Chapter Four
Present Value
The Present Value
of One Future Payment
• Would you rather have $100 today or $105 in
one year?
• What does your answer depend on?
• What happens to your choice as the interest
rate rises? As the interest rate falls?
• Present value is the amount of money you
would need to invest today to yield a given
future return.
Copyright © Houghton Mifflin Company. All rights reserved.
4|2
Present Value
(cont’d)
Present value is based in two ideas
1. You can determine how much money you have
available at different times
2. Interest is earned on past interest
(compounding)
Copyright © Houghton Mifflin Company. All rights reserved.
4|3
The Power of Compounding
• Compare $300 today vs. $350 in two years
• What does it depend on?
– Today, your principal (P) represents your entire
investment
– What rate of interest will you earn?
• In one year: (1 + i) × P
– Where i is annual interest rate
• In two years: (1 + i)2 × P
– Compounding is earning interest on interest that
was earned in prior years
Copyright © Houghton Mifflin Company. All rights reserved.
4|4
Compounding
(cont’d)
Extend as many years as you would like
Value after N years = (1 + i)N × P
Compounding enables your money to keep working for
you, even without additions to the principal
Copyright © Houghton Mifflin Company. All rights reserved.
4|5
Compounding
(cont’d)
Compounding makes a
huge difference
P = $1,000, i = 8%
1 year: $1,080
5 years: $1,469
10 years: $2,159
25 years: $46,902
100 years: $2,199,761
Copyright © Houghton Mifflin Company. All rights reserved.
4|6
Compounding
(cont’d)
How much does the interest rate matter?
$1,000 × 1.08100 = $2,199,761
$1,000 × 1.07100 = $867,716
1% makes a BIG difference!!!
Copyright © Houghton Mifflin Company. All rights reserved.
4|7
Discounting
• In discounting, we consider an amount to be
received in the future and ask how much it is
worth today.
• Example: Would you rather have $350 in one
year, or $300 today?
• What does it depend on? It is the same
question as “compounding”… only in reverse!
• How much would you be willing to give up today
to receive $350 in the future?
Copyright © Houghton Mifflin Company. All rights reserved.
4|8
Discounting
(cont’d)
F = (desired) future value
What P today is worth F in one year?
P = F/(1 + i) because (1 + i) × P = F
The term (1 + i) is the discount factor.
The term i is the rate of discount.
Copyright © Houghton Mifflin Company. All rights reserved.
4|9
Why Discount?
Key question
What is your rate of discount?
What would you do with $10,000 today?
What is the expected return?
Copyright © Houghton Mifflin Company. All rights reserved.
4 | 10
Discounting (cont’d)
• For a given F, what happens to P as i
rises?
• Is $350 in one year worth more or less
today as i rises? As i falls?
Key results
• Present value is inversely related to the
rate of discount
• Present value is inversely related to the
discount factor
Copyright © Houghton Mifflin Company. All rights reserved.
4 | 11
The General Form
of the Present-Value Formula
P
F1
(1  i)
1

F2
(1  i)
2
 ... 
FN
(1  i )
N
.
• Used to calculate the present value of almost any
financial security
• Higher future amounts will yield a higher present value
• Higher rates of discount or discount factors will yield a
lower present value
Copyright © Houghton Mifflin Company. All rights reserved.
4 | 12
Different Types of Securities
• A way of describing the payments promised by a
financial security
Example: N = 1 year
F = $10,000
Time
(years)
0
Payment
Copyright © Houghton Mifflin Company. All rights reserved.
N
F
4 | 13
Present Value of a Perpetuity
Perpetuity = financial security that never matures. It
pays interest forever, does not repay principal (eg. a
share of stock in a corporation)
Example: N = 1 year
F = $10,000
Perpetuity
Time
(years)
Payment
0
1
2
3
4
.
.
.
F
F
F
F
.
.
.
Copyright © Houghton Mifflin Company. All rights reserved.
4 | 14
Perpetuities (cont’d)
Would you ever want to own a perpetuity?
Is the present value really infinite?
How much would you pay for it?
To calculate: P = F / i
The present value of each successive payment
is less than the last…Therefore, even the
present value of a perpetuity is finite.
Copyright © Houghton Mifflin Company. All rights reserved.
4 | 15
Perpetuities (cont’d)
Present value of perpetuities
are also affected by the
rate of discount.
Compare the sensitivity of P to i
where F = $1,000
i = .08
P = $12,500
i = .05
P = $20,000
i = .02
P = $50,000
Copyright © Houghton Mifflin Company. All rights reserved.
4 | 16
Fixed Payment Securities
• Fixed-payment security:
The dollar payments are
the same every year so
that the principal is
amortized
• Amortization: The
process of repaying a
loan’s principal gradually
over time
Copyright © Houghton Mifflin Company. All rights reserved.
4 | 17
Fixed Payment Securities (cont’d)
Payment Timeline
Time
(years)
Payment
0
1
2
3
4
.
.
.
N
F
F
F
F
.
.
.
F
To calculate present value:
1 N
1 (
)
1 i
P=F×
i
Copyright © Houghton Mifflin Company. All rights reserved.
4 | 18
Present Value of a Coupon Bond
Coupon Bond: Pays a regular interest
payment until maturity, when face value is
repaid (e.g. most corporate & government
bonds)
Time
(years)
0
Payments
Interest
Face Value
1
2
3
4
.
.
.
N
F
F
F
F
.
.
.
F
V
Copyright © Houghton Mifflin Company. All rights reserved.
4 | 19
Present Value of a Coupon Bond (cont’d)
To calculate present value:
1 N
1 (
)
V
1

i
P = [F ×
]+
N
i
(1  i)
interest payments are
fixed-payment security
Copyright © Houghton Mifflin Company. All rights reserved.
present value of
face value
4 | 20
Payments More Than
Once Per Year
• Many securities require payments more
frequently
• Semi-annually: Government & corporate bonds
• Quarterly: Many stock dividends
• Monthly: Consumer & business loans
• Because of compounding, this frequency must
be accounted for in calculating present value
Copyright © Houghton Mifflin Company. All rights reserved.
4 | 21
Payments More Than
Once Per Year (cont’d)
• Time period needs to be adjusted to account
for payment frequency
• Assume that interest compounds each
period and N = number of periods to maturity
Example: 30 year mortgage at 9%
 N = 360 (12 months x 30 years)
 i = 0.0075 (0.09/12 months)
Copyright © Houghton Mifflin Company. All rights reserved.
4 | 22
Present Value & Decision Making
Comparing alternative offers
• A magazine subscription costs $50 for 1
year or $95 for 2 years. Which is better?
• Comparing coupon bonds: use one as
an alternative for the other; use the
interest rate on one bond as the rate of
discount on other bonds in the
secondary market
Copyright © Houghton Mifflin Company. All rights reserved.
4 | 23
Buying or Leasing a Car
Question: Given a choice, would you buy or
lease a car?
Answer: Financially there is not much
difference between buying and
leasing. Money paid today is worth
more than money paid in the future.
Copyright © Houghton Mifflin Company. All rights reserved.
4 | 24
Buying or Leasing a Car
(cont’d)
Other factors matter in deciding to lease
Pros
– option value (check car out & see how used car
prices change)
– avoid transactions costs of selling
Cons
– added transactions if you keep car;
– limited mileage;
– car dealers love them because of higher turnover, so
it must be bad for you!
Copyright © Houghton Mifflin Company. All rights reserved.
4 | 25
Interest-Rate Risk
• Why does the price of a security change
when the market interest rate changes?
• This uncertainty is interest-rate risk
• Reflects a change in opportunities…
suppose you buy a bond paying 6% but
market rates rise to 8%. Does your bond
price rise or fall?
Copyright © Houghton Mifflin Company. All rights reserved.
4 | 26
Interest-Rate Risk (cont’d)
• Bond price = present
value of bond
• Present value of bond
inversely related to i
• Bond price inversely
related to i
Copyright © Houghton Mifflin Company. All rights reserved.
4 | 27
Interest-Rate Risk (cont’d)
Interest Rate Change Example
• A $1000 bond matures in one year and
pays $50 interest. Other 1-year bonds also
have interest rates of 5%.
P = $1050/1.05 = $1000
• What if market interest rate fell (just after
you bought it) to 4%?
P = $1050/1.04 = $1009.62
Copyright © Houghton Mifflin Company. All rights reserved.
4 | 28
Interest-Rate Risk (cont’d)
Example (continued)
• What if the market interest rate rose (just
after you bought it) to 10%?
P = $1050/1.10 = $954.55
Why did P change?
Market opportunity!
Copyright © Houghton Mifflin Company. All rights reserved.
4 | 29
Using Present Value Formula
to Calculate Payments
• Sometimes we already know present
value, but are concerned with size of
payments to be made to the lender…
Example: Mortgage loan
• P = $100,000
• monthly payments for 30 years (N = 360)
• annual interest rate = 6% (therefore i =
.06/12 = 0.005)
Copyright © Houghton Mifflin Company. All rights reserved.
4 | 30
Using Present Value Formula
to Calculate Payments
1 N
1 (
)
Example (cont.):
1 i
P=F×
i
i
so
F=P×
1 N
1 (
)
1 i
0.005
F = $100,000 ×
= $599.55
1 360
1 (
)
1.005
Copyright © Houghton Mifflin Company. All rights reserved.
4 | 31
Looking Forward or
Backward at Returns
Why would investors look forward or
backward at security returns?
– To calculate an expected return based on a
forecast they have received
– To evaluate different loan offers
– To determine the interest rate one will
receive on an annuity in retirement
– To compare a stock’s return to the average
(historical) market return
Copyright © Houghton Mifflin Company. All rights reserved.
4 | 32
Looking Forward or
Backward at Returns (cont’d)
To calculate, still solve for i in the formula
• Thus far, i = rate of discount
• Backward looking: i = past return
• Forward looking
(1) i = expected return (accounts for
probability of default)
(2) i = yield to maturity (average annual
return if held to maturity without default)
Copyright © Houghton Mifflin Company. All rights reserved.
4 | 33
Looking Forward or
Backward at Returns (cont’d)
Payments made by security on right-hand side of
equation, price on left-hand side, solve for i
One payment in one year:
(1 + i) × P = F, so i = (F/P) – 1
Copyright © Houghton Mifflin Company. All rights reserved.
4 | 34
Looking Forward or
Backward at Returns (cont’d)
One payment in more than one year (N years) in
the future:
(1 + i)N × P = F, so i = (F/P)1/N – 1
Copyright © Houghton Mifflin Company. All rights reserved.
4 | 35
Looking Forward or
Backward at Returns (cont’d)
• With a fixed-payment security, you know the payment
amount & price of security, and are looking for the
implied interest rate
1 N
1 (
)
1 i
P=F×
i
• Cannot determine i as a function of just P and F, so
we must “guess, test & revise”
• Same holds true for coupon bonds
Copyright © Houghton Mifflin Company. All rights reserved.
4 | 36
Looking Forward or
Backward at Returns (cont’d)
• The same method can be used when there
are multiple payments per year
• In solving for i, note that i is not at an annual
rate, so you must multiply by number of
periods per year
Copyright © Houghton Mifflin Company. All rights reserved.
4 | 37
Policy Insight:
Annual Percentage Yield (APY)
• Annual Percentage Yield = The annual interest
rate that would give you the same amount you
would earn with more frequent compounding than
with the stated annual interest rate
• The U.S. government requires banks to
report APY on savings.
• APY offers a way to compare investment
with different periods of compounding.
• Example: Which is better to invest in?
A: 8.0% compounded annually
B: 7.95% compounded monthly
Copyright © Houghton Mifflin Company. All rights reserved.
4 | 38
Policy Insight:
Annual Percentage Yield (APY) (cont’d)
Example (cont.)
A: $1000 × 1.081 = $1080
B: $1000 × [1+(.0795/12)]12 = $1082.46
Option B is a better investment
To compare easily, define:
APY = [1 + (i/x)]x – 1
where compounding occurs x times per year
APY(A) = .08; APY(B) = .08246.
Copyright © Houghton Mifflin Company. All rights reserved.
4 | 39