Time Value of Money
Maryam Mirzaei
Assistant Professor
German University of Technology
Spring 2021
Maryam Mirzaei Assistant Professor (German University of Technology) Time Value of Money Spring 2021 1 / 37
Outline
1 Time Value of Money
Cash flows and Assets
Valuing an Asset
Present Value / Future Value
The Annuity
The Perpetuity
2 Money Market
3 Valuing a Bond
Zero-Coupon Bonds
Coupon Bonds
4 Self-study Questions
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Time Value of Money
Money today is worth more than the same amount of money
received in the future.
i What is the future value of an amount invested or borrowed today?
ii What is the present value of an amount to be paid or received at a
certain time in future?
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Cash flows and Assets
What Is An Asset?
. .
From a business perspective, an asset is a sequence of cash flows.
Assett = CFt + CFt+1 + CFt+2 + :::
t = 0 1
CF1
2
CF2
5
CF3
6
CF4
T
CFT
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Valuing an Asset
To value an asset, we value a sequence of given cash flows:
. .
Value of Assett = Vt(CFt; CFt+1; CFt+2; :::)
t = 0 1
CFt
2
CFt+1
5
CFt+2
6
CFt+3
T
CFT
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Valuing an Asset
| i | Simple interest rate |
| ii | Compounding |
AssignmentTutorOnline
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Simple Interest Rate
Suppose that an amount is paid into a bank account, where it is to earn
“interest”.
The “future value” of this investment consists of the initial
deposit, called the principal “P”, plus all the interest earned since
the money was deposited in the account.
After one year:
Interest = rP,
where: r > 0 is the interest rate.
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Simple Interest Rate
The value of investment:
. .
After 1 year: V (1) = P + rp = (1 + r)P
After 2 years: V (2) = (1 + 2r)P
. .
The value of investment at time t, V(t), is:
V (t) = (1 + rt)P
(1+rt): growth factor
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Simple Interest Rate
Exercise 1:
A sum of $10,000 paid in to a bank account for 61 days to attract simple
interest rate will produce 10,020 at the end of the term. What is the
interest rate (r)?
Maryam Mirzaei Assistant Professor (German University of Technology) Time Value of Money Spring 2021 9 / 37
Simple Interest Rate
Exercise 1:
A sum of $10,000 paid in to a bank account for 61 days to attract simple
interest rate will produce 10,020 at the end of the term. What is the
interest rate (r)?
V (t) = (1 + rt)P
P = 10; 000,
t = 61days and
V (t) = 10; 020
10; 020 = (1 + r × (61=365))10; 000
1 + r × (61=365) = 1:002
r = 0:0119688
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Simple Interest Rate
Exercise 2:
How much should you deposit in an account with a simple interest rate of
8% if $1,200 is needed after 4 months (121 days)?
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Simple Interest Rate
Exercise 2:
How much should you deposit in an account with a simple interest rate of
8% if $1,200 is needed after 4 months (121 days)?
V (t) = (1 + rt)P
P =?,
t = 121days
V (t) = $1; 200, and
r = 8%
1; 200 = (1 + 0:08 × (121=365))P
1; 200 = 1:02652P
P = $1; 168
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Compounding Interest Rate
Suppose that an amount is paid into a bank account, where it is to earn
“interest”.
This time the interest rate will be added to the principal.
This is compounding interest rate. In the case of monthly compounding:
The first interest payment of (r=12)P will be added to the principal by
(1 + r=12)P. After two months, the interest payment will be:
r=12(1 + r=12)P
And, after one year:
(1 + r=12)12P
If the interest rate remains the same, after t years, the future value of the
principal value (P) will be:
V (t) = (1 + r=m)tmP
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Compound Interest Rate
Exercise 3:
Calculate the future value of a deposit of $1000 if the interest rate is 10%
compounded annually and semi-annually after 2 years.
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Compound Interest Rate
Exercise 3:
Calculate the future value of a deposit of $1000 if the interest rate is 10%
compounded annually and semi-annually after 2 years.
V (t) = (1 + r=m)tmP
P = 1000,
t = 2year, and
r = 10%
Annually: 1; 000(1 + 0:1)2 = 1; 210
Semi-annually: 1; 000(1 + 0:1=2)4 = 1; 215
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Compound Interest Rate
Exercise 4:
Which of the following investments will generate a higher future value
after one year:
A deposit of $1,000 attracting interest at 15% compounded daily, or at
15.5% compounded semi-annually.
P = 1000,
t = 1year, and
r = 15%
Daily: 1; 000(1 + 0:15=365)1×365 = 1; 161:8
Semi-annually: 1; 000(1 + 0:155=2)1×2 = 1; 161:01
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Compound Interest Rate
Exercise 5:
What initial investment subject to annual compounding at 12% is needed
to produce $1,000 after 2 years?
V (t) = 1000,
t = 2year, and
r = 12%
1; 000 = (1 + 0:12)2P
P = 797:19
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Present Value / Future Value
What determines the value of $1 over t years?
Opportunity cost of capital (r)
$1 today should be worth more than $1 in the future.
$1 in year 0= $1 × (1 + r)in year 1
$1 in year 0= $1 × (1 + r)2in year 2 .
. .
$1 in year 0= $1 × (1 + r)tin year T.
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Present Value / Future Value
What determines the today (present) value of $1 in year T?
Opportunity cost of capital (r)
$1 in year T should be worth less than $1 today.
$1=(1 + r) in year 0 = $1 in year 1
$1=(1 + r)2 in year 0 = $1 in year 2
. . .
$1=(1 + r)t in year 0 = $1 in year T.
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The Effect of Discounting
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The Annuity
An annuity is a sequence of finitely many payments of a constant amount
due at equal time intervals. Annuity pays fixed cash flow CF for specific
period of time (T).
In the case of Annuity:
PV = CF
1+r + (1+ CFr)2 + ::: + (1+CFr)T
(1 + r)PV = CF + 1+ CFr + :::(1+CF r)T-1
| rPV = CF – | CF (1+r)T |
PV = CF
r
–
CF
r
1
(1+r)T
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The Annuity
· · ·
(periods)
0
$1
1
$1
2
$1
3
$1
n – 1
$1
n
$1
Ani Sni
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The Annuity – Exercise
You won a lottery that pays $10,000 a year for 20 years. How much is the
present value, if r=10%.
Answer:
PV = 10; 000 × 01:1(1 – 1:1120 )
10; 000 × 8:514 = 85; 135
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The Perpetuity
Perpetuity pays fixed cash flow (CF) forever.
In the case of Perpetuity:
PV = CF
1+r + (1+ CFr)2 + (1+ CFr)3 + :::
(1 + r)PV = CF + 1+ CFr + (1+ CFr)2 + :::
rPV = CF
PV = CF
r
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Exercise:
Calculate the present value of $500 receivable annually for 13 years and
$2,200 receivable after 15 years, assuming an interest rate of 11 percent.
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Money Market
The money market consists of risk-free (more precisely, default-free)
securities.
An example of risk-free securities:
Bond: A financial security promising the holder a sequence of guaranteed
future payments.
Risk-free means: these payments will be delivered with certainty.
(Nevertheless, even in this case risk cannot be completely avoided, since
the market prices of such securities may fluctuate unpredictably.)
Different types of bonds:
i Treasury bills and notes,
ii Mortgage and debenture bonds,
iii Commercial papers.
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Zero-Coupon Bonds
Bond: A debt obligation with a maturity of more than a year.
Principal: The Principal or face value is the amount of the debt excluding
interest.
Yield to maturity: The estimated annual rate of return for a bond
assuming that the investor holds the asset until its maturity date.
Coupon rate: The nominal annual rate of interest expressed as a
percentage of the principal value, which equals to the earning an investor
can expect to receive from holding a particular bond.
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Zero-Coupon Bonds
The simplest case of a bond is a zero-coupon bond, which involves just a
single payment.
In case of zero-coupon bonds, the issuing institution (e.g., a government, a
bank or a company) promises to exchange the bond for a certain amount
of money F (face value), on a given day T (maturity date).
Present value of such a bond:
V (0) = F(1 + r)-1
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Zero-Coupon Bonds
Example:
Suppose that a bond with face value F = 100 dollars is maturing in one
year, and the annual compounding rate r is 12%. Then the present value
of the bond should be:
V (0) = F(1 + r)-1 = $89:29
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Zero-Coupon Bonds
In reality, the opposite happens: Bonds are freely traded and their prices
are determined by market forces, whereas the interest rate is implied by
the bond prices,
| r = | F V (0) |
– 1
This formula gives the implied annual compounding rate.
Example:
If a one-year bond with face value of $100 is being traded at $91, then the
implied rate is 9.89%.
r = 100
91 – 1
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Coupon Bonds
Bonds promising a sequence of payments are called coupon bonds.
These payments include:
•The face value due at maturity, and
•coupons paid regularly (annually, semi-annually, or quarterly).
We assume that interest rates are constant to calculate the price of a
coupon bond by discounting all the future payments.
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Coupon Bonds
EXAMPLE:
Assume a bond with face value F = $100 maturing in five years (T = 5),
with coupons of C = $10 (paid annually), the last one at maturity. This
means a sequence of payments of 10, 10, 10, 10, 110 dollars at the end of
each consecutive year. Given the continuous compounding rate r= 12%,
the price of the bond:
V (0) = 10e-r + 10e-2r + 10e-3r + 10e-4r + 110e-5r
V (0) = 90:27 If r = 0:12
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Dividend Valuation Model
The value of a share now (P0):
The sum of the stream of future discounted dividends plus the value of the
share as and when sold in some future years, n.
P0 = D1
(1+ke) + (1+Dk2e)2 + (1+Dk3e)3 + ::: +
| (1+Dkne)n + | (1+Pkne)n |
If the lifespan is infinite, and the annual dividend is constant:
D1 = D2 = D3
Then we have:
P0 = P (1+Dkte)t = Dke1
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Dividend Growth: DGM
Dividend Growth Model (DGM): In this case, the value of a share is the
sum of all discounted dividends, growing at the annual rate, g:
P0 = D0(1+g)
(1+ke + D(1+ 0(1+keg)2)2 + D(1+ 0(1+keg)3)3 + ::: + D(1+ 0(1+keg)n)n
If D0 is the dividend recently paid for this year, D0(1 + g) is the dividend
to be paid in one year’s time (D1), and so on.
The present value for such a series growing to infinity:
P0 = D0(1+g)
(ke-g) =
D1
(ke-g)
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Dividend Growth (DGM): Example
XYZ plc currently earns $16 per share. It retains 75 per cent of its profits
to reinvest at an average return of 18 per cent. Its shareholders require a
return of 15 per cent. What is the ex-dividend value of XYZ’s shares?
What happens to this value if investors suddenly become more risk-averse
by seeking a return of 20 per cent?
Refer to: Pike & Neale, Corporate Finance & Investment.
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Self-study Questions
i What is the difference between simple interest and compound
interest?
ii You won a lottery that pays $100,000 a year for 20 years. How much
is the present value, if r=10%.
iii What if the payments last for 50 years?
iv What about forever?
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References
| i | Pike & Neale, Corporate Finance and Investment Decisions and Strategies 9th edition. Brealey, Myers, & Allen, Principles of Corporate Finance. |
| ii |
USEFUL WEBSITES:
D Financial Times: www.FT.com
D Guardian: www.guardian.co.uk/money
D The Economist: www.economist.com
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