Leeds University
Business School
LUBS1035
Foundations of
Finance
Sina Erdal
Lecture 2: Introduction to Valuation: The Time Value of Money
Time Value of Money:
Intuition
Assume that today is October 4th,
2017. You can have £10,000
now…
Which would you choose?
What is worth more?
£10,000 now or £10,000 in three
years time?
Or, you can have £10,000
when you graduate in 2020.
Why?
Too easy?
Assume that today is October 4th ,
2017. You can have £10,000
now…
Which would you choose?
What is worth more?
£10,000 now or £15,000 in three
years time?
Or, you can have £15,000
when you graduate in 2020.
Why?
Overview of Lecture
Future Value and Compounding
Present Value and Discounting
Tying it all Together
The first step…
Future Value and
Compounding
Future Value and Compounding
Future value (FV)
The amount an
investment is worth
after one or more
periods.
Investing for a Single Period
Year 0
• You invest £100 at 10%
Year 1
• Your money grows
to £100 + 10% of
£100 = £110
Investing for a Single Period
In general:
V1  V0 (1  r )
Where V1 is the value at time t = 1;
r is the interest rate
Some Terminology
Compounding
• The process of accumulating interest
on an investment over time to earn
more interest.
Interest on Interest
• Interest earned on the reinvestment
of previous interest payments.
Some Terminology
Compound Interest
• Interest earned on both the initial
principal and the interest reinvested
from prior periods.
Simple Interest
• Interest earned only on the original
principal amount invested.
Example (see example 4.1)
Interest on Interest
Suppose you locate a two-year investment opportunity that
pays 10 per cent interest per year. You decide to invest
£100.
Required:
Calculate what the total investment will be worth at the end of
the two years.
Calculate how much simple interest you have received.
Calculate how much interest on interest you have received
Calculate how much compound interest you have received.
Investing for More than One Period
Year 0
• Invest
£100
at 10%
Year 1
• £100(1.1)
= £110
Year
£100 invested plus £10
simple interest
• £110(1.1)
2 = £121 £100 invested plus
£20 (2 x £10) simple
interest plus £1
interest on interest
Interest on Interest
Solution:
Calculate what the total investment will be worth at the end of the
two years.
• The total investment is worth £121 (£100 x 1.1 x 1.1)
Calculate how much simple interest you have received.
• The simple interest is £20 (£100 x 10% x 2)
Calculate how much interest on interest you have received.
• The interest on interest is £1 (£100 x 10% x 10%)
Calculate how much compound interest you have received.
• The compound interest is £21 (£121 - £100)
Investing for More than One Period
Year 0
Invest £100
@10%
Year 1
£100 x 1.1 =
£110
Year 2
£110 x 1.1 =
£121
Investing for More than One Period
Year 0
Invest £100
@10%
Invest £100
@10%
Year 1
£100 x 1.1 =
£110
£100 x 1.1 =
£110
Year 2
£110 x 1.1 =
£121
£100 x 1.1 x 1.1
= £121
Investing for More than One Period
Year 0
Invest £100
@10%
Invest £100
@10%
Invest £100
@10%
Year 1
£100 x 1.1 =
£110
£100 x 1.1 =
£110
£100 x 1.1 =
£110
Year 2
£110 x 1.1 =
£121
£100 x 1.1 x 1.1
= £121
£100 x 1.12 =
£121
Investing for More than One Period
Year 0
Invest £100
@10%
Invest £100
@10%
Invest £100
@10%
Invest V0 @ r%
Year 1
Year 2
£100 x 1.1 =
£110 x 1.1 =
£110
£121
£100 x 1.1 =
£100 x 1.1 x 1.1
£110
= £121
£100 x 1.1 =
£100 x 1.12 =
£110
£121
V0 x (1 + r) = V1 V0 x (1 + r)2 = V2
FV = Vt = V0 (1+r)t
Future Value: A Generalisation
Vt = V 0
t
(1+r)
Where Vt is the value at time t;
r is the interest rate
Example:
Suh-Pyng Ku has put €500 in a savings account
at Barclays. The account earns 7 percent,
compounded annually.
Required:
Calculate how much Ms. Ku will have at the end
of three years.
Calculate how much is simple interest.
Calculate how much is compound interest.
Example:
Suh-Pyng Ku has put €500 in a savings account at Barclays. The
account earns 7 percent, compounded annually.
How much will Ms. Ku have at the end of three years?
How much is simple interest? How much is compound interest?
The answer is:
Total Amount:
Simple interest:
Compound interest:
Example:
Suh-Pyng Ku has put €500 in a savings account at Barclays. The
account earns 7 percent, compounded annually.
How much will Ms. Ku have at the end of three years?
How much is simple interest? How much is compound interest?
The answer is:
Total Amount:
€500  1.07  1.07  1.07 = €500 (1.07)3 = €612.52
Simple interest:
€500  (.07 x 3) = € 105
Compound interest:
€612.52 - €500 = € 112.52
Figure 4.1 Future Value, Simple Interest and
Compound Interest
•
Simple interest is
constant - £10 each
year
•
Compound interest
increases each year
because interest is
compounded on itself
•
Compound interest
can be a powerful
investment tool
The Power of Compounding
Simple Interest
r = 8 per cent
Compound Interest
r = 8 per cent
Invest £1 for 200 Years
Invest £1 for 200 Years
Interest = £0.08
Value of investment at end of
Value of investment at end of 200 years
200 years:
FV = £1(1.08)200
£1 + (200 x £0.08)
= £4,838,949.59
= £1 + £16
= £17
A Big Difference!
Figure 4.2 Future Value of £1 for Different
Periods and Rates
• Future Values are
critically dependent
on the interest rate
• The higher r, the
higher the FV
• After 10 years at 10
per cent, £1 is worth
£2.59
Over a long period, the effects can
be dramatic…
• After 10 years at 20
per cent, £1 is worth
£6.19
The Power of Interest
Compound Interest
r = 8 per cent
Compound Interest
r = 12 per cent
Invest £1 for 200 Years
Invest £1 for 200 Years
Value of investment at end of Value of investment at end of
200 years
200 years
FV = £1(1.08)200
FV = £1(1.12)200
= £4,838,950
= £6,975,968,872
A BIG Difference!
Where is this?
Manhattan Island
In 1626, Peter Minuit
allegedly bought Manhattan
Island for 60 guilders’ worth
of trinkets from Native
Americans.
It is reported that 60 guilders
was worth about $24 at the
prevailing exchange rate.
Was this a good deal for the
Native Americans?
Manhattan Island
If the Native Americans had sold the trinkets
at a fair market value and invested the $24
at 5 percent (tax free), it would now, about
380 years later, be worth more than $2.7
billion.
Today, Manhattan is undoubtedly worth
more than $2.7 billion, so at a 5 percent
rate of return the Native Americans got the
worst of the deal.
What if the interest rate was10 percent?
Manhattan Island
C = $24
t = 380 years
r = 10%
FV = $24(1  r)t = 24  1.1380  $129 quadrillion
$129 quadrillion is more than all the real estate
in the world is worth today.
Example 4.2
Compound Interest
You’ve located an investment that pays 12 per
cent per year. That rate sounds good to you, so
you invest €400.
How much will you have in three years?
How much will you have in seven years?
At the end of seven years, how much interest will
you have earned?
How much of that interest results from
compounding?
Example 4.2
Compound Interest
The future value of €400 at 12 per cent after three years is
calculated as follows:
€400  1.123 = €561.97
After seven years, you will have:
€400  1.127 = €884.27
Thus, you will more than double your money over seven
years.
Total interest earned is €884.27 - €400 = €484.27
Simple interest is €400 x 0.12 x 7 = €48 x7 = €336
Interest from compounding = €484.27 - €336 = €148.27
Test your knowledge
Calculating Future Values
Assume you deposit 10,000 Swedish
Kroner today in an account that pays
5% interest.
How much will you have in five years?
Your solution
Future Value:
Your solution
Future Value:
10,000 x (1.05) 5
= 10,000 x 1.2763
= SKr12,763
The next step…
Present Value and
Discounting
Remember Future Value?
Future value (FV)
The amount an
investment is worth
after one or more
periods.
What about Present Value?
Present value (PV)
The current value of
future cash flows
discounted at the
appropriate discount rate.
Present Value: The Single Period Case
Year 1
• Receive £1,
interest rate
is 10%
Year 0
• What is the
Year 0
equivalent value
of £1 received
in Year 1?
Present Value: The Single Period Case
Year 1
• Receive £1,
interest rate
is 10%
Year 0
• V0 = £1/1.1
= £0.909
Some More Terminology
Present Value
• The current value of future cash
flows discounted at the appropriate
discount rate.
Discount
• Calculate the present value of some
future amounts.
See Example 4.5
Single Period PV
Suppose you need £100 to buy
final year textbooks in two
years time.
You can earn 10 per cent on
your money.
How much do you have to
invest today?
Present Values for Multiple Periods
Year 2
• Receive
£100, r =
10%
Year 1
• £100/1.1 =
£90.91
Year 0
• £90.91/1.1 =
£82.65
Investing for More than One Period
Year 2
Receive £100
Receive £100
Receive £100
Receive V2
Year 1
£100/1.1 =
£90.91
£100/1.1 =
£90.91
£100/1.1 =
£90.91
V0 = V1/(1 + r)
Year 0
£90.91/1.1 =
£82.65
£100/1.21 =
£82.65
£100/1.12 =
£82.65
V0 = V2/(1 + r)2
PV = V0 = Vt / (1+r)t
Present Value: A Generalisation
Vt
PV  V0 
t
(1  r )
Where Vt is the value at time t;
r is the interest rate
Table 4.3
Present Value Interest Factors
This figure
is 1/(1.1) =
0.9091
This figure
is 1/(1.1)3
= 0.7513
This figure
is 1/(1.1)2
= 0.8264
Test your knowledge
Calculating Present Value
Assume you win £10,000 on the
National Lottery but it is payable in
5 years.
Assuming an interest rate of 5%,
what is your win worth today?
Your solution
Present Value:
10,000 /(1.05)
5
= 10,000 x 1/(1.05) 5
= 10,000 x 0.7835
= 7,835
We can check
this figure using
the present
value tables
Table 4.3
Present Value Interest Factors
Table 4.3
Present Value Interest Factors
This figure
is 1/(1.05)5
= 0.7835
Figure 4.3 Present Value of £1 for Different
Periods and Rates
• As the length of time
increases, PV falls
• As the interest rate
increases, PV
declines
• These effects are
magnified over time
Overview of Lecture
Tying it all Together
Tying it all Together
PV  r )  FVt
t
PV = FVt /  r )
t
 FVt  r ) ]
t
 FVt (1  r )
t
Tying it all Together
Compounding
FV = PV x (1 + r)t
Discounting
PV = FV / (1 + r)t
Example 4.8
Evaluating Investments
Your company proposes to buy an asset for
£335.
This investment is very safe. You would sell
off the asset in three years for £400.
You know you could invest the £335
elsewhere at 10 per cent with very little
risk.
What do you think of the proposed
investment?
The wrong answer!
• £400 is £65 more than £335, therefore it is a good
investment.
• That is correct from a financial accounting perspective;
£65 represents an accounting ‘profit’
• However, it is not a correct corporate finance answer
• Why?
• Because, it fails to take account of the time value of
money
• We need to convert to either present values, or future
values to make a meaningful comparison
Example 4.8
Evaluating Investments – Present Values
This is not a good investment.
Why not?
Because £400 in three years time has a
present value of:
£400/(1.1)3 = £300.53
£300.53 is less than the cash you have now.
Because the proposed investment is only
worth £300.53 now, it is not as good as other
alternatives we have
Example 4.8
Evaluating Investments – Future Values
This is not a good investment. Why not?
Because you can invest the £335 elsewhere
at 10 per cent. If you do, after three years it
will grow to a future value of:
£335  (1  r )t = £335  1.13
= £335  1.331
= £445.89
Because the proposed investment pays out
only £400, it is not as good as other
alternatives we have
If you can invest at 5 per cent, can you now
answer this question?
Assume that today is October 4,
2017. You can have £10,000
now…
Which would you choose?
What is worth more? £10,000 in
2017 or £15,000 in 2020?
£15,000/(1.05)3 = £12,957.
Wait for your money.
Or, you can have £15,000
when you graduate in 2020.
If you can invest at 15 per cent, can you now
answer this question?
Assume that today is October 4,
2017. You can have £10,000
now…
Which would you choose?
What is worth more? £10,000 in
2017 or £15,000 in 2020?
£15,000/(1.15)3 = £9,863.
Take your money now.
Or, you can have £15,000
when you graduate in 2020.
Activities for this Lecture
Reading
• Chapter 2 The Time Value of Money
Assignment
• Class 2 Preparation Questions