Portfolio Management
2.1 Portfolio Optimisation
Outline
1. Optimisation Theory
2. Markowitz Model
1. Optimisation Theory
Optimisation
Optimisation is a mathematical method to find the values of a set of variables
(control variables) that minimise or maximise a function (objective function)
subject to some constraints.
❖
A Typical Optimisation Problem
f(x)
min f (x)
global maximum
x
local maximum
subject to g(x) = 0
h(x)  0
❖
Maximum/Minimum (Extremum)
-
-
Maximum/minimum value of a
function within a neighbourhood.
local minimum
global minimum
constraint
Called global min/max if it is the min/
max in the whole domain, local min/
max otherwise.
3
x
1. Optimisation Theory
Unconstrained Optimisation
❖
Unconstrained Optimisation
min f (x)
x
-
1st order necessary condition
@f (x)
=0
@x
-
Example
min (x1
x
5)2 + 3(x2
10)2
1st order condition:
@f (x)
= 2(x1
@x1
5) = 0,
@f (x)
= 6(x2
@x2
10) = 0
) x⇤1 = 5, x⇤2 = 10
4
1. Optimisation Theory
Constrained Optimisation
❖
Optimisation with Equality Constraints
min f (x)
x
subject to
g(x) = 0
-
This problem can be rewritten as an unconstrained problem by using
Lagrangian, L.
min L = min f (x)
x,
-
x,
1st order necessary condition
@f (x)
@L
=
@x
@x
0
g(x),
0 @g(x)
@x
@L
= g(x) = 0
@
5
: Lagrangian multiplier
=0
1. Optimisation Theory
Constrained Optimisation
❖
Example (Minimum Variance Portfolio)
1 0
min w ⌃w
w 2
subject to
w0 1 = 1
-
-
Lagrangian, L
1 0
L = w ⌃w
2
(w0 1
1)
1st order necessary conditions
@L
= ⌃w
1=0 ) w=⌃ 1 1
@w
@L
= w0 1 1 = 0
) 10 w = 10 ⌃ 1 1 = 1
@
1
⌃
1
1
⇤
⇤
)
= 0 1 , w = 0 1
1⌃ 1
1⌃ 1
6
1. Optimisation Theory
Constrained Optimisation
❖
General Constrained Optimisation
min f (x)
x
subject to
g(x) = [g1 (x), · · · , gL (x)]0 = 0
h(x) = [h1 (x), · · · , hM (x)]0  0
-
Lagrangian, L
L = f (x) +
-
0
1 g(x)
+
0
2 h(x)
1st order necessary condition (Karush-Kuhn-Tucker (KKT) condition)
@f (x)
+
@x
g(x) = 0
h(x)  0,
0 @g(x)
1
@x
2
0 @h(x)
2
+
@x
0,
=0
2i hi (x)
7
= 0,
i = 1, · · · , M
2. Markowitz Model
Feasible Set
❖
Two-Asset Portfolio
-
The feasible set forms a line.
µp = ↵µ1 + (1
2
p
p
❖
2
=
=
↵2 12
(
+ (1
↵)µ2
↵)2 22
𝝆=-1
+ 2⇢↵(1
|↵ 1 (1 ↵) 2 |
↵ 1 + (1 ↵) 2
↵)
𝝆=1
1 2
𝝆=-1
if ⇢ = 1
if ⇢ = +1
1
N(>2)-Asset Portfolio
-
The feasible set is a solid 2-D
region.
-
It is convex to the left.
2
1
3
* Matlab: L3_Script1.m
9
2. Markowitz Model
Efficient Frontier
❖
Minimum Variance Set
-
❖
Minimum Variance Portfolio
-
❖
The left boundary of the feasible
set.
efficient frontier
The portfolio with the minimum
variance
feasible set
Efficient Frontier
-
minimum variance
portfolio
Minimum variance set above the
minimum variance portfolio.
minimum
variance set
10
2. Markowitz Model
Markowitz Model
❖
Problem
1 0
min w ⌃w
w 2
subject to w0 1 = 1
)
1 0
min w ⌃w
w 2
subject to Aw = b
w 0 µ = µp
❖
where A = [1 µ]0 , b = [1 µp ]0
Solution
1 0
L = w ⌃w
2
@L
= ⌃w
@w
@L
= Aw
@
⇤
= (A⌃
b)0
(Aw
A0 = 0 ) w = ⌃
b=0
1
A0 )
1
b,
1
A0
) Aw = A⌃
w⇤ = ⌃
11
1
A0 (A⌃
1
A0 = b
1
A0 )
1
b
2. Markowitz Model
Efficient Frontier
❖
Portfolio Variance
-
The variance of the optimal portfolio return has the form
2
p
⇤0
= w ⌃w⇤
= (⌃
1
A0 (A⌃
= b0 (A⌃
-
Let Q = (A⌃
1
A0 )
1
1
A0 )
1
A0 )
1
b
1
b)0 ⌃(⌃
1
A0 (A⌃
1
A0 )
1
b)
. Then,
2
p
= b0 Qb

q11
= [1 µp ]
q12
q12
q22

1
µp
= q22 µ2p + 2q12 µp + q11
-
That is, the portfolio variance is a quadratic function of the portfolio return,
µ plain.
and the efficient frontier is a curve convex to the left on the
12
2. Markowitz Model
Two-Fund Theorem
Consider two optimal solutions
wi⇤ = ⌃
1
A0 (A⌃
If we define w3⇤ = ↵w1⇤ + (1
w3⇤
=⌃
1
1
A0 )
1
bi ,
bi = [1 µpi ]0 , i = 1, 2
↵)w2⇤ , then
0
A (A⌃
1
0
A)
1
b3 ,

1
b3 =
↵µp1 + (1
↵)µp2
Therefore, given any arbitrary
µp3 = ↵µp1 + (1
↵)µp2 for some ↵,
the optimal solution can be constructed as a linear combination of w1⇤ and w2⇤ .
w3⇤ = ↵w1⇤ + (1
13
↵)w2⇤
2. Markowitz Model
Inclusion of a Risk Free Asset
❖
Risk Free Asset + Risky Asset
-
Consider a portfolio of a risk-free asset, and a risky asset F. Let α denote the
weight on the risky asset. Since the variance of the risk free asset is 0, the
return and risk of the portfolio have the form
µp = ↵µF + (1
p
=↵
↵)rf
F
µ
-
Therefore, the feasible set forms a
straight line.
µp = rf +
-
µF
rf
F
F
(
p
rf
The slope is the Sharpe Ratio.
SR =
µF
F , µF )
rf
F
14
0<↵<1
lending
↵>1
borrowing
2. Markowitz Model
Inclusion of a Risk Free Asset
❖
Efficient Frontier
-
❖
As you can see from the figure on
the right, a new efficient frontier
can be constructed from the riskfree asset and the risky asset
portfolio tangent to the efficient
frontier constructed by risky
assets only (tangent portfolio).
efficient frontier
One-Fund Theorem
-
feasible set
rf
There is a single fund T of risky
assets such that any efficient
portfolio can be constructed as a
combination of the fund T and the
risk-free asset.
15
risk-free asset
T: tangent portfolio
2. Markowitz Model
Markowitz Model with Risk Free Asset
❖
Problem
1 0
min w ⌃w
w 2
subject to w0 µ + (1
❖
1 0
min w ⌃w
w 2
subject to w0 µ = µp
µ:=µ rf
!
µp :=µp rf
w0 1)rf = µp
Solution
1 0
L = w ⌃w
2
@L
= ⌃w
@w
@L
= w0 µ
@
⇤
(w0 µ
µp )
µ=0
)
w=⌃
µp = 0
)
µ0 ⌃
1
1
µ = µp
1
µ
⌃
µ
p
⇤
w = 0 1
µ⌃ µ
µp
= 0 1 ,
µ⌃ µ
16
µ
2. Markowitz Model
Efficient Frontier & Tangent Portfolio
❖
❖
Efficient Frontier
-
As shown below, the standard deviation of the optimal portfolio return is a
linear function of the mean return.
-
Therefore, the efficient frontier becomes a straight line.
p
µp
0
⇤
⇤
w ⌃w = p
p =
µ0 ⌃ 1 µ
Tangent Portfolio
-
Tangent portfolio is a point on the efficient frontier where the sum of the
risky portfolio is 1.
-
This point can be found by dividing the risky weights by their sum.
⌃ 1µ
w⇤
Tangent Portfolio: wT = 0 ⇤ = 0 1
1w
1⌃ µ
17
2. Markowitz Model
Portfolio Choice
A unique portfolio on the efficient frontier needs to be chosen.
❖
Utility Maximisation
-
Need to define a utility function, which is a nontrivial task.
-
Optimal portfolio depends on the choice of the utility function and the
results can be less intuitive. Example:
❖
max w0 µ
w
2
w0 ⌃w,
: risk aversion coefficient
Risk/Return Constraints
-
Some constraints can be imposed to uniquely determine a portfolio.
-
e.g., target return for variance minimisation, risk tolerance (defined as
variance or short fall risk) for return maximisation.
-
Suitable for policy implementation and more practical.
18
2. Markowitz Model
Allocation to the Risk Free Asset
❖
One-Step Approach
-
Include the risk free asset in the optimisation problem. e.g.,
1 0
w = argmin w ⌃w
2
s. t. w0 µ = µp
w1 < 0.1
⇤
optimal portfolio: [w⇤ , 1
w⇤ 0 1]
-
The portfolio weights can be determined in one step.
-
Constraints should be stated in the context of the entire portfolio.
-
Information on the risky portfolio is indirect.
-
Optimisation needs to be carried out for any change of the constraints.
19
2. Markowitz Model
Allocation to the Risk Free Asset
❖
Two-Step Approach (One Fund Theorem)
i. Construct an optimal risky portfolio (tangent portfolio). e.g.,
0
µ
w
⇤
w = argmax p
w0 ⌃w
s.t. w0 1 = 1, w1  0.1
ii. Determine the allocation between the risk free asset and the risky portfolio.
µp
⇤0
aµT = µp , µT = w µ ) a =
µT
optimal portfolio: [aw⇤ , 1
a]
-
Constraints in the first step should be in the context of the risky portfolio.
-
Easier to interpret the results with direct information on the risky portfolio:
Risky assets and the risk free asset are often treated separately.
-
The optimal risky portfolio can be recycled.
20
3. Markowitz & CAPM
Market Portfolio
Assume that
-
all the investors are mean-variance optimisers;
-
they have the same estimate of the distribution of assets;
-
risk free rate is unique for all the investors.
One fund theorem implies that every investor will invest in an optimal risky
portfolio and a risk free asset. If the above assumptions hold, the optimal risky
portfolio will be the same for everyone, and if the market is in equilibrium, the
optimal risky portfolio must be equal to the market portfolio. This implies that
there is no need to solve optimisation problem to find the optimal risky portfolio;
the solution will always result in the market portfolio.
21
3. Markowitz & CAPM
Capital Market Line
❖
Capital Market Line (CML)
-
If the risky portfolio is the market portfolio, the efficient frontier will be
µM
µ=
M
-
❖
This line is also called capital market line. The CML implies that any
efficient portfolio has an expected return proportional to the risk measured
by standard deviation. The slope µM / M is called the price of risk.
CML
µ
M
(
M , µM )
✴
rf
22
Note: In the graph, mu is the
expected return before
subtracting the risk free rate.
3. Markowitz & CAPM
Capital Asset Pricing Model
❖
CAPM
-
❖
Under the same assumptions required for the market portfolio to be the
optimal risky portfolio, CAPM implies that
µi =
i µM ,
ri =
i rM
i
=
iM
2
M
+ ei
where ei is a idiosyncratic risk which satisfies
E(ei ) = 0, V (ei ) =
2
ei ,
23
COV (ei , rM ) = 0.
3. Markowitz & CAPM
Security Market Line
❖
Security Market Line
-
CAPM implies a linear relationship between the expected return and beta.
-
Security market line (SML) is a graphical representation of this relationship.
-
Under CAPM, all assets should fall on the SML.
SML
µ
M
(1, µM )
✴
rf
24
Note: In the graph, mu is the
expected return before
subtracting the risk free rate.
3. Markowitz & CAPM
CAPM and Risk of an Asset
❖
Risk of an Asset
2
i
2
i M
2
ei
=
2
i M
+
2
ei
: systemic risk
: idiosyncratic risk
-
An efficient portfolio will bear only the systemic risk and fall onto the CML.
-
Assets with nonzero idiosyncratic risk will drift to the right from the CML.
µ
asset with
systemic risk only
CML
M
systemic
risk
idiosyncratic
risk
rf
25
asset with
idiosyncratic risk
3. Markowitz & CAPM
Proof of CAPM
Consider a portfolio of asset i and the market portfolio, M.
M
i
rf
26
3. Markowitz & CAPM
Proof of CAPM
Consider a portfolio of asset i and the market portfolio, M . If the weight on i is a,
µp = aµi + (1
2
p
= a2
2
i
a)µM
+ 2a(1
a)
iM
+ (1
a)2
2
M
If we construct a curve by changing a, the curve will
be drawn inside of the feasible region and tangent to
the CML at the market portfolio (a = 0):
dµp
d p
=
a=0
µM
M
27
M
i
rf
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<latexit
3. Markowitz & CAPM
Proof of CAPM
Note
dµp
dµp /da
,
=
d p
d a /da
d p2
=2
da
dµp
= µi
da
1 d p2
a
d p
=
=
da
2 p da
where
p |a=0
=
µM ,
dµp
d p
M
d p
p
da
=
µi =
(µi
a=0
iM
iM
2
M
28
µM =
2
i
µM )
+ (1
M
2
M
=
i µM
2a)
Therefore,
µM
,
M
has been used. Solve for µi to obtain
iM
p
+ (a
1)
2
M
4. Markowitz & Utility
Utility Function
❖
Utility Function1)
-
Individuals will make an investment
decision in a way to maximise their
welfare (utility).
U(w)
MU1 > MU2
MU2
-
Utility function is a function of wealth
that quantifies utility.
-
If individuals are risk averse, utility
function will be concave.
-
Concave utility function also implies
decreasing marginal utility, i.e., the
more wealth you own, the less utility
you will gain from another unit of
wealth.
1) Wealth can also be interpreted as consumption.
29
MU1
w1
E(w)
w2
Utility function
w
4. Markowitz & Utility
Certainty Equivalent
❖
Certainty Equivalent
-
Certainty equivalent C of a random
wealth w is defined to be a certain
wealth that has the same utility
level as the expected utility of w:
U(w)
U[E(w)]
E[U(w)]
U (C) = E[U (w)]
-
If individuals are risk averse,
C < E(w)
w1
E[U (w)] < U [E(w)]
C
E(w)
w2
Utility function
30
w
4. Markowitz & Utility
Indifference Curve
❖
Indifference Curve
-
❖
An investor is indifferent among investment opportunities as long as they
bring the same expected utility. Indifference curve is a collection of random
wealths with the same expected utility drawn on the standard deviationexpected return diagram. Indifference curve has a convex shape if the
utility function is concave.
μ
Quadratic Utility Example
Increasing utility
rp2
U (rp ) = rp
2
: risk aversion coefficient
h
i
E[U (rp )] = E rp
rp2
2
= µp
2
µ2p
2
Indifference
Curves
C3
C2
2
p
C1
C1 < C2 < C3: Certainty equivalent
=C
𝜎
31
4. Markowitz & Utility
Portfolio Choice
❖
Expected Utility Maximisation
-
An optimal portfolio can be obtained so that the expected utility is
maximized.
max E[U (rp )] = max E[U (w0 r)]
w
w
μ
-
-
The optimal portfolio is the point
where the indifference curve is
tangent to the feasible set.
more risk averse
infeasible
Optimal portfolios for different risk
aversion coefficients form the
efficient frontier.
not optimal
𝜎
32
4. Markowitz & Utility
Choice of Utility Function (Quadratic)
❖
Quadratic Utility Function
U (rp ) = rp
-
rp2
2
Marginal utility becomes negative when rp > 1/γ
dU (rp )
MU =
=1
drp
M U < 0 if rp >
-
1
Risk aversion increases with wealth. Absolute risk aversion:
A(rp ) =
-
rp
U 00 (rp )
=
0
U (rp )
1
rp
>0
This is counterintuitive and quadratic utility function is not commonly
used.
33
4. Markowitz & Utility
Choice of Utility Function (Exponential)
❖
Exponential Utility Function
U (rp ) =
-
-
rp )
Exponential utility is a constant absolute risk aversion (CARA) utility:
A(rp ) =
-
exp(
U 00 (rp )
=c
0
U (rp )
If the return is normally distributed, expected utility becomes:
E[U (rp )] = E[ exp(
Z
=
exp(
=
=
=
Z
rp )]
rp ) p
1
p
exp
2⇡ p
⇣
⇣
exp
µp
exp
⇣
⇣
µp
1
2⇡
exp
p
✓
(rp
(rp
µp +
2 p2
⌘⌘ Z
1
2
p
2 p
2⇡
⌘⌘
2
2
p
34
µp )
2
2
p
2 2
p)
exp
p
2
◆
⇣
drp
µp
(rp
2
2
p
⌘
µp +
2 p2
!
drp
2 2
p)
!
drp
4. Markowitz & Utility
Choice of Utility Function (Exponential)
❖
Exponential Utility Function
-
Maximizing the expected utility is equivalent to
max µp
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w
2
2
p
max w0 µ
or
w
2
w0 ⌃w
-
Don’t be confused with the quadratic utility function we saw before.
-
1st order condition:
-
µ
⌃w = 0
Therefore,
⇤
w =
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35
1
⌃
1
µ
4. Markowitz & Utility
Utility Maximisation vs Mean-Variance
For the exponential utility function, utility maximization and mean-variance rule
are equivalent. More generally, mean-variance criterion is equivalent to the
expected utility approach if 1) the utility function is quadratic or 2) returns are
normally distributed. Many other utility maximization problems are also well
approximated by the mean-variance rule.
Quadratic Utility
-
1 2 1 2
E[U (rp )] = µp
µp
2
2 p
For a given expected return, the expected utility will be maximised when
the variance is minimised.
Normal Return
-
Distribution is completely defined by the mean and variance. Therefore,
E[U (rp )] = f (µp ,
p)
Expected utility will increase with the mean and decrease with the variance.
36
5. Practical Framework
More Practical Framework
In practice, there are usually some constraints that need to be satisfied when
optimizing portfolio.
Also, it is not always obvious what utility function and risk aversion coefficient
should be chosen: implementing utility maximization can be tricky.
The next few slides show more practical portfolio optimization problems.
37
5. Practical Framework
Objective Functions
❖
Return Maximisation
max w0 µ
(w),
w
❖
Variance Minimisation
min w0 ⌃w
w
❖
Return/Risk (Sharpe Ratio) Maximisation
w0 µ
(x)
max p
w
w0 ⌃w
❖
38
(w) : Transaction cost
5. Practical Framework
Constraints
❖
Budget Constraint
w0 1 = 1
❖
❖
Individual Asset Level Constraints
wl  w  wu
-
Weight bounds:
-
Weight change bounds: xl  w w0  xu
w0 : current weights.
Sub-portfolio Level Constraints
wkl  w0 pk  wku , k = 1, . . . , K
(
1 if asset i 2 sub-portfolio k
pk (i) =
0 otherwise
39
5. Practical Framework
Constraints
❖
Maximum Variance
w0 ⌃w 
❖
Target Return
w0 µ
❖
2
max
µmin
Shortfall Risk
Prob(rp
)
1
(P0 )
p
rmin )
P0
w0 ⌃w
0
wµ
rmin
Prob
✓
1
✓
rp
µp
p
rmin
(P0 )
µp
p
p
rmin
◆
 µp
µp
p
P0
rmin
(x) : Standard normal cdf
❖
Target Beta, Target Duration,…
-
Many constraints can be framed as linear or quadratic constraints.
40
◆
P0
5. Practical Framework
Transaction Cost
Let x denote the change of weight, x = w
w0 .
• Linear transaction cost
If transaction cost is c for both buy and sell of all assets
(w) = c|x|0 1
If buy and sell transaction costs are di↵erent,
(
c b xi ,
if xi 0 (buy)
(xi ) =
cs xi , if xi < 0 (sell)
cb , cs : buy and sell transaction costs
• Quadratic transaction cost
(w) = x0 Cx
for some matrix C.
41
5. Practical Framework
Transaction Cost
The absolute value function in the linear transaction cost can be eliminated by the following transformation. First, introduce two variables
x+ and x such that
x = x+
x ,
x+ , x
0
Then,
(x) = (x+ , x ) = cb x+ + cs x
This holds because one of x+
i and xi will always be zero. For examples, if xi > 0, both (x+
i , xi ) = (xi , 0) and (xi + dxi , dxi ) for
some dxi > 0 can be a solution. However, since (xi , 0) = cb xi <
(xi + dxi , dxi ) = cb (xi + dxi ) + cs dxi , the optimisation will choose
(xi , 0).
42
5. Practical Framework
Example I: Problem
❖
Objective
-
❖
❖
Maximise Sharpe ratio.
Constraints
-
No short sale is allowed.
-
No more than 30% can be allocated to each asset.
-
2nd and 3rd assets combined together cannot exceed 40%.
-
Probability of the return falling below 0 should be less than 5%.
Transaction Costs
-
Buy/sell: 30 basis points
-
43
5. Practical Framework
Example I: Formulation
w0 µ
max p
w
(w)
w0 ⌃w
subject to
0  wi  0.3,
pw  0.4,
1
p
i = 1, · · · , N
p = [0 1 1 0 · · · 0]
(0.95) w0 ⌃w  w0 µ
(w) = 0.003|w
44
w0 |0 1
5. Practical Framework
Example II: Problem
❖
Objective
-
❖
❖
Construct an optimal portfolio from the DJIA stocks and a risk free asset.
Constraints
-
(Short sale) Short sale is not allowed.
-
(Diversification) Weight on each stocks cannot exceed 10% of the risky
portfolio.
-
(Shortfall) Probability of the portfolio return falling below -0.25% should be
no more than 30%.
Assumptions
-
No transaction cost.
-
Risk free rate is 0.
45
5. Practical Framework
Example II: Strategy
❖
Stage I: Tangent Portfolio
-
❖
Find the tangent portfolio that satisfies the short sale and the diversification
constraints.
Stage II: Allocation to Risk-Free Asset
-
Determine allocation between the tangent portfolio and a risk-free asset so
that the shortfall constraint is satisfied.
46
5. Practical Framework
Example II: Formulation
❖
Tangent Portfolio
w0 µ
wT = argmax p
w0 ⌃w
s. t. 10 w = 1
0  w  0.1
❖
Allocation to Risk-Free Asset
where
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Prob(rp
rmin )
P0
1
rmin
(P0 )WT
rmin
) WT 
1 (P )
µT
0 T
) WT µT
T
WT : Weight on the tangent portfolio
µT = wT0 µ,
rmin =
T
0.0025,
= wT0 ⌃wT
P0 = 0.7
1
Note that µT
(P0 ) T < 0 is assumed above. Otherwise, there
are infinitely many solutions of WT that satisfies the shortfall risk.
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47
5. Practical Framework
Example: Input Estimation
❖
Sample Period: 1994.01 - 2013.12
❖
Moving Average
T
X
1
µi =
rit
T t=1
2
i
ij
=
=
1
T
1
1
T
1
T
X
(rit
µi ) 2
(rit
µi )(rjt
t=1
T
X
t=1
48
µj )
5. Practical Framework
Example II: Results
❖
Efficient Frontier
-
Subject to short sale constraint (without diversification constraints)
49
5. Practical Framework
Example II: Results
❖
Tangent Portfolio
-
Subject to short sale and diversification constraints
Tangent Portfolio
50
5. Practical Framework
Example II: Results
❖
Optimal Portfolio
Tangent Portfolio
Optimal Portfolio
51
5. Practical Framework
Example II: Results
❖
Tangent Portfolio Weights
0.10
0.08
0.05
0.02
0.00
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
52
5. Practical Framework
Example II: Results
❖
Tangent Portfolio vs. Optimal Portfolio
Optimal
Tangent
Optimal
WT
100.00
22.66
𝜇
15.17
3.44
𝜎
15.64
3.54
P(r>-0.25%)
63.13
70.00
Tangent
53
5. Practical Framework
Example II: Results
❖
Portfolio Return Distribution
-
5%-95% interval for different tangent portfolio weights.
WT=0.2
WT=0.4
WT=0.6
WT=0.8
WT=1.0
54
5. Practical Framework
Example: Results
❖
Out-of-Sample (2014.01-2014.12) Performance
S&P500
Tangent
Optimal
Cumulative
Return
14.31
13.75
3.03
Mean Return
1.15
1.11
0.25
Stdev
2.47
2.44
0.55
Sharpe Ratio
0.46
0.45
0.45
55
6. Reading List
Further Readings
Luenberger. Appendix B. Calculus and Optimisation.
Luenberger. Ch. 6 and 7.
Markowitz, H., 1952. Portfolio selection. The journal of finance, 7(1), pp.77-91.
Kroll, Y., Levy, H. and Markowitz, H.M., 1984. Mean-variance versus direct utility maximization. The
Journal of Finance, 39(1), pp.47-61.
Optional
M.S. Lobo, Robust and convex optimization with applications in finance. Ph.d thesis. Stanford University,
2000.
* The chapters of Luenbeger are based on the 1st edition and it might be different from the 2nd edition.
56