to#depart#from#an#indexed#
por=olio.#
Therefore,#other#things#held#equal,#
the#greater#the#residual#variance#of#
an#asset#(σ2),#the#smaller#its#
posiAon#in#the#opAmal#risky#
por=olio.#
Why?#
The Sharpe ratio indicates that:
a higher alpha makes a security more desirable
alpha, the numerator of the Sharpe ratio, is a fixed number that
is not affected by the standard deviation of returns, the denomi-
120
nator of the Sharpe ratio. Hence, an increase in alpha increases
the Sharpe ratio.
Since the portfolio alpha is the portfolio-weighted average of the
securities’ alphas, then, holding all other parameters fixed, an increase in a security’s alpha results in an increase in the portfolio
Sharpe ratio.
The index portfolio is only efficient if all alpha values are zero.
When adding a security to the overall risky portfolio that security brings with it systematic risk, which is compensated for by
the market risk premium as well as firm specific risk which is
compensated for by the alpha.
Therefore, if the alpha is zero, adding the security to the overall
portfolio makes it less efficient…i.e. there is no risk premium to
compensate for the addition risk associated with the new security.
When market analysis uncovers positive alphas, the index (market) portfolio is no longer efficient. Why? Because there are
more positive gains out there that the market index does not capture
Optimal Risky Portfolio:
• Weight'in'Por,olio:'
• W*M$=$1'$W*A$
$
• Return'
• E(Rp)$=$+$W*A$αA$+$(W*M$+$W*A$βA)$E(RM)$
$
• Risk'
• δ2p$=$(W*M$+$W*A$βA)2$δ2p$$+$(W*A$δA)2$
If all securities have an alpha of zero the optimal weight of the
active portfolio in the overall portfolio is zero and the weight in
the index (market) portfolio will be 1.
121
Optimal risky portfolio of the single index model:
Combination of:
• Market-index portfolio, the passive portfolio denoted by M
• If we are only interested in diversification we will hold the
index portfolio. In theory, all diversifiable risk will have
been reduced to zero in this portfolio. We call this the Passive Portfolio / Passive Investment
• Active portfolio denoted by A
• However, we may feel that specific securities have the potential to offer extra firm specific returns (i.e. positive alpha). If so, we would want to include this in our portfolio.
We call this the Active Portfolio. It is made possible by security analysis.
of your active portfolio is now 50% higher than originally anticipated. Explain the significance of this change.
3. Explain why adding a security with a zero alpha to an efficient
index portfolio will make the portfolio less attractive.
Solution:
1. The initial position of the active portfolio in the overall risky
portfolio is given by:
W0A = 46%
2. The adjusted position of the active portfolio in the overall
risky portfolio is given by:
Example:
W*A = 57.7%
Assume that you have a market (index) portfolio with an expected return to variance ratio (E(RM) / δ2M ) of 26%. Assume
further that you have an active portfolio with a portfolio alpha to
portfolio variance ratio (αA / δ2A ) of 12%.
The weight of the active portfolio in the optimal risky portfolio
would increase from 46% to 57.7% and the weight in the passive
portfolio would decrease to from 54% to 42.3%.
1. What would the initial position of your active portfolio in
your optimal risky portfolio be?
The correlation between the active and the passive portfolio is
greater when the beta of the active portfolio is higher. This implies less diversification benefit from the passive portfolio, hence
a lower position in it.
2. How would the initial position of your active portfolio in your
optimal risky portfolio change if you discovered that the beta
122
3. Adding a security to the portfolio adds systematic risk to the
portfolio, but this risk is compensated for by the market premium (through beta). The security also adds company specific
risk to the portfolio variance, if the security’s beta is zero, there
is no compensation for this risk. This makes the portfolio less
attractive.
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123
Chapter 15
The Capital Asset Pricing Model
Section 1
The Capital Asset Pricing Model
Capital market theory (CMT):
by each investor is small relative to the aggregate wealth of all investors.
• CMT builds on Portfolio Theory leading to the Markowitz Efficient Frontier.
2. Single-period investment horizon – This implies that investors
are not concerned with anything that happens after the single period; it would not affect his current investment decision (not
very realistic)
• Recall that in Portfolio Theory we assumed that we can derive
a set of risky assets and an aggregate efficient frontier.
• We also assumed that all investors will want to maximise their
utility in terms of risk and return.
• To do this they will select a risky portfolio on the efficient frontier which is tangent to their utility map.
• CMT takes this one step further and develops a model for pricing all risky assets
• This leads to the development of the CAPM which we could
use to determine the required rate of return for any risky asset.
Assumptions:
1. Individual investors are price takers – this ensures that no single investor can influence the market price of securities through
his actions. This assumption pre-supposes that the wealth held
3. Investments are limited to traded financial assets – Investment
is therefore confined to stock, bonds and other tradable securities. Lending and borrowing at the risk free rate is considered
part of these investments.
4. No taxes and transaction costs – In reality these are relevant
factors that investors would consider in their investment decision. The tax bracket in which investors fall may dictate their investment behavior and transaction costs have a direct bearing on
the final performance measure of an investment.
5. Information is costless and available to all investors – this assumption is far from true; there is a high cost attached to macroeconomic and security analysis.
6. Investors are rational mean-variance optimizers – If all investors are mean-variance optimizers it would imply that all of them
use the Markowitz portfolio selection model. Their preferences
125
would be described by the utility function. People will take on
risk relative to the potential reward.
7. There are homogeneous expectations – This would imply that
all investors would use the same expected returns and covariance
matrix to generate their efficient frontier and the optimal risky
portfolio. A further implication is that all inputs used in estimating these returns and risk measures use the same set of inputs.
8. For every Borrower in the market, there is a lender – This
would imply that collectively, all investors hold the market portfolio./
Characteristics of equilibrium under CAPM:
E (r)
M
rf
QM
2
The Market portfolio will be on the CML,
tangent to the Efficient Frontier. All investors
hold “M”, the market portfolio in their optimal
risky portfolio or some combination of M and
the risk free asset.
3
The equilibrium risk premium on the market
portfolio is determined by the average level of
risk aversion of investors' and the market
portfolio’s variance (risk).
E(rM)-rf = Āδ2M
4
CML
CAL
E (rM)
1
All investors will hold the proportion of risky
assets in their portfolio in proportion to that
assets weight in the market portfolio.
Q
The risk premium on the an asset within the
portfolio will be determined by the Beta
Coefficient of that asset. (remember: the beta
coefficient is the asset’s covariance with the
market relative to market risk)
E(ri)-rf = βi [E(rM)-rf ]
1. Because all investors are assumed to use the same assumptions
to build their portfolios they will end up having the same risky
portfolio; this portfolio will be a mirror image of the market portfolio as far as the percentage assets allocation within the portfolio is concerned. As an example, if the risky portfolio of investor
A had 3% of security XYZ in its portfolio, investor B will have
126
the same percentage of security XYZ in its portfolio and the market portfolio will contain the magnitude of XYZ as a percentage
of its portfolio. Of course the rand or dollar value of the portfolios may differ.
2. If all investors hold the market portfolio, there is no need for
complicated security analysis; the market portfolio, by virtue of
its tangency with the CML, is the efficient market index portfolio.
We introduced the “Separation Property” of portfolio selection
when we looked at Optimal Risky Portfolios earlier; the same
principal can be applied here. The investor s’ portfolio selection
decision is segregated into…
a decision to adopt the risky portfolio (in this case the market
portfolio) and
a decision between the percentage of the risk free asset to include in the complete portfolio i.e. what percentage of the risky
portfolio would be substituted with the risk free asset. This will
be determined by the investors level of risk aversion as we saw in
the discussion under the construction of a risky portfolio.
different investment managers may well come up with different
“efficient risky portfolios” because in reality the input variables
will differ.
3. The equilibrium risk premium in the market portfolio is determined by the average risk aversion of investors in the market
portfolio and the market risk (variance). One of the assumptions
of CAPM is that risk-free investments involve borrowing and
lending among investors and that for every borrower there was a
lenders. Therefore in aggregate borrowers and lenders cancel out
i.e. they net to zero. If the investor opts to invest “y” in the optimal portfolio “M” it would be denoted by..
y- E(rM)-rf/ Aδ2M
If borrowers and lenders “cancel each other out” the average position in the risky portfolio will be equal to 1; the average risk aversion will be Ā; the above equation then becomes…
E(rM) – rf = Ā δ2M
i.e The equilibrium risk premium in the market portfolio is determined by the average risk aversion of investors in the market
portfolio and the market risk (variance).
A passive investor will view the market index portfolio as a reasonable approximation of an efficient risky portfolio. Remember
127
Understanding)the)Market)Por=olio)
E (r)
Includes)all)risky)assets)–)stocks,)binds,)
op5ons,)real)estate)etc.)therefore))a)
Completely)Diversified)Por2olio)i.e.)no)
unique)risk)
Propor5on)of)individual)stock)in)the)
Market)Por=olio)is)determined)by)its)
value)rela5ve)to)that)of)the)market)
i.e.)the)propor5on)is)value9
weighted.))This)must)be)the)case)if)
the)market)is)in)equilibrium.)
CML
E (rM)
M
CML)=)The)(new))Relevant)
efficient)fronGer)viz.)only)
risky)assets)
rf
QM
Lending Portfolio =
Borrowing Portfolio
The)price)at)which)a)stock)is)included)
in)the)op5mal)por=olio)is)the)only)
point)of)concern)for)the)investor)
Q
All)investors)must)hold)the)same)
market)por2olio))(on)average))
because)they)use)the)same)input)
data)to)determine)the)op5mum)
por=olio.)
The)equilibrium)risk)premium)on)the)
market)por=olio)is)determined)by)
the)average)level)of)risk)aversion)of)
investors')and)the)market)por=olio’s)
variance)(risk).)
E(rM)-rf = Āδ2M
Expected return-beta relationship under CAPM:
A stock’s contribution to the variance of the market portfolio is what
matters
! Portfolio risk is what matters to investors and what governs risk
premium they demand.
! Therefore, what matters is the contribution of the risk to the overall
portfolio from holding a particular asset, in this cases assume that
assets is a stock (share) in GE
! We can measure this contribution by looking at the covariance of the
stock with the market. We get…
WGE [ w1Cov(r1,rGE) = +w2Cov(r2, rGE) + (wGECov(rGE,rGE) + (wnCov(rn,rGE)]
! Notice that the covariance of GE with all other assets in the market
dominates the variance of the GE stock. GE’s contribution to the
variance of the market portfolio is determined by its covariance with the
market.
GE’s contribution to the variance of the market portfolio
9-11
= WGE Cov(rGE,rM)
Reward–to-risk ratio’s of the stock and portfolio
! We know from previous discussions that the excess return of an asset
(or the risk premium of the asset) is driven by the weight of the asset
within the portfolio. Therefore the contribution of holding a percentage of
GE in the portfolio to the risk premium of the market portfolio is..
WGE [ E(rGE) – rf]
! The reward –to-risk ratio for investment in GE is therefore..
GE’s contribution of risk premium
GE’s contribution to portfolio variance
[ E(rGE) – rf]
Cov(rGE,rM)
=
! The reward –to-risk ratio for the portfolio is…
Market risk premium
Market variance
=
[ E(rM) – rf]
δ 2M
9-12
! CAPM states that in equilibrium all investments will offer the same
reward–to– risk ratio. Therefore…
[E(rGE) – rf]
Cov(rGE,rM)
=
[ E(rM) – rf]
δ 2M
! We can determine the risk premium of an individual stock (let’s say GE)
from this equation…
E(rGE) – rf
=
Cov(rGE,rM) [ E(r ) – r ]
M
f
δ 2M
! The contribution of an asset’s stock to the variance of the market
portfolio is also know as the stock’s beta…we can therefore
E(rGE) = rf +
βGE [ E(rM) – rf]
E(rM)
= rf +
βM
CAPM!!
[ E(rM) – rf]
128
The Security Market Line:
The SML and Forecast Performance:
E (r)
SML
E (rM)
M
[ E(rM) – rf] = Slope
The SML is a graphical depiction of
the individual asset risk premium as
a function of asset risk. How does
this differ from the CML?
18%
Risk Premium – βGE [ E(rM) – rf]
16%
Beta (market) –
Slope -
rf
βM=1
β
Analyst(Forecast(
return(
βM
SML
α=#+2#
Fair(Return(
s1pulated(by(SML(
(CAPM(
M
E (rM) = 14%
rf = 10%
[ E(rM) – rf]
SML = Fairly priced assets i.e.
return is commensurate with risk
1
E(rGE) = rf +
E (r)
Assets in Market Equilibrium
βGE [ E(rM) – rf]
9-14
SML – interested in the properties of the security more than we
are interested in the properties of the portfolio. Have a different
measure of risk to the CML - Beta
CML – measure of risk was standard deviation
Measure of stock variance to MP & Risk premium f(beta) – Relationship as confirmed by CAPM.
βM=1
1.5
β
• This(stock(is(under+priced.#CAPM(says(it(should(return(16%,(but(the(analyst(
forecast(18%.(
• Based(on(the(analyst(forecast,(this(stock(is(a(good#buy##
• UnderGpriced(stock(plot(above#the#SML#
• Difference(between(fair(return(and(analyst(forecast(return(is(the(alpha((here(+2%)(
E (r)
Fair(Return(
s1pulated(by(SML(
(CAPM(
SML
M
E (rM) = 17%
15%
α=#$8%#
rf = 10%
7%
Analyst(Forecast(
return(
0.71
βM=1
β
• This(stock(is(over$priced.#CAPM(says(it(should(return(15%,(but(the(analyst(forecast(
7%.(
• Based(on(the(analyst(forecast,(this(stock(is(a(bad#buy##
• OverHpriced(stock(plot(below#the((SML(
• Difference(between(fair(return(and(analyst(forecast(return(is(the(alpha#(here((H8%)(
129
The SML and Actual Performance:
• The$SML$defines$all$“Fairly$priced$assets”$
E (r)
Actual'Asset'
Return'
18%
16%
SML
α=#+2#
• Nega9ve$Alpha$(over>priced$stock)$
• This$stock$is$over&priced.+CAPM$says$it$should$return$15%,$but$the$analyst$
forecast$7%.$
• Based$on$the$analyst$forecast,$this$stock$is$a$bad+buy++
• Over>priced$stock$plot$below+the$$SML$
Fair'Return'
s/pulated'by'SML'
(CAPM'
M
E (rM) = 14%
rf = 10%
βM=1
1.5
• Posi9ve$Alpha$(under>priced$stock)$
• This$stock$is$under&priced.+CAPM$says$it$should$return$16%,$but$the$
analyst$forecast$18%.$
• Based$on$the$analyst$forecast,$this$stock$is$a$good+buy++
• Under>priced$stock$plot$above+the+SML+
β
• Actual$Performance$
• Posi;ve+Alpha++=$over$–$performance$
• Nega;ve+Alpha+=$under>performance$
$
Over*Performed#
E (r)
Fair'Return'
s/pulated'by'SML'
(CAPM'
SML
• You can treat the SML as a passive investment. / Use SML as a
Benchmark
M
E (rM) = 17%
15%
• Based on the analyst forecast you can adjust your portfolio (active Investment).
α=#$8%#
rf = 10%
7%
Actual'Asset'
return'
0.71
βM=1
Under$Performed#
Potential investment strategy:
• Increase assets with a positive alpha.
β
• Reduce assets with a negative alpha.
• When all non-positive alphas have been included / excluded
the average portfolio held will resemble the market portfolio
again.
130
Estimation of alpha:
A security has an forecast rate of return of 0.13 and a beta of 2.1.
The market expected rate of return is 0.09 and the risk-free rate
is 0.045. The alpha of the stock is ?
• Es#mate(of(Expected(return(based(on(SML:(
E(ri)(=(rf(+(βi((rM=rf)((((((((((((
E(ri)(=(4.5%(+(2.1((9%=4.5%)((((((((((((
E(ri)(=(13.95%(
• Es#mate(of(alpha(
α(=(Forecast(Return(–(CAPM(Expected(Return(
α(=13%–(13.95%(=(=0.95%(
E(r)&=&D1&+&P1&–&P0&/&P0&
&&
0.18&=&P1&-75&+&6&/&75&&
&
P1&=&82.50&
&
E(ri)&=&rf&+&βi[E(rM)-rf]&
&
E(ri)&=&6%&+&1.2(10%)&
&
E(ri)&=&18%&
&
Can these two assets exist in equilibrium?
Por$olio' Expected'return' Beta'
A"
20" 1.4"
B"
25" 1.2"
Estimation of share price:
E (r)
SML
A share sells for $75 today. It will pay a dividend of $6 per share
at the end of the year. Its beta is 1.2. What do investors expect
the stock to sell for at the end of the year? Assume a risk free
rate of 6% and E(rM) = 16%
A???
20%
16%
E (rM)
B ???
M
Por-olio"A"has"a"
higher"beta"than"
Por-olio"B,"but"the"
expected"return"for"
Por-olio"A"is"lower"
than"the"expected"
return"for"Por-olio"B.""
Thus,"these"two"
por-olios"cannot"
exist"in"equilibrium."
rf
βM=1
1.2
1.4
β
131
Under CAPM is it possible to have the following example?
Por$olio' Expected'return' Standard'Dev.'
A"
30"
35"
B"
40"
25"
Is the following consistent with CAPM?
Por$olio'
Risk%Free%
Market%
A%
Expected'return' Beta'
10%
0.0%
18%
1.0%
16%
0.9%
E (r)
If"the"CAPM"is"valid,"the"expected'rate'of'return'
compensates'only'for'systema:c'(market)'risk,"represented"
by"beta,"rather"than"for"the"standard"devia>on,"which"
includes"nonsystema>c"risk."""
"
Thus,"PorFolio"A’s"lower"rate"of"return"can"be"paired"with"a"
higher"standard"devia>on,"as'long'as'A’s'beta'is'less'than'B’s.'
E(r)%=%10%+%β%×%(18%–%10)%
E(r)%=%10%+%[1.5%×%(18%–%10)]%
E(r)%=%22%%
%
SML
22%
A = 16%
M
E (rM)
rf
βM=1
1.5
The%expected%return%for%
PorKolio%A%is%16%;%that%is,%
PorKolio%A%plots%below%
the%SML%(!%A%=%–6%),%and%
hence,%is%an%overpriced%
porKolio.%%%
%
This%is%inconsistent%with%
the%CAPM.%
%
β
Impact of selected relaxed assumptions:
Assump5on:%Investors%can%borrow%and%lend%at%
the%risk%free%rate%%
AssumpAon:&
Borrow%at%the%Risk.Free%Rate%and%invest%in%the%%Risky%asset,%essen5ally,%Lending
at 7% and Borrowing at 9%
E&(r&)&
(CAL)&
P&
E&(rp&)&=&15%&
rB&=&9%&
rf&=&7%&
S&(y≤&1)&=&0.36&
S&(y&<&&1)&=&0.27&
Sharpe'ra(o'
Slope'=&[E(rp')'–'rf]/'σ'p'
'
F&
Slope'=&(0.15'–'0.09)/'0.22'
'
Slope'=&0.27'
σp&=&22%&
σ
132
Assump&on:*Investors*invest*in*a*risk*free*
asset*and*the*risky*por7olio*
E (r)
E(ri))=))E(r2))+)Betai(E(rm))–)E(R2)))SML
M
E (rM)
Remember….
• CAPM%is%based%on…%
– …%the%Markowitz%full%covariance%
model%in%which%the%theore;cal%
market%por=olio%consists%of%every%
available%asset.%
– …an%assump;on%of%mean@
variance%efficiency%i.e.%The%
op;mal%por=olio%mean@variant%
efficient,%implying%a%loca;on%on%a%
the%efficient%fron;er.%
E (rM) - E (rZ)
E (rZ)
βM=1
β
Zero%Beta)Asset)
Assump&on:*There*are*no*transac&on*costs*and*
investors*have*heterogeneous*expecta&on*
– …an%assump;on%about%expected%
return@beta%rela;onship%which%
cannot%be%observed.%%
Given%that%%the%theory%is%
based&on&expected&
returns,%which%are%not%
observable…%
%
1. How%do%you%%test%the%
validity%of%CAPM?%
2. %How%do%you%test%for%
mean%variance%
efficiency?%
With an Index Model….
E (r)
SML
E (rM)
• We# look# at# actual# returns# based# on# historical# data…this# is#
observable.#
• The#rela6onship#between#excess#return#and#market#variance#(i.e.#the#
Cov#(Ri,#RM)#is#the#same#as#that#of#CAPM#
M
• The#only#difference#in#the#above#rela6onship#between#the#2#models#
is# that# the# market# under# CAPM# is# the# (unobservable)# market# of#
every#single#risky#porHolio#whereas#that#under#the#index#model#the#
market#is##the#market#index#(a#proxy#for#the#market#porHolio).#
E (rf)
or
E (rZ)
βM=1
β
133
With an Index Model….
•
The$excess$return$rela.onship$under$the$2$models$is$also$almost$iden.cal.$$
– CPM$:$E(ri)$–$rf$=$σi[E(rM)$–$rf]$
– Index:$$E(ri)$–$rf$=$α$+$σi[E(rM)$–$rf]$
•
The$only$difference$is$the$alpha….$
– under$CAPM$alpha$is$assume$to$be$zero.$$
– Under$the$Index$model$alpha$is$assume$to$average$out$to$zero.$$
•
Actual$data$shows$that$alpha$is$cantered$around$zero$and$slightly$posi.vely$
skewed$as$a$result$of$pressure$on$porPolio$managers$to$deliver$posi.ve$
alpha.$$
•
However$this$skew$is$rela.vely$small,$which$explains$how$difficult$it$is$to$
outperform$the$market…giving$support$to$CAPM$theory.$On$a$sample$basis$
alpha$tends$to$zero,$even$under$the$Index$model.$
Example(of(alpha(distribu3on(
134
Chapter 16
Arbitrage Pricing Theory and Multifactor
Models of Risk and Return
Section 1
Arbitrage Pricing Theory and Multifactor Models of Risk
Review&'&Building&a&Single&Factor&Model&
Return
Review&'&Building&a&Single&Factor&Model&
Firm-specific (unique)
ri&=&E&(ri)&&+& ei&
“e” has a:
Mean = 0
SD = δ
Expected
return
Unexpected
return
m i&
ri&=&E&(ri)&&+&&&&&&&+&e
Uncorrelated
“m” has a:
Mean = 0
SD = δ
Expected
return
Unexpected
return
Return emanating from the inclusion
of a common macro-economic
factor
Whilst&“m”&is&
common&to&all&
securiAes,&ei&will&be&
unique&to&firm&“i”,&
therefore,&by&
definiAon&m&and&ei&
are&not&correlated&
with&each&other;&
any&shock&resulAng&
from&the&
macroeconomic&
factor&will&not&
impact&the&
unexpected&frim'
specific&return.&
Risk
Because returns m and ei are
uncorrelated we can write the
variance as…
Variance
δi2$=$$δ2m$$+$δ2ei$
Systematic factors
(market factors)
Unsystematic factors
(firm specific factors)
Covariance
Cov(ri,$rj)$=$Covm$+$ei,$m+ej$
e is uncorrelated across
firms
Cov(ri,$rj)$=$Covm,m$=$δ2m$
Significantly reduced number of estimates because we are
now relating the variance of each security to the market
and not to each other as in the Markowitz model
136
What would the impact/change on expected return be based on
the following assumptions ?
Review&'&Building&a&Single&Factor&Model&
Beta
Systema8c#Risk#<#β2iδ2m#
##
Firm#Specific#Risk##<#δ2ei###
Return
ri#=#E#(ri)#+#βm#+#ei#
Variance
We can ignore ei
and ej we assume
them to be
uncorrelated.
δ2i#=#β2iδ2m+#δ2ei###
Assume:
• the Macro- economic factor is GDP
• we expect GDP of 5%
• βi = 1.5
Covariance
• Actual GDP = 3%
Cov(ri,#rj)#=#Cov#(βi#m+ei,#βj#m+ej)#
Cov(ri,#rj)#=#βi#βj#δ2m###########################
[δax,by = ab δxy]
8-4
Single Factor Model equation:
ri = E(ri) + βiF + ei
Result:
Given a beta of 1.5, Ri will be 1.5( 5%-3%) or 3% lower than anticipated.
The$security’s$surprise$
return$is$/3%$from$the$
macro$factor$PLUS$any$
firm$specific$return.$
ri!=!Return!on!security!!
βi=!Factor!sensi1vity!or!factor!loading!or!factor!beta!
F!=!Surprise!in!macro;economic!factor!
!(F!could!be!posi1ve!or!nega1ve!but!has!expected!value!of!
zero)!
ei%=!Firm!specific!events!(zero!expected!value)!
137
Multifactor Models:
Is the single factor a good representation of systematic risk to
which expected returns are exposed? Even if an index is used as a
proxy for the market?
Multifactor models can provide a better description of security
returns. This means that the risk measures that you put in place
could be better and more effective.
They allow us to better explain the individual stock returns,
build models of equilibrium security pricing and can expose various macroeconomic risks and manage these risks. (Hedging
strategies become more effective) – creating an opposite factor
exposure
Mul$factor+Model+Equa$on+
ri = E(ri) + βiGDPGDP + βiIRIR + ei
+
+
+
ri++=+Return+for+security+I$
Surprise+Return+
$
βGDP$=+Factor+sensi$vity+,+loading+or+beta+for+GDP++
+
βIR++=+Factor+sensi$vity,+loading+or+beta+for+Interest+
Rate+
$
$ei$=+Firm+specific+events+
Compare+SAA+to+Eskom+–+will+a+single+index+model+do?+
+
+
10-9
Surprise returns (Expected return = 0) – anything that differs
from what we assumed when we calculated our securities
Negative Beta – e.g. IR
Return on security = Expected return – Any surprise return on
that seurity
138
Required:
Mul$factor+SML+Models+
• Calculate the risk premium that is attributable to each factor.
E(ri ) = rf + βiGDPRPGDP + βiIRRPIR
!
!
!!
!
• Calculate the total factor risk premium.
• Assume a T Bill rate of 4%.what would the Expected return be
for Company A
βiGDP+=+Factor+sensi$vity+for+GDP++
+
RPGDP=+Risk+premium+for+GDP+
+
βiIR!=+Factor+sensi$vity+for+Interest+Rate+
RPIR+=+Risk+premium+for+Interest+Rate+
10-11
Example:
Assumptions:
• Company A has a GDP beta of 1.2 and an interest rate beta of
-0.3.
Risk%Premium%for%GDP:%
βGDPRPGDP=%1.2%x6%%
βGDPRPGDP=%7.2%%
%
Risk%Premium%for%Interest:%
βIRRPIR=%<0.3%x(<7%)%
βIRRPIR=%2.1%%
%
Total%Factor%Premium:%
βGDPRPGDP%+%βIRRPIR%=%9.3%%
Expected(Return:(
E((r)(=(rf+(βGDPRPGDP(+(βIRRPIR((
E(r)(=(4%+1.2(6%)(+(>0.3)(>0.7)(
E(r)(=(13.3%(
%
• The risk premium for one unit of exposure to GDP is 6%
• The risk premium for one unit of exposure to interest rate risk
is -7%
139
Arbitrage Pricing Theory:
• Arbitrage occurs if there is a zero investment portfolio with a
sure profit.
• Since no investment is required, investors can create large positions to obtain large profits.
• Regardless of wealth or risk aversion, investors will want an infinite position in the risk-free arbitrage portfolio.
• In efficient markets, profitable arbitrage opportunities will
quickly disappear.
• Occasionally mispricing can occur/arise.
Well-diversified portfolios:
rP#=#E#(rP)#+# PF#+#eP#
!
F#=#some#factor#
#
! For#a#well4diversified#por:olio,#eP##
◦ approaches#zero#as#the#number#of#securiAes#in#the#
por:olio#increases##
◦ and#their#associated#weights#decrease,#therefore#we#
can#write…#
rP#=#E#(rP)#+# PF#
• Two assets must have the same beta before we can create the
arbitrage portfolio.
• Efficient market hypothesis says that this position is unstable
and so will be quickly eliminated.
Expected return from the portfolio + Surprise return from a
macro-economic factor +Surprise return from the market
140
Returns(as(a(Func,on(of(the(Systema,c(
Factor(
Return (%)
A
Return (%)
Returns(as(a(Func,on(of(the(Systema,c(Factor:(
An(Arbitrage(Opportunity(
B
A
Return (%)
A"="1"
A"="1"
(0.10 + 1 x F) x R1M Long A
C
-(0.08 + 1 x F) x R1M Short C
C="1"
10
10
10
F
0
+0.20 x R1M = R20K
8
F
0
• Risk Free Profit because factor
F
risk cancels out over the long and
short positions
0
rA"="E"(rA)"+" AF(
• Net Investment is zero
rA"="10%+1F(
Well5diversified(
Single5Stock(
• As long as well diversified portfolios
with equal betas have different
expected returns an arbitrage
10-18
opportunity exists
10-17
Well diversified portfolio – Non-factor risk (systematic risk) is
diversified out of the portfolio. Return determined by systematic
risk only and the beta is 1. Expected return of the portfolio is
10%. At 10% the systematic factor is zero.
An#Arbitrage#Opportunity#
• Construct portfolio D with 50% A
and 50% F
Expected Return (%)
• Beta of D = (0.5X0)+(0.5X1) = 0.5\
A
10
D
7
6
r f= 4
C
• Return of D = (0.5*4) + (0.5*10%) =
7%
Risk Premium
• Beta is the same as Portfolio C but
return is higher – an arbitrage
opportunity exists
F
0.5
1
Beta with respect to the
Macro-factor
Strategy:)
• Long#D#
• Short#C#
10-19
Therefore we can say that to rule out the arbitrage opportunities, well diversified portfolos must lie on the line F-A
141
Therefore we can say that to rule out the arbitrage opportunities, well diversified portfolios must lie on the line F-A
APT$and$CAPM$
APT$
Security)Market)Line)(Mkt.)Index))
E(rp) = rf + [E(rM) – rf]βp
Expected Return (%)
E(rM)
[E(rM - rf)]
rf
• There are a sufficient
number of assets to
construct a welldiversified portfolio–
all asset are on the
SML
• No arbitrage
opportunity exists
1
Beta with respect to
Index
• Equilibrium$means$no$arbitrage$
opportuni8es.$
CAPM$
• Model$is$based$on$an$inherently$
unobservable$“market”$
porDolio.$
• APT$equilibrium$is$quickly$
restored$even$if$only$a$few$
investors$recognize$an$arbitrage$ • Rests$on$meanHvariance$
efficiency.$The$ac8ons$of$many$
opportunity.$
small$investors$restore$CAPM$
equilibrium.$
• The$expected$return–beta$
rela8onship$can$be$derived$
without$using$the$true$market$
• CAPM$describes$equilibrium$for$
porDolio.$
all$assets.$
10-23
10-20
APT Model:
APT applies to well diversified portfolios and not necessarily to
individual stocks.
With APT it is possible for some individual stocks to be mispriced - not lie on the SML.
APT can be extended to multifactor models.
142
Example(
Example(
Table&1&
Table&1&
Factor&
Ifla+on(
Industrial(Produc+on(
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Factor&Beta&
Factor&Risk&Premium&
1.2(
6%(
0.5(
8%(
0.3(
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Table&2&
Factor&
Ifla+on(
Industrial(Produc+on(
Oil(Prices(
QuesAon:&
Expected&Rate&of&Change&Actual&Rate&of&Change&
5%(
4%(
3%(
6%(
2%(
0%(
Suppose(that(the(market(expected(the(values(for(the(3(macro(factors(given(
in(column(1(of(table(2,(but(the(actual(values(turn(out(as(given(in(column(2(of(
the(same(table.(Calculate(the(revised(expected(returns(on(the(stock(once(the(
surprises(become(known(
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Factor&Risk&Premium&
1.2(
0.5(
0.3(
6%(
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Actual&Rate&of&Change&
5%(
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2%(
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Factor&
Ifla+on(
Industrial(Produc+on(
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Expected&Rate&of&Change&
SoluAon&
E(r)(=(6%(+((1.2(×(6%)(+((0.5(×(8%)(+((0.3(×(3%)(=(18.1%(
(
Surprises&in&the&macroeconomic&factors&will&result&in&surprises&in&the&return&
of&the&stock:&
Unexpected(return(from(macro(factors(=((
[1.2(×((4%(–(5%)](+([0.5(×((6%(–(3%)](+([0.3(×((0%(–(2%)](=(–0.3%(
(
E((r)(=18.1%(−(0.3%(=(17.8%(
(
Beta * risk premium: Explains Return
Beta * (Actual – Expected): Explains Risk
143
Where%Should%We%Look%for%Factors?%
• Need%important%systema:c%risk%factors%
– Chen,%Roll,%and%Ross%used%industrial%produc:on,%expected%
infla:on,%unan:cipated%infla:on,%excess%return%on%
corporate%bonds,%and%excess%return%on%government%bonds.%
– Fama%and%French%used%firm%characteris:cs%that%proxy%for%
systema:c%risk%factors.%
Fama$French*Three$Factor*Model*
• SMB*=*Small*Minus*Big*(firm*size)*
• HML*=*High*Minus*Low*(book$to$market*raAo)*
• Are*these*firm*characterisAcs*correlated*with*actual*
(but*currently*unknown)*systemaAc*risk*factors?*
rit = α i + βiM RMt + βiSMB SMBt + βiHML HMLt + eit
144
Chapter 17
Financial Risk Management
Risk v.s. Return
Higher risk <=> higher returns
Section 1
Introduction
Financial Risk Management:
Market risk:
The practice of creating economic value in a firm by using financial instruments to manage exposure to risk.
The risk of losing value in our investment due to changes in market factors:
• Firm only adds value when performing activities shareholders
couldn't undertake themselves
• So risk management only add value when mitigating risks shareholders couldn't mitigate themselves
• In perfect markets, hedging of risks has no value as price of
bearing risk within firm is same as price of bearing risk outside
it
• Risks unique to firm are the best candidates for FRM - diversification
• Changes in asset prices, interest rates (yield curves, credit
spreads), currencies, inflation and commodity prices
Measurement: Value at Risk (VaR)
Credit risk:
Risk arising from failure of counterparty to meet its obligation
• Pre-settlement risk: default risk that arises prior to settlement
• Settlement risk: default risk that arises at settlement
1. Exposure: in event of default how large will outstanding obligation be?
2. Default probability
3. Recovery rate: if counterparty defaults, how much of obligation can be recovered e.g. Through bankruptcy proceedings
146
Liquify risk:
Basel Committee on Banking supervision:
Arises from situations in which party wanting to trade in an asset
unable to do so because nobody in the market wants to trade
that asset
• Set international standards for banks to guard against losses as
a result of financial and operational risk
Asset liquidity risk: risk arising from the inability to liquidate assets at a reasonable price
• Value at Risk: measures how much is an entity at risk of losing
with a given probability level ad over a specific time horizon
Funding liquidity risk: risk arising from the inability to raise cash
to meet obligations at a reasonable cost
• Expected shortfall: alternative method that Basel is currently
trying to implement - given that an entity’s loss exceeds a certain threshold, what amount is it most likely to lose?
Operational risk:
Normal distribution:
Risks arising from current or future losses due to failed or inadequate people, processes or systems or external events
Problems:
Failed/ inadequate people: employee fraud or collusion, mistakes,
strikes, lack of knowledge or expertise, death and disability
Failed/ inadequate processes: accounting error, project risk, transaction error
• Allows any value, even negative, but share prices cannot be
negative
• Does not account for compounding
So use Lognormal
Failed/ inadequate systems: data quality, security breaches, system failure, system capacity constraints
External events: government action, natural disaster
147
Lognormal distribution:
Example:
Describes random variable that grows every instant by a rate that
is itself a normal random variable.
Hypothetical returns on Telkom SA Limited given in table below,
together with the simple returns and continuously compounded
returns:
Progression of lognormal random variable reflects continuous
compounding.
Day
Price
100.0
102.5
107.0
103.0
105.0
1
2
3
4
5
Simple Return Log Return
2.50%
4.39%
-3.74%
1.94%
2.47%
4.30%
-3.81%
1.92%
Whole Week
5.00%
4.88%
Calculate the simple return over the last day as:
Sum of Returns
5.09%
4.88%
102.50-100 = 2.5%
100
Notice from table, sum of log-returns is same as log returns over
whole week (calculated using the prices from day 1 and day 5).
Example:
Suppose stock price of Telkom SA Limited yesterday was R100
per share, and share price today is R102.50.
(
(
(
100e0.0247+0.043-0.0381+0.0192 = 105
Log-return is rate r that satisfies:
Same does not apply to simple returns.
er = 1.025
i.e. r = ln(1.025) = 2.47%
This property is convenient because we assume that share returns are normally distributed.
Sum of normal is still normal
Difference is usually small, but lognormal return has important
properties.
148
Suppose future daily log-returns on ABC Co.’s shares are independent of each other, and normally distributed with mean of
1% and a standard deviation of 2%
Lets say tomorrow’s return is r1 and the next day’s return is r2,
because of property of log-return, we know that two-day return r
is just r1 + r2.
We know that r is also normally distributed with a mean of
2(0.01) and a variance of 0.022 + 0.022 = 2(0.02)2.
The standard deviation of the two day return r is √2 x (0.02).
General rule:
If log-returns over next t days are all independent and identically
normally distributed with mean of μ and standard deviation σ,
then t-day log-return is normally distributed with mean tμ and
deviation √t.σ
149
Section 2
Value at Risk
Value at risk (VaR)
• Measure of how the market value of an asset or of a portfolio
of assets likely to decrease over a certain time period under
usual conditions.
The pre-specified probability: Level of certainty that we attach
to expected loss figure. Very important to realize with VaR there
is always (small) chance real loss could exceed VaR. Pre-specified
probability can also be stated in terms of a confidence level,
which is just one minus probability.
Or more formally:
• The maximum loss over a target horizon, such that there is a
low, pre-specified probability that the actual loss will be larger.
Example:
Investment bank holding portfolio reports that it has 1-day VaR
of R2 million at a 95% confidence level.
VaR therefore asks:
“What is the worst case scenario?”
The maximum loss figure: This is the VaR figure (and is reported
as positive number). VaR is described as measure of downside
risk.
Provided that normal conditions prevail, this implies bank can
expect that, with a probability of 95%, change in value of its portfolio would not result in a decrease of more than R2 million over
1 day
(
(
(
(
(
OR
with probability of 5%, value of its portfolio will decrease by
(lose) R2 million or more over 1 day.
The target horizon: This is time period over which loss is being
measured. VaR is most often calculated over short time periods
(1- week or 1-day).
150
Common VaR calculation models:
Delta-Normal method:
• Assumes that underlying risk factors have normal distribution.
• Variance-covariance also known as Delta-Normal: assumes risk
factor returns always (jointly) normally distributed and that
change in portfolio value are linearly dependent on all risk factor returns
• Historical simulation: assumes asset returns in future will have
same distribution as they had in past (historical market data)
• Monte Carlo simulation: future asset returns more or less randomly simulated to derive approximation of distribution
• Uses these assumptions to determine distribution of underlying portfolio profits and losses, which is also normal.
• Once distribution obtained, we use standard mathematical
properties of normal distribution to determine loss that will be
equalled or exceed x percent of the time, i.e. Value at Risk.
• Method typically easy to calculate due to simplifying assumptions.
Drawbacks:
All models calculate VaR using same formula – difference lies in
how distribution of investment returns derived.
Assumption that portfolio value linearly related to risk factors
• Can be dangerous assumption when portfolio contains instruments with non-linear exposure to risk factors. E.g. Options.
Assumption of a normal distribution of asset returns
• Unfortunately, there is large body of empirical evidence that
suggests that this assumption is not realistic.
151
Historical Simulation method:
Monte Carlo Simulation method:
• Calculated by taking the current portfolio and subjecting it to
actual changes in market over past period.
• Rather than simulating observed risk factor changes like Historical simulation, we choose distribution believed to adequately approximate possible changes in risk factors.
• Results indicates what our losses and gains would have been if
our current portfolio was held in past periods instead.
• We make the assumption that this distribution will repeat itself in future.
• From this distribution’s mean and standard deviation we can
then compute VaR.
• Then use random number generator to generate thousands of
hypothetical changes in risk factors.
• Use these to construct thousands of hypothetical profits and
losses on current portfolio and, from this, distribution of profits and losses (law of large numbers).
• Use this distribution to calculate VaR.
Benefits:
• Simple to implement (preferred method by majority of banks)
• Does not assume normal distribution of returns (more realistic)
• Able to approximate options and option-like instruments without difficulty
1. Choose risk factors that will affect instrument/situation
2. Assign distributions/values to each factor
3. Create/Determine the ‘function’ that relates the risk factors to
the instrument/situation
4. Generate random numbers from 0 to 1 for each of the factors
Drawbacks:
5. Use RN’s and the CDF’s of each factor, obtain a value for each
factors.
• Requires large data set to compute
6. Calculate function value for that iteration
• Can be computationally intensive
7. Go back to step 4 and repeat process n times
• Relies on assumption that past distributions will be repeated
8. Average all the simulated path values
152
Benefits:
• Does not assume normal distribution of returns
• Able to approximate options and option-like instruments without difficulty
Drawbacks:
• Complex method requiring substantial computational power.
• Hence, not preferred by most financial institutes.
• Selecting appropriate distribution and relevant parameters requires high degree of expertise and judgement.
!
Simulating Quarterly Profits (look @ Excel sheet)
!
Assign Distributions to Profit Factors:
"
"
"
!
Quantity (Q)
Variable Costs per unit (VC)
Price per unit (P)
~ Uniform(8000, 12000)
~ Normal (7,2)
~ Normal(10,3)
Fixed Costs are R5000
Profit Function:
153
Section 3
The Delta-Normal Method
Step 1: Mapping the Exposures
First step involves mapping exposures to risk factors using sensitivities to obtain approximate linear relationship between risk
factors and changes in value of portfolio.
It is the combination of these three components that determine
how much we can lose on our portfolio.
The Risk factor: this is primary source of uncertainty. Examples
of risk factors are stock prices, credit spreads, exchange rates,
etc.
The Exposure: This is how much exposure we have to particular
risk factors.
For a well-diversified portfolio of stocks, use beta of portfolio to
obtain relationship between changes in value of portfolio P and
changes in level of stock market M :
∆P = β ∆M
We are treating each rand invested in portfolio as β Rands invested directly in market.
E.g. If we had a portfolio worth R1 million with a beta of 1.25 we
would treat it for purposes of calculating VaR as R1.25 million exposed to the market.
The Sensitivity: This is ‘link’ between risk factor and exposure,
and tells us the extent to which any unit change in risk factor
will affect our portfolio (stronger if more sensitive).
154
Step 2: Obtain distribution of risk factors
For purposes of simplicity, we will always assume that changes in
risk factors (stock prices, indices, bond yields etc.) are normally
distributed.
Will therefore need to know the mean and standard deviations
of the risk factors, as well as correlations between them when
more than one risk factor present.
Sometimes, you may be given distribution of daily returns and
have to calculate distribution of weekly or monthly returns using
square-root of time rule.
Step 3: Obtain distribution of changes in value of position
Derive using variance and covariance rules.
Step 4: Get the VaR statement
Once we have calculated mean µ∆V and standard deviation σ∆V,
(1-x)% VaR is given by:
VaR = µ∆V + σ∆V αx%
Always check that distribution of risk factors corresponds with
horizon over which VaR is being calculated.
Example: Need to calculate VaR for 1-day period but mean and
standard deviation of the distribution given in months. Need to
convert relevant variables to 1-day period so horizons match.
If log-returns over next t days are all independent, and identically normally distributed with mean of µ and standard deviation
σ, then the t -day log-return is normally distributed with mean tµ
and standard deviation √t. Σ
155
Basic example:
Suppose portfolio manager manages a portfolio which consists of
a single asset. The return of the asset is normally distributed
with annual mean return 10% and annual standard deviation
30%. The value of the portfolio today is R100 million.
With 1% probability what is the maximum loss at the end of the
year?
VaR = µ∆V + σ∆V α1%
(
= (100 x 0.1) + (100 x 0.3) x – 2.326
(
= -59.78
Thus, annual 99% confidence VaR for our portfolio is R59.78 million.
Note: have defined αx% to be negative, so VaR (being a loss figure) will typically be negative.
We often report the VaR number as a positive number, so don’t
get confused.
156
Section 4
Multiple Exposures
The Square root of time rule:
Q: Assume that the returns on portfolio are normally distributed
with a mean monthly return of 3% and standard deviation of
12%. You wish to calculate the 5% annual VaR for this portfolio.
What is the adjustment that needs to be made to the variables in
order to match time periods?
More frequently, however, we need to convert from annual or
monthly values to daily values to compute VaR.
Q: Assume we have a normally distributed asset with an annual
mean of 22% and standard deviation of 32%. We want to calculate the 1-week VaR so we need to convert as follows:
If we assume
A:(
1
1-week mean = 22 x 52
Stationarity: The distribution of the portfolio changes will remain the same for the next 12-months. We can then get
(
1-week standard deviation =
= 0.423%
1
× 32
52
= 4.44%
(
= 12 x 3% = 36%
Independence: The changes in the value of the portfolio over
each month are independent of the other months.
= 12 ×12
= 41.6%
157
Multiple Exposures:
What is the 5% one-month VaR of your combined position?
Assume we have $110 million invested in a well-diversified portfolio of equities. Returns on our equities are perfectly correlated
with the S&P-500 index, and the portfolio has beta of 1. In addition, the portfolio pays a fixed dividend of 1.4% per year. We also
have a short position of 200 (hypothetical) S&P-500 Index futures contracts which mature in 6 months. We have computed
the value of the position to be $55.643 million.
Our first exposure is clearly a $110m well-diversified portfolio.
Lastly, we have a long position of 500 FTSE-100 Index futures
contracts invested in the UK market with a current value of
₤29.696 million.
The expected monthly return on the S&P-500 is 1% and that on
the FTSE-100 is 1.25%. The monthly standard deviation on the
S&P-500 is 6.1% and that of FTSE-100 is 6.5%. The correlation
between the two indices is 0.55 and the current exchange rate is
1.6271 $/₤.
The portfolio has a beta of 1 and is perfectly correlated with the
S&P 500 index so we treat the investment exposure as $110m in
the S&P 500.
Keep in mind at this stage that the portfolio pays dividends.
Our second position is a short position in the S&P-500, worth
$55.643 million.
Given that both of these initial investments are in the same underlying asset (S&P-500) we can net them off against each other:
$110m – $55.643m = $54.357m net exposure in the S&P500
Lastly, we have a long position in FTSE-100 futures..
We will need to bring these two exposures together to compute a
VaR for our combined position, however, so we need to convert
the FTSE-100 exposure to dollars before proceeding:
Our FTSE-100 position is worth ₤29.696m which converts at exchange rate to $48.319m. (exch rate = 1.62712)
158
We have been told that the returns are normally distributed and
the mean and standard deviations have been given.
The only factor left to consider is the dividends payable on the
portfolio. As the dividends are certain this will influence the final
expected portfolio value.
µΔV( = 54.357 x (0.01) + 48.319 x (0.0125) + 0.128
(
= $1.2759m
σ2ΔV= (54.3572 x 0.0612) + (48.3192 x 0.0652) + (2 x 54.357 x 48.319
x 0.061 x 0.065 x 0.55)
(
= 32.3138
Our monthly dividend is therefore:
0.014 * 110
= 0.128m
12
σΔV= $5.6845m
Our VaR is therefore:
The change in our value is therefore based on changes in each of
the underlying positions plus the dividend which is received with
certainty:
where X1 is the exposure to the S&P 500 and X2 is the exposure
to the FTSE-100.
From this, our mean and deviation can be expressed in dollar
terms (millions) as follows:
VaR = (1.2759 - 1.645*5.6845) = -8.0751
So, the 5% 1-month VaR of the portfolio is $8.0751m
Concept Check: If we were to calculate the VaR for our exposures separately, they would not add up to the answer above.
Why?
Correlation between assets needs to be considered.
159
VaR in terms of relative losses:
Up until now looked at VaR as absolute loss number expressed in
monetary terms.
Therefore instead of ΔV we use ΔV:( (
(
(
V
(
(
(
(
X
ΔV X 1
0.128
=
(Δ ALSI ) + 2 (Δ S&P ) +
V
V
V
V
When comparing investments of different sizes, however, how
do we determine whether the difference is significant or not?
Example: A VaR of $8.0751m on a $50m portfolio is a lot riskier
than the same VaR on a $100m portfolio.
Computing Relative VaR
Instead of calculating the absolute loss, we can use the relative
loss to ensure a more accurate comparison.
For our previous example, we could have found the relative VaR
by:
VaR = (0.01243 - 1.645*0.05536) = -7.864%
8.0751/102.68 = 7.864% of portfolio value.
Alternatively, we can work with relative losses from our initial calculations so that the final answer is expressed as a relative figure.
160
Risk Decomposition:
When holding multiple positions we might be interested in decomposing our combined VaR to find the contributions from
each individual position.
E.g. For our earlier example, how much VaR stems from portfolio, how much from the short position in S&P futures and how
much from the long position in FTSE futures?
The Final VaR of the whole portfolio was calculated as $8.0751m.
The FTSE position has therefore increased the risk from
$4.7825m to $8.0751m.
Initial VaR
-$9.8096
Final VaR
-$ 8.0751
Intermediate VaR
-$ 4.7825
Impact of S&P Futures
Impact of FTSE Futures
Calculating VaR for just the portfolio we would find a value of
$9.8096m. (Remember Dividends!)
We could then calculate the VaR of the combined position of the
portfolio and short position in the S&P 500 futures. (Remember
Dividends!)
While intuitive, the biggest problem with this approach is that
the risk contributions of different parts of the complete portfolio will depend on the order in which the positions are listed.
Concept check: Why does the above statement hold?
Self Assessment: Re-order exposures & re-calculate
This would yield a VaR of $4.7825m. The addition of the short position has therefore reduced VaR by $5.0271m
A short $55.643m position index futures on S&P 500
A well diversified $110m portfolio of American shares
A long ₤29.696 position index futures on FTSE-100
161
Summary:
!
Calculating VaR for share portfolio only:
"
"
!
Exposure of $110m to S&P 500 (Remember Dividends!)
Calculate $ Mean and Std Dev from S&P distribution
Calculate VaR of the combined position of the
share portfolio and short pos. in S&P 500 futures.
"
"
Used combined exposure: $110m – $55.643m = $54.357m
Again, only looking at S&P distribution (Remember div’s!)
Marginal risk decomposition:
Found that incremental risk decomposition suffers from problem related to order in which investments added to overall portfolio.
Can avoid problems with incremental risk decomposition by
looking at how much risk changes if we increase size of exposure
by one unit.
E.g. What would impact be on VaR if we increased holdings in
FTSE by $1?
This will give us marginal risk decomposition.
Calculating VaR for just the portfolio we would find a value of
$9.8096m. (Remember Dividends!)
Calculating the VaR of the combined position of the portfolio
and short position in the S&P 500 futures would yield a VaR of
$4.7825m. (Remember Dividends!)
The addition of the short position has therefore reduced VaR by
$5.027m
!
First, we write VaR in terms of portfolio elements.
!
Let
"
"
XC1 = S&P equity portfolio
XF1 = Short futures position in the S&P 500
Since we aggregated them in our previous discussion, we’ll
also indicate the net S&P position as X1 ( = XC1 + XF1).
"
X2 = Long FTSE futures position
162
The 95% VaR can be written as follows:
Thus to find the risk contribution of each investment we find
the partial derivatives with respect to each element:
A convenient property of VaR is that it can be written as:
Notice that the risk contributions sum to the total VaR.
If we increase the amount of the FTSE portfolio from X2 to
X*2, the amount of risk in the portfolio changes by:
If we multiply top and bottom by X2 we find:
So:
= (0.01117 ) − 1.645 ×
54.357 × 0.0612 + 48.319 × 0.55 × 0.061 × 0.065
54.357 2 × 0.0612 + 2 × 54.357 × 48.319 × 0.55 × 0.061 × 0.065 + 48.319 2 × 0.065 2
= 0.01117 − 1.645 ×
0.202 + 0.1054
10.994 + 11.455 + 9.864
= -0.0778
= the increase in portfolio risk for each percentage increase in
the amount of X2 held in the portfolio.
163
Therefore the marginal risk contribution from the original portfolio is:
$110 x -0.0778 = -$8.564m (with some rounding error)
= (0.0125 ) − 1.645 ×
48.319 × 0.065 2 + 54.357 × 0.55 × 0.065 × 0.061
54.357 2 × 0.0612 + 2 × 54.357 × 48.319 × 0.55 × 0.061 × 0.065 + 48.319 2 × 0.065 2
= 0.0125 − 1.645 ×
And:
0.204 + 0.1185
10.994 + 11.455 + 9.864
= -0.081
Therefore the marginal risk contribution from the original portfolio is:
= (0.01) − 1.645 ×
= 0.01 − 1.645 ×
54.357 × 0.0612 + 48.319 × 0.55 × 0.061 × 0.065
54.357 2 × 0.0612 + 2 × 54.357 × 48.319 × 0.55 × 0.061 × 0.065 + 48.319 2 × 0.065 2
$48.319 x -0.081 = -$3.908m (with some rounding error)
0.202 + 0.1054
10.994 + 11.455 + 9.864
= -0.079
Therefore the marginal risk contribution from the original portfolio is:
-$55.643 x -0.079 = +$4.397m (with some rounding error)
Baring’s Bank:
!
Long $7.7b NIKKEI futures
!
Short $16b JGB Futures
!
σNK = 5.83%, σJGB = 1.18%, ρ = 0.114
!
Var95% = 1.645·σp = $835m
!
Var99% = 2.326·σp = $1.18b
!
Actual loss was $1.3b – exceeding both the 95% and 99% VaR
164
Example:
You are responsible for a portfolio comprising three investments,
two of which are held in derivative instruments. Your current equity portfolio investment of R45m comprises a position in the
Satrix Top 40 ETF. In addition, you hold a long position in 150
index futures contracts on the Top 40 Index with each futures
contract having an exposure of R4371. You have also attempted
to diversify your risk and to this end have sold 200 put options
on the S&P 500 Index. Each option contract has an exposure of
$2062 to the S&P 500 Index. The contract multiplier for both
the options and futures contracts is 10.
The expected monthly rate of return on the Top 40 is 1.1% and
that of the S&P 500 is 1.25%. The standard deviation of the
monthly returns of the Top 40 index is 4.45%, and that of the
S&P 500 is 5.85%. The correlation between the monthly rates of
return is estimated to be r = 0.55. Ignore currency risk.
x%
αx%
0.01%
-3.719
1%
-2.326
2.50%
-1.96
5%
-1.645
10%
-1.282
25%
-0.674
50%
0
Solution:
First determine the exposures:
R45m invested in Satrix Top 40 ETF. Since the ETF overall return is the same as the Top 40’s, can treat as a direct position in
the Top 40 index
Long position in 150 Top 40 index futures contracts:
Given that exposure per contract is R4371 and contract multiplier is 10
This means that Top 40 futures exposure is:
a) Assuming the exchange rate is 7.15 R/$, what is your 1-month
95% VaR?
b) Calculate the marginal risk contribution of each of your exposures.
150 x 10 x 4371 = R6 556 500
Net Exposure to Top 40 is thus:
R45m + R6 556 500 = R51 556 500
165
Short position on 200 S&P Put options:
VaR = R0.936m – 1.645 x R3.549m = -R4.902m
Given that exposure per option is $2062 and contract multiplier
is 10
Therefore the 1-month 95% VaR is R4.902m.
This means that S&P 500 exposure is:
200 x 10 x 2062 = $4 124 000
Marginal Risk Decomposition for 3 Factors:
Satrix Top 40 ETF
Long Top 40 Index futures
Convert to Rand terms :
Short S&P 500 Index futures
$4 124 000 x 7.15 = R29 486 600
Now determine Portfolio Mean and Std Deviation:
µ∆V( = (51.557 x 0.011) + (29.487 x 0.0125)
All Subscripts “1” represents Top 40
(
= 567 121.5 + 368 582.5
All Subscripts “2” represents S&P 500
(
= R935 704
σ2∆V (= (51.557 x 0.0445)2 + (29.487 x 0.0585)2 + (2 x 51.557 x 29.487
x 0.0445 x 0.0585 x 0.55)
(
If we consider the Satrix Fund exposure and the Top 40 Futures
exposure i.t.o the previous derivative formula, we can see that we
will get the same marginal risk sensitivity rate for both exposures
respectively.
= R12.592m
σ∆V = R3.549m
Why?
The input values used in the equation are identical!
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Thus, only decompose risk to two factors, namely:
!
• Top 40 Net exposure
= 0.0125 – 1.645 x (29.487 x 0.05852) + (51.557 x 0.0445 x 0.0585 x 0.55)
3.549
= -0.0685
• S&P 500 exposure
!
!
For the Top 40, the marginal risk contribution is:
= 0.011 – 1.645 x (51.557 x 0.04452) + (29.487 x 0.0445 x 0.0585 x 0.55)
3.549
= -0.0559
!
For the S&P 500, the marginal risk contribution is:
Therefore the rand risk contribution is:
0.0685 x R29.487 = R2.020m
!
Final Check: Sum of Marginal Values = Total VaR
R2.882m + R2.020m = R4.902
!
Therefore the rand risk contribution is:
0.0559 x R51.557 = R2.882m
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