The Time Value of Money
Warren Buffett’s Advice
On spending, “If you buy things you do not
need, soon you will have to sell things you
need”
On savings, “Do not save what is left after
spending, but spend what is left after saving”
On earnings, “Never depend on single income,
make investment to create a second source”
On investment, “Do no put all eggs in one
basket”
On taking risks, “Never test the depth of river
with both the feet”
Lecture Outline
Future Value and Compounding
Present Value and Discounting
Discount Rate
Number of Periods
Annuities and Perpetuities
Formulas for Annuities and Perpetuities
EAR and APR
Interest Rates and Inflation
Basic Definitions
Present Value – earlier money on a time line
Future Value – later money on a time line
Interest rate – “exchange rate” between
earlier money and later money
The Timeline
Assume that you are lending $10,000 today and
that the loan will be repaid in two annual
$6,000 payments.
5
Three Rules of Time Travel
6
The 1st Rule of Time Travel
A dollar today and a dollar in one year
are not equivalent.
It is only possible to compare or combine
values at the same point in time.
7
The 2nd Rule of Time Travel
To move a cash flow forward in time, you must
compound it.
• Future Value of a Cash Flow
n
n
rC
n
rrrCFV ) (1
times
) (1 ) (1 ) (1
8
The 3rd Rule of Time Travel
To move a cash flow backward in time, we
must discount it.
Present Value of a Cash Flow
PV C (1 r )
n
C
(1 r )
n
9
Combining Values Using
the Rules of Time Travel
Suppose we plan to save $1000 today, and
$1000 at the end of each of the next two
years. If we can earn a fixed 10% interest
rate on our savings, how much will we
have three years from today?
10
The time line would look like this:
11
12
13
14
0
3 4 521
$10,000
$5,000
Assume that an investment will pay you
$5,000 now and $10,000 in five years.
◦ The time line would like this:
15
0
3 4 521
$6,209 $10,000
$5,000
$11,209
÷ 1.10
5
You can calculate the present value of the
combined cash flows by adding their values
today.
16
0
3 4 521
$5,000 $8,053
x 1.10
5
$10,000
$18,053
You can calculate the future value of the
combined cash flows by adding their
values in Year 5.
17
0
3 4 521
$11,209 $18,053
÷ 1.10
5
0
3 4 521
$11,209 $18,053
x 1.10
5
Present
Value
Future
Value
19
The Power of Compounding
20
Rule of 72
21
Valuing a Stream of Cash Flows
Based on the first rule of time travel we
can derive a general formula for valuing a
stream of cash flows: if we want to find
the present value of a stream of cash
flows, we simply add up the present
values of each.
22
Present Value of a Cash Flow Stream
PV PV (C
n
)
n 0
N
C
n
(1 r)
n
n 0
N
23
Example
24
Annuities and Perpetuities Defined
Annuity – finite series of equal payments
that occur at regular intervals
◦ If the first payment occurs at the end of the
period, it is called an ordinary annuity
◦ If the first payment occurs at the beginning of
the period, it is called an annuity due
Perpetuity – infinite series of equal
payments
Valuing a Perpetuity
A perpetuity is a constant stream of cash
flows lasting forever.
Mathematically,
Suppose you bought the perpetuity today
and sold it for P1 in a year. The PV of the
perpetuity is the PV of the cash flow C+P1
But the PV of the perpetuity must be the
same at each point of time because the
future cash flows are identical. So
Thus, or or
Valuing a Perpetuity
Alternatively…
Mathematically, summing a geometric series
that goes to infinity:
Perpetuity is:
A =
r =
What is P?
Value a Growing Perpetuity
A growing perpetuity provides a cash flow of
$C in one year, and grows at a rate of g each
subsequent year. The cash flows from a
growing unit perpetuity looks like this (g=5%)
Mathematically,
Suppose you bought the growing perpetuity
today and sold it for P1 in a year. The PV of
the perpetuity is the PV of the cash flow
C+P1
The price in 1 year must be (1+g) times its
price now, because its future cash flows are
all g% larger. So,
Thus, or
Valuing a Growing Perpetuity
Valuing an Annuity
An Annuity is a constant stream of cash
flows lasting for a fixed number of years.
The PV formula for an annuity is simply
time 0 perpetuity minus time t perpetuity
Valuing an Annuity
The PV of time 0 perpetuity is
The PV of time t perpetuity is
The value of the annuity is the difference:
What if the first payment occurs today
(year 0) rather than in a year?
What is the future value (at time 6) of the
annuity?
Valuing a Growing Annuity
A growing annuity is a stream of cash flows that
grows at a constant rate, say, g, for a fixed
number of periods. The cash flows of a growing
unit annuity look like this:
Again, the value of the growing annuity is
calculated by noting that it is a growing
perpetuity at year 0 minus a growing perpetuity
at year t
Valuing a Growing Annuity
The PV of time 0 growing perpetuity is
The time t growing perpetuity is and
its PV is
The value of the annuity is the difference:
Annuity (1) – Lottery Example
Annuity (2)
Suppose you deposit $50 a month into an account
that has annual interest of 9%, based on monthly
compounding. How much will you have in the
account in 35 years?
◦ Monthly rate = .09 / 12 = .0075
◦ Number of months = 35(12) = 420
◦ 35(12) = 420 N
◦ 9 / 12 = .75 I/Y
◦ 50 PMT
◦ CPT FV = 147,089.22
Finding the Payment (1)
Finding the Payment (2)
o 10 N
o 10,000,000 FV
o 5 I/Y
o CPT PMT = -795,046
Finding the Number of Payments (1)
o 8 I/Y
o 300,000 PV
o -45,000 PMT
o CPT N = 9.9
Finding the Rate
Future Values for Annuities
Perpetuity
Company A has an existing perpetual bond that
pays $1 quarterly, which is currently priced at
$40. If the company plans to raise $100 by
issuing a new perpetual bond that pays
quarterly, how much dividend per quarter
should the company pays?
Current required return:
◦ 40 = 1 / r
◦ r = .025 or 2.5% per quarter
Dividend for new perpetual bond:
◦ 100 = C / .025
◦ C = 2.50 per quarter
Growing Perpetuity
A company will pay an annual dividend next
year of $3. You expect its dividend to grow
at the rate of 5% per year forever. You
calculate that the expected return on their
equity should be 10%. What should be the
company’s price per share?
Important Notes
The previous examples are direct
applications of the annuity/perpetuity
formula.
In trickier scenarios, we need to be
careful about the timing of the cash flows.
◦ When do cash flows occur?
◦ How often do they occur?
What is the difference between an ordinary
annuity and an annuity due?
Ordinary Annuity
PMT PMTPMT
0 1 2 3
i%
PMT PMT
0 1 2 3
i%
PMT
Annuity Due
Annuity Due (1) – Lottery Example
Bob has just won the lottery, paying 20
equal installments of $50,000 each. He
receives his first payment now, i.e. at year
0. If the interest rate is 8 percent, what is
the present value of the lottery?
OR
Annuity Due (2)
You are saving for a new house and you put
$10,000 per year in an account paying 8%. The
first payment is made today. How much will you
have at the end of 3 years?
◦ Set Type=1
◦ 3 N
◦ -10,000 PMT
◦ 8 I/Y
◦ CPT FV = 35,061.12
Delayed Annuity - Saving For
Retirement Example
You are offered the opportunity to put
some money away for retirement. You will
receive four annual payments of $500
each beginning at year 6. How much
would you be willing to invest today if you
desire an interest rate of 10%?
An Infrequent Annuity
Charlie receives an annuity of $450, payable
once every two years. The annuity stretches
out over 20 years. The first payment occurs
at date 2, that is, two years from today. The
annual interest rate is 6 percent. What is the
present value of the annuity?
The (annual) interest rate over two-year
period, denoted by R, is
So,
Multiple Annuities
An insurance agent approaches Debra and
would like to sell her the following contract: (1)
Debra pays $5,000 per year for the coming 15
years. (2) In return, she will receive $7,000 a
year for the following 15 years.
Assume that interest rates will remain at a
constant 9%. How much profit does the
insurance company make? In order for the deal
to generate zero profits, what an annual
payment does Debra need to receive?
The insurance company’s profit:
For zero profit, we need:
Multiple Annuities (con’t)
Decisions, Decisions
Your broker calls you and tells you that he has this
great investment opportunity. If you invest $100
today, you will receive $40 in one year and $75 in
two years. If you require a 15% return on
investments of this risk, should you take the
investment?
◦ CF
0
= 0; C01 = 40; C02 = 75
◦ I = 15
◦ CPT NPV = 91.49
◦ No – the broker is charging more than you would be
willing to pay.
Perpetuity
◦ A constant stream of cash flows that lasts forever.
Growing perpetuity
◦ A stream of cash flows that grows at a constant rate
forever.
Annuity
◦ A stream of constant cash flows that lasts for a fixed
number of periods.
Growing annuity
◦ A stream of cash flows that grows at a constant rate for
a fixed number of periods.
Summary
We presented four simplifying formulae:
r
C
PV :Perpetuity
gr
C
PV
:Perpetuity Growing
T
rr
C
PV
)1(
1
1:Annuity
T
r
g
gr
C
PV
)1(
1
1 :Annuity Growing
Summary
