Chapter 5
The Time Value
of Money
Laurence Booth, Sean Cleary
and Pamela Peterson Drake
Outline of the chapter
5.1 Time is money
5.2 Annuities and
perpetuities
•Simple versus
compound
interest
•Future value of
an amount
•Present value of
an amount
•Ordinary
annuities
•Annuity due
•Deferred
annuities
•Perpetuities
5.3 Nominal and
effective rates
•APR
•EAR
•Solving for the
rate
5.4 Applications
•Savings plans
•Loans and
mortgages
•Saving for
retirement
5.1 Time value of money
Simple interest
Simple interest is interest that is paid only
on the principal amount.
Interest = rate × principal amount of loan
Simple interest: example
A 2-year loan of $1,000 at 6% simple interest
 At the end of the first year,
interest = 6% × $1,000 = $60
 At the end of the second year,
interest = 6% × $1,000 = $60
and loan repayment of $1,000
Compound interest
Compound interest is interest paid on both
the principal and any accumulated interest.
accumulated
Interest = rate × principal +
interest
Compounding
 Compounding
is translating a present
value into a future value, using compound
interest.
Future Present
Future value
=
×
value
value
interest factor
 Future
value interest factor is also referred
to as the compound factor.
Terminology and notation
Term
Notation
Meaning
Future value
FV
Value at some specified
future point in time
Present value
PV
Value today
Interest
i
Compensation for the use
of funds
Number of periods
n
Number of periods
between the present value
and the future value
Compound factor
(1 + i)n
Translates a present value
into a future value
Compare:
simple versus compound
Suppose you deposit $5,000 in an account
that pays 5% interest per year. What is the
balance in the account at the end of four
years if interest is:
1. Simple interest?
2. Compound interest?
Simple interest
Year
Beginning
Add interest
Ending
1
$5,000.00
+ (5% × $5,000) =
$5,250.00
2
$5,250.00
+ (5% × $5,000) =
$5,500.00
3
$5,500.00
+ (5% × $5,000) =
$5,750.00
4
$5,750.00
+ (5% × $5,000) =
$6,000.00
Compound interest
Year
Beginning
Compounding
Ending
1
$5,000.00
× 1.05 =
$5,250.00
2
$5,250.00
× 1.05 =
$5,512.50
3
$5,512.50
× 1.05 =
$5,788.13
4
$5,788.13
× 1.05 =
$6,077.53
Interest on interest
How much interest on interest?
Interest on interest = FVcompound – FVsimple
Interest on interest = $6,077.53 – 6,000.00 = $77.53
Comparison
0
$5,000.00
$5,000.00
1
$5,250.00
$5,250.00
2
$5,500.00
$5,512.50
3
$5,750.00
$5,788.13
4
$6,000.00
$6,077.53
$5,000
$6,000
$6,078
$5,750
$5,788
$5,500
$5,513
$6,000
Compound interest
$5,250
$5,250
$7,000
Balance in the account
Year
End of year balance
Simple
Compound
interest
interest
$5,000
$5,000
Simple interest
$4,000
$3,000
$2,000
$1,000
$0
0
1
2
3
Year in the future
4
Try it: Simple v. compound
Suppose you are comparing two accounts:
 The Bank A account pays 5.5% simple
interest.
 The Bank B account pays 5.4% compound
interest.
If you were to deposit $10,000 in each, what
balance would you have in each bank at the
end of five years?
Try it: Answer
1.
2.
Bank A: $12,750.00
Bank B: $13,007.78
A note about interest
Because compound interest is so common,
assume that interest is compounded unless
otherwise indicated.
Short-cuts
Example:
Consider $1,000 deposited for three years
at 6% per year.
The long way
FV1 = $1,000.00 × (1.06) = $1,060.00
FV2 = $1,060.00 × (1.06) = $1,123.60
FV3 = $1,123.60 × (1.06) = $1,191.02
or
FV3 = $1,000 × (1.06)3 = $1,191.02
or
FV3 = $1,000 × 1.191016 = $1,191.02
Future value factor
Short-cut: Calculator
Known values:
PV = 1,000
n=3
i = 6%
Solve for: FV
Input three known values,
solve for the one unknown
Known:
Unknown:
PV, i , n
FV
HP10B
BAIIPlus
HP12C
TI83/84
1000 +/PV
3N
6 I/YR
FV
1000 +/PV
3N
6 I/YR
FV
1000 CHS PV
3n
6i
FV
[APPS] [Finance]
[TVM Solver]
N =3
I%=6
PV = -1000
FV [Alpha] [Solve}
Short-cut: spreadsheet
Microsoft Excel or Google Docs
=FV(RATE,NPER,PMT,PV,TYPE)
TYPE default is 0, end of period
=FV(.06,3,0,-1000)
or
A
1
6%
2
3
3
-1000
4
=FV(A1,A2,0,A3)
Problems Set 1
Problem 1.1
Suppose you deposit $2,000 in an account
that pays 3.5% interest annually.
1. How much will be in the account at the
end of three years?
2. How much of the account balance is
interest on interest?
23
Problem 1.2
If you invest $100 today in an account that
pays 7% each year for four years and 3%
each year for five years, how much will you
have in the account at the end of the nine
years?
24
Discounting
Discounting
 Discounting
is translating a future value
into a present value.
 The discount factor is the inverse of the
1
compound factor:
𝑛
1+𝑖
 To translate a
value, PV=
future value into a present
FV
1+𝑖 𝑛
Example
Suppose you have a goal of saving $100,000
three years from today. If your funds earn
4% per year, what lump-sum would you have
to deposit today to meet your goal?
Example, continued
Known values:
FV = $100,000
n=3
i = 4%
Unknown: PV
Example, continued
$100,000
1
PV =
=$100,000 ×
3
1+0.04
1+0.04 3
PV = $100,000 × 0.8889964
PV = $88,899.64
Check:
FV3 = $88,899.64 × (1 + 0.04)3 = $100,000
Short-cut: Calculator
HP10B
BAIIPlus
HP12C
TI83/84
100000 +/PV
3N
4 I/YR
PV
100000
+/- PV
3N
4 I/YR
PV
100000 CHS PV
3n
4i
PV
[APPS] [Finance] [TVM
Solver]
N =3
I%=4
FV = 100000
PV [Alpha] [Solve]
Short-cut: spreadsheet
Microsoft Excel or Google Docs
=PV(RATE,NPER,PMT,PV,TYPE)
TYPE default: end of period
=PV(.06,3,0,-1000)
or
A
1
6%
2
3
3
100000
4
=PV(A1,A2,0,A3)
Try it: Present value
What is the today’s value of $10,000
promised ten years from now if the discount
rate is 3.5%?
Try it: Answer
Given:
FV = $10,000
N = 10
I = 3.5%
Solve for PV
PV = $10,00010 = $7,089.19
1+0.035
Frequency of compounding
 If interest
is compounded more than once
per year, we need to make an adjustment
in our calculation.
 The stated rate or nominal rate of interest
is the annual percentage rate (APR).
 The rate per period depends on the
frequency of compounding.
Discrete compounding:
Adjustments
 Adjust
the number of periods and the rate
per period.
 Suppose the nominal rate is 10% and
compounding is quarterly:
rate per period is 10%  4 = 2.5%
 The number of periods is
number of years × 4
 The
Continuous compounding:
Adjustments
 The compound
 The discount
 Suppose
factor is eAPR x n.
factor is
1
eAPR x n
.
the nominal rate is 10%.
 For five years, the continuous compounding
factor is e0.10 x 5 = 1.6487
 The continuous compounding discount factor
for five years is 1 ÷ e0.10 x 5 = 0.60653
Try it: Frequency of
compounding
If you invest $1,000 in an investment that
pays a nominal 5% per year, with interest
compounded semi-annually, how much will
you have at the end of 5 years?
Try it: Answer
Given:
PV = $1,000
n = 5 × 2 = 10
i = 0.05  2 = 0.25
Solve for FV
FV = $1,000 × (1 + 0.025)10 = $1,280.08
Problem Set 2
Problem 2.1
Suppose you set aside an amount today in an
account that pays 5% interest per year, for
five years. If your goal is to have $1,000 at
the end of five years, what would you need
to set aside today?
40
Problem 2.2
Suppose you set aside an amount today in an
account that pays 5% interest per year,
compounded quarterly, for five years. If your
goal is to have $1,000 at the end of five
years, what would you need to set aside
today?
41
Problem 2.3
Suppose you set aside an amount today in an
account that pays 5% interest per year,
compounded continuously, for five years. If
your goal is to have $1,000 at the end of five
years, what would you need to set aside
today?
42
0
|
1
2
3
4
5
CF
CF
CF
CF
CF
|
|
|
PV?
5.2 Annuities and
Perpetuities
|
|
FV?
What is an annuity?
 An
annuity is a periodic cash flow.
 Same amount each period
 Regular intervals of time
 The different
types depend on the timing
of the first cash flow.
Type of annuities
Type
First cash flow
Examples
Ordinary
One period from
today
Mortgage
Annuity due
Immediately
Lottery payments
Rent
Deferred annuity
Beyond one period
from today
Retirement savings
Time lines: 4-payment annuity
Ordinary
Annuity due
Deferred annuity
0
1
2
3
4
5
|
|
|
|
|
|
CF
CF
CF
CF
FV
CF
CF
CF
CF
CF
PV
CF
PV
PV
FV
CF
CF
FV
Key to valuing annuities
 The key to valuing
annuities is to get the
timing of the cash flows correct.
 When in doubt, draw a time line.
Example: PV of an annuity
What is the present value of a series of three
cash flows of $4,000 each if the discount rate
is 6%, with the first cash flow one year from
today?
0
1
2
3
|
|
|
|
$4,000
$4,000
$4,000
4
Example: PV of an annuity
The long way
0
1
2
3
|
|
|
|
$4,000
$4,000
$4,000
$3,773.58
3,559.99
3,358.48
$10,692.05



4
Example: PV of an annuity
In table form
Year
1
2
3
Discount Present
Cash flow
factor
value
$4,000.00
0.94340 $3,773.58
$4,000.00
0.89000
3,559.99
$4,000.00
0.83962
3,358.48
2.67301 $10,692.05
PV = $4,000.00 × 2.67301 = $10,692.05
Example: PV of an annuity
Formula short-cuts
1−
PV = $4,000 ×
1
3
1+0.06
0.06
PV = $4,000 × 2.67301
PV = $10,692.05
Example: PV of an annuity
Calculator short cuts
Given:
PMT = $4,000
i = 6%
N=3
Solve for PV
Example: PV of an annuity
Spreadsheet short-cuts
=PV(RATE,NPER,PMT,FV,TYPE)
=PV(.06,3,4000,0)
Note: Type is important for annuities
• If Type is left out, it is assumed a 0
• 0 is for an ordinary annuity
• 1 is for an annuity due
Example: FV of an annuity
What is the future value of a series of three
cash flows of $4,000 each if the discount rate
is 6%, with the first cash flow one year from
today?
0
1
2
3
|
|
|
|
$4,000
$4,000
$4,000
4
Example: FV of an annuity
The long way
0
1
2
3
|
|
|
|
$4,000.00
$4,000.00
$4,000.00

4,240.00

4,494.40
$12,734.40
4
Example: FV of an annuity
In table form
Year
1
2
3
Compound
Cash flow
factor
Future value
$4,000.00
1.1236
$4,494.40
$4,000.00
1.0600
4,240.00
$4,000.00
1.0000
4,000.00
3.1836
$12,734.40
PV = $4,000.00 × 3.1836 = $12,734.40
Example: FV of an annuity
Calculator short cuts
CALCULATOR
Given:
PMT = $4,000
i = 6%
N=3
Solve for FV
Example: FV of an annuity
Spreadsheet short-cuts
=FV(RATE,NPER,PMT,PV,type)
=FV(.06,3,4000,0)
Annuity due
Consider a series of three cash flows of
$4,000 each if the discount rate is 6%, with
the first cash flow today.
1. What is the present value of this annuity?
2. What is the future value of this annuity?
The time line
0
1
2
3
|
|
|
|
$4,000.00
$4,000.00
$4,000.00
PV?
This is an annuity due
FV?
Valuing an annuity due
Present value
End of year Compoun Present value of
Year cash flow d factor
cash flow
0
$4,000.00
1.00000
$4,000.00
1
$4,000.00
0.94340
3,773.58
2
$4,000.00
0.89000
3,559.99
2.83339
$11,333.57
Future value
End of year
Year cash flow
0
$4,000.00
1
$4,000.00
2
$4,000.00
Factor Future value
1.19102
$4,764.06
1.12360
4,494.40
1.06000
4,240.00
3.37462 $13,498.46
Valuing an annuity due:
Using calculators
Present value
Future value
PMT = 4000
N=3
I = 6%
BEG mode
Solve for PV
PMT = 4000
N=3
I = 6%
BEG mode
Solve for FV
Valuing an annuity due:
Using spreadsheets
Present value
=PV(RATE,NPER,PMT,FV,TYPE)
=PV(0.06,3,4000,0,1)
Future value
=PV(RATE,NPER,PMT,FV,TYPE)
=PV(0.06,3,4000,0,1)
Any other way?
There is one period difference between an
ordinary annuity and an annuity due.
Therefore:
PVannuity due = PVordinary annuity × (1 + i)
and
FVannuity due = FVordinary annuity × (1 + i)
Valuing a deferred annuity
A
deferred annuity is an annuity that
begins beyond one year from today.
 That
means that it could begin 2, 3, 4, … years
from today, so each problem is unique.
Valuing a deferred annuity
0
4-payment
ordinary annuity,
then discount
value one period
PV0
4-payment annuity
due, then discount
value two periods
PV0
1
2
3
4
5
|
|
|
|
CF
CF
CF
CF
←PV1
←PV2
Example: Deferred annuity
What is the value today of a series of five
cash flows of $6,000 each, with the first cash
flow received four years from today, if the
discount rate is 8%?
0
PV?
1
2
3
4
5
6
7
8
|
|
|
|
|
CF
CF
CF
CF
CF
9
10
Example, cont.
Using an ordinary annuity:
PV3 = $23,956.26
Discount 3 periods at 8%
PV0 = $19,017.25
Using an annuity due:
PV4 = $25,872.76
Discount 4 periods at 8%
PV0 = $19,017.25
Example: Deferred annuity
Calculator solutions
HP10B
BAIIPlus
TI83/84
0 CF
0 CF
0 CF
0 CF
6000 CF
6000 CF
6000 CF
6000 CF
6000 CF
8i
NPV
0 CF ↑ 1
0 CF ↑ 1 F1
0 CF ↑ 1 F2
0 CF ↑ 1 F3
6000 CF ↑ 1 F4
6000 CF ↑ 1 F5
6000 CF ↑ 1 F6
6000 CF ↑ 1 F7
6000 CF ↑ 1 F8
8i
NPV
[2nd] {
0 0 0 6000 6000
6000 6000 6000}
STO [2nd] L1
[APPS] [Finance]
[ENTER] 7
NPV(.08,0,L1)
[ENTER]
Example: Deferred annuity
Spreadsheet solutions
A
B
Year Cash flow
1
1
$0
2
2
$0
3
3
$0
4
4
$6000
5
5
$6000
6
6
$6000
7
7
$6000
8
8
$6000
1. =PV(0.08,3,0,PV(0.08,5,6000,0))
2. =PV(0.08,4,0,PV(0.08,5,6000,0,1))
3. =NPV(0.08,A1:A9)
Perpetuities
A perpetuity is an even cash flows that
occurs at regular intervals of time, forever.
The valuation of a perpetuity is simple:
PV =
𝐶𝐹
𝐶𝐹
𝐶𝐹
𝐶𝐹
+
+
+…
1
2
3
1+𝑖
1+𝑖
1+𝑖
1+𝑖 ∞
𝐶𝐹
PV =
𝑖
Problem Set 3
Problem 3.1
Which do you prefer if the appropriate
discount rate is 6% per year:
1. An annuity of $4,000 for four annual
payments starting today.
2. An annuity of $4,100 for four annual
payments, starting one year from today.
3. An annuity of $4,200 for four annual
payments, starting two years from today.
73
5.3 Nominal and effective
rates
APR & EAR
 The
annual percentage rate (APR) is the
nominal or stated annual rate.
 The
APR ignores compounding within a year.
 The APR understates the true, effective rate.
 The effective
annual rate (EAR)
incorporates the effect of compounding
within a year.
APR EAR
EAR = 1 +
𝐴𝑃𝑅 𝑚
𝑚
−1
Suppose interest is stated as 10% per years,
compounded quarterly.
EAR = 1 +
0.10 4
4
EAR = 10.3813%
− 1 = 1.0254 − 1
EAR with continuous
compounding
EAR = 𝑒 𝐴𝑃𝑅 − 1
Suppose interest is stated as 10% per years,
compounded continuously.
EAR = e0.1 −1=1.05171−1
EAR = 10.5171%
77
Frequency of compounding
 If interest
is compounded more frequently
than annually, then this is considered in
compounding and discounting.
 There are two approaches
1. Adjust the i and n; or
2. Calculate the EAR and use this
Example: EAR &
compounding
Suppose you invest $2,000 in an investment
that pays 5% per year, compounded
quarterly. How much will you have at the
end of 4 years?
Example: EAR &
compounding
Method 1:
FV = $2,000 (1 + 0.0125)16 = $2,439.78
Method 2:
EAR = (1 + 0.05 4)4 – 1 = 5.0945%
FV = $2,000 (1 + 0.050945)4 = $2,439.78
Try it: APR & EAR
Suppose a loan has a stated rate of 9%, with
interest compounded monthly. What is the
effective annual rate of interest on this loan?
Try it: Answer
EAR =
0.09 12
1+
−1
12
1.007512 − 1
EAR =
EAR = 9.3807%
Problem Set 4
Problem 4.1
What is the effective interest rate that
corresponds to a 6% APR when interest is
compounded monthly?
84
Problem 4.2
What is the effective interest rate that
corresponds to a 6% APR when interest is
compounded continuously?
85
5.4 Applications
Saving for retirement
Suppose you estimate that you will need $60,000
per year in retirement. You plan to make your first
retirement withdrawal in 40 years, and figure that
you will need 30 years of cash flow in retirement.
You plan to deposit funds for your retirement
starting next year, depositing until the year before
retirement. You estimate that you will earn 3% on
your funds.
How much do you need to deposit each year to
satisfy your plans?
Deferred annuity time line
0
1
2
3
4
5
6
7
8
|
|
|
|
|
|
|
|
D
D
D
D
D
D
D
D
D = Deposit (39 in total)
W = Withdrawal (30 in total)
…
39
40
41
42
43
|
|
|
|
|
W W W W W
…
79
|
W
Deferred annuity time line
0
1
2
3
4
5
6
7
8
|
|
|
|
|
|
|
|
…
39
40
41
42
43
|
|
|
|
|
|
W
W
W
W
W
←
Ordinary annuity
PV
↓
Ordinary annuity
D
D
D
D
D
D
D
→
FV
D …
D
…
79
Two steps
Step 1: Present value of ordinary annuity
N = 30; i = 3%; PMT = $60,000
PV39 = $1,176,026.48
Step 2: Solve for payment in an ordinary
annuity
N = 39; i = 3%; FV = $1,176,026.48
PMT = $16,280.74
What does this mean?
If there are 39 annual deposits of $16,280.74
each and the account earns 3%, there will be
enough to allow for 30 withdrawals of
$60,000 each, starting 40 years from today.
Balance in retirement account
$1,400,000
$1,200,000
Balance in
the
retirement
account
$1,000,000
$800,000
$600,000
$400,000
$200,000
$0
1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69
Year into the future

Practice problems
Problem 1
What is the future value of $2,000 invested
for five years at 7% per year, with interest
compounded annually?
Problem 2
What is the value today of €10,000 promised
in four years if the discount rate is 4%?
Problem 3
What is the present value of a series of five
end-of-year cash flows of $1,000 each if the
discount rate is 4%?
Problem 4
Suppose you plan to save $3,000 each year
for ten years. If you earn 5% annual interest
on your savings, how much more will you
have at the end of ten years if you make your
payments at the beginning of the year
instead of the end of the year?
Problem 5
Sue plans to deposit $5,000 in a savings
account each year for thirty years, starting
ten years from today. Yan plans to deposit
$3,500 in a savings account each year for
forty years, starting at the end of this year. If
both Sue and Yan earn 3% on their savings,
who will have the most saved at the end of
forty years?
Problem 6
Suppose you have two investment
opportunities:
Opportunity 1: APR of 12%, compounded
monthly
Opportunity 2: APR of 11.9%, compounded
continuously
Which opportunity provides the better
return?
Problem 7
If you can earn 5% per year, what would you
have to deposit in an account today so that
you have enough saved to allow withdrawals
of $40,000 each year for twenty years,
beginning thirty years from today?
Problem 8
Suppose you deposit ¥50000 in an account
that pays 4% interest, compounded
continuously. How much will you have in the
account at the end of ten years if you make
no withdrawals?
The end