Time Value of Money

4
Financial Management (13th edt)
By E.F. Bringham and M.C. Ehrhardt
4-1

Time Value of money


Receiving $1 today is worth more than $1 in the future
 Today’s dollar can be invested to have more dollars tomorrow  Opportunity cost is the interest we could have earned on $1 if received earlier
 Opportunity cost rate is the rate of return on the best available alternative investment of equal risk.

4-2

1

Compounding and Discounting


Compounding
 Translating today’s $1 into its equivalent FV

 FVn = PV ( 1 + i )n
 FV1= 100(1+.10)1=110


Discounting
 Translating tomorrow's $1 into its equivalent PV
 PVn = FV÷( 1 + i )n
 PV1= 110÷ (1+.10)1=100
4-3

Compounding and Discounting
When making investment decisions, managers usually calculate PV.

4-4

2

FVIFi,n (FVIF Table)

FVn = PV ( 1 + i )n

4-5

FV– Single Sum
If you deposit $100 in an account earning 6%, how much would you have after 1 year?
PV = 100

FV =

0

106

1

Tabular and Mathematical Solution:
FV = PV (FVIF i, n ) (use FVIF table)
FV = 100 (FVIF .06, 1 ) = 100 (1.06)= $106
FV = PV (1 + i)n = 100 (1.06)1 = $106
4-6

3

FV– Single Sum
If you deposit $100 in an account earning 6%, how much would you have after 5 years?
PV = 100

FV =

0

133.82

5

Tabular and Mathematical Solution:

FV = PV (FVIF i, n ) (use FVIF table)
FV = 100 (FVIF .06, 5 ) = 100 (1.3382)= $133.82
FV = PV (1 + i)n = 100 (1.06)5 = $133.82
4-7

FV– Single Sum
If you deposit $100 in an account earning 6% with quarterly compounding, how much would you have after 5 years?
PV = 100

0

FV =

134.68

5

Tabular and Mathematical Solution:
FV = PV (FVIF i, n ) (use FVIF table)
FV = 100 (FVIF .015, 20 ) (can’t use FVIF table)
FV = PV (1 + i/m)n×m = 100 (1.015)20 = $134.68
4-8

4

FV– Single Sum
If you deposit $100 in an account earning 6% with monthly compounding, how much would you have after 5 years?
PV = 100

FV =

0

134.89

5

Tabular and Mathematical Solution:
FV = PV (FVIF i, n ) (use FVIF table)
FV = 100 (FVIF .0005, 60 ) (can’t use FVIF table)
FV = PV (1 + i/m)n×m = 100 (1.0005)60 = $134.89
4-9

FV– Continuous Compounding
What is the FV of $1,000 earning 8% with continuous compounding, after 100 years?
PV = 1,000

FV =

0

2.98Mil

100

Calculator Solution:
FV = PV (e i × n)
FV = 1000 (e.08×100) =1000 (e8)= $2,980,957 e=2.71828 4-10

5

PVIFi,n (PVIF Table)

PVn = FV/( 1 + i )n

4-11

PV– Single Sum
If you receive $100 one year from now, what is the
PV of that $100 if your opportunity cost is 6%?
PV = 94.34

0

FV =

100

1

Tabular and Mathematical Solution:
PV = FV (PVIF i, n ) (use PVIF table)
PV = 100 (PVIF .06, 1)= 100 (0.9434)= $94.34
PV = FV÷(1+ i)n = 100÷ (1.06)1 = $94.34
4-12

6

PV– Single Sum
If you receive $100 five years from now, what is the PV of that $100 if your opportunity cost is 6%?
PV = 74.73

FV =

0

100

5

Tabular and Mathematical Solution:

PV = FV (PVIF i, n ) (use PVIF table)
PV = 100 (PVIF .06, 5)= 100 (0.7473)= $74.73
PV = FV÷(1+ i)n = 100÷ (1.06)5 = $74.73
4-13

PV– Single Sum
If you sold land for $11,933 that you bought 5 years ago for $5,000, what is your annual rate of return?
PV = 5,000

0

FV =

Tabular and Mathematical Solution:
PV = FV (PVIF i, n )
»5,000 = 11,933 (PVIF ?, 5 )
» PV = FV / (1 + i)n
» 5,000 = 11,933 / (1+ i)5
» 0 .419 = ((1/ (1+i)5)
» 2.3866 = (1+i)5
»(2.3866)1/5 = (1+i)

11,933

5

Thus, i = 0.19
4-14

7

PV– Single Sum
Suppose you placed $100 in an account that pays
9.6% interest, compounded monthly. How long will it take for your account to grow to $500?
PV = 5,000

FV =

0
Tabular and Mathematical Solution:

11,933

5

PV = FV / (1 + i)n
100 = 500 / (1+ .008)n
» 5 = (1.008)n

» ln 5 = ln (1.008)n
» ln 5 = (n) ln (1.008)
» 1.60944 = (n) 0.007968

n = 202 months

4-15

Multiple CFs Example
Consider the following CFs:
Time
0
1
2
3

Notation
CF0
CF1
CF2
CF3

CFs
– $ 2,000
+ $ 1,000
+ $ 1,500
+ $ 2,000

The interest rate (r) is 10% per period.
4-16

8

The Time Line of CFs
−$2,000

$1,000

$1,500

$2,000

t=0

t =1

t=2

t=3

NPV = - $2,000 +

$1,000 $1,500 $2,000
+
+
(1.10)1 (1.10)2 (1.10)3

NPV  $2000  $3,651.39  $1,651.39
4-17

FV (at t = 3) of Multiple CFs
- $2,000 × (1.10)3
+ $1,000 × (1.10)2

+ $1,500 × (1.10)1

+ $2,000 × (1.00)0
$2,198.00
−$2,000 t=0 $1,000

$1,500

$2,000

t =1

t=2

t=3
4-18

9

Annuity


A series of equal payments occurring over a specified number of times with an equal interval of time






In an annuity immediate (ordinary), payments occur at the end of each period
In an annuity due, payments occur at the beginning of each period
In deferred annuity, payment begins more than one period from today

4-19

PV of an Annuity Immediate

PVImm. 

PMT
{1 - (1  r)-n } r PV (Immediate) = PMT (PVIFA

i, n )

4-20

10

PVIFAi,n (PVIFA Table)
PVIF

The PV of an annuity of $1 per period n i=2%

i=4%

i=6%

i=8%

1
2
3
4
5
6
7
8
9

0.980
1.942
2.884
3.808
4.713
5.601
6.472
7.326
8.162

0.962
1.886
2.775
3.630
4.452
5.242
6.002
6.733
7.435

0.943
1.833
2.673
3.465
4.212
4.917
5.582
6.210
6.802

0.926
1.783
2.577
3.312
3.993
4.623
5.206
5.747
6.247

i=10% i=10%
0.909
1.736
2.487
3.170
3.791
4.355
4.868
5.335
5.759

0.909
0.826
0.751
0.683
0.621
0.564
0.513
0.467
0.424
4-21

PV of Your Bank Loan
Cynthia Smart agrees to repay her
PMT
loan in 24 monthly
PVImm. 
{1 - (1  r)-n } r installments of $250 each. If the interest rate on the loan is
$250
0.75% per month
PVImm =
{1 - (1 + 0.0075)-24 }
0.0075
(9% per annum), what is the present
=$5,472.29
value of her loan payments? 4-22

11

PV of an Annuity Due

PVDue 

PMT
{1 - (1  r)-n }(1  r) r PV (Due) = PMT (PVIFA

i, n )

(1 + i)

4-23

PV of Annuity Examples
Q-1: You win the Lottery in the USA. The prize is $50
MM. However American lotteries pay the prize out in equal yearly payments over 20 years. Therefore you will get $2.5 MM per year for 20 years. Assuming that interest rates are 7% compounded annually, and that the first payment arrives in exactly 1 year, what is the
PV of your lottery prize?
Q-2: Repeat question 1, but assume that the first payment of $2.5 MM is made today. You will still receive
20 payments (including the payment made today) on an annual basis. What is the PV of your lottery prize?
4-24

12

Solution of PV of Annuity Example
PMT= $2.5 MM; n =20; r = 0.07; PV=?
Putting the values in the formula, we get:

$2.5
{1 - (1 + 0.07)-20 }
0.07
$2.5
⇒ PVImm =
(1 - 0.2584)
0.07
⇒ PVImm = $26.485 MM (App.)
PVImm =

4-25

Solution of PV of Annuity Example
PMT
{1 - (1 + r )-n }(1 + r ) r $2.5
PVDue =
{1 - (1 + 0.07)-20 }(1 + 0.07)
0.07
$2.5
⇒ PVDue =
{1 - 0.2584}(1.07)
0.07
⇒ PVDue = $28.338 MM (App.)
PVDue =

4-26

13

FV of an Annuity Immediate

FV(Imm.) 

PMT
{(1  r)n - 1} r FV (Immediate) = PMT (FVIFA

i, n )

4-27

FVIFAi,n (FVIFA Table)
The FV of an Annuity of $1 Per Period n i=2%

i=4%

i=6%

i=8%

i=10%

i=10%

1
2
3
4
5
6
7
8
9

1.000
2.020
3.060
4.122
5.204
6.308
7.434
8.583
9.755

1.000
2.040
3.122
4.246
5.416
6.633
7.898
9.214
10.583

1.000
2.060
3.184
4.375
5.637
6.975
8.394
9.897
11.491

1.000
2.080
3.246
4.506
5.867
7.336
8.923
10.637
12.488

1.000
2.100
3.310
4.641
6.105
7.716
9.487
11.436
13.579

1.000
1.100
1.210
1.331
1.464
1.611
1.772
1.949
2.144
4-28

14

FV of Your Savings
Suppose you save
$1,500 per year for
15 years, beginning one year from today.
The savings bank pays you 8% interest per year. How much will you have at the end of 15 years?

FV(Imm) 

PMT
{(1  r)n - 1} r $1,500
{(1.08)15 - 1}
0.08
 $40,728.17


4-29

Saving for Retirement
You wish to retire 25 years from today
PMT
FV(Imm) 
{(1  r)n – 1} with $2,000,000 in r the bank. If the bank pays 10%
PMT
$2.0 
{(1  0.10)25 – 1} interest per year,
0.10
how much should you save each year
PMT = $23,336.14 to reach your goal?

4-30

15

FV of an Annuity Due

FVDue 

PMT
{(1  r)n – 1}(1  r) r FV (Due) = PMT (FVIFA

i, n )

(1 + i)

4-31

FV of a Continuous Annuity

FV(Continuous)  PV(e)rn

Where, e= 2.71828

4-32

16

Loan Amortization Schedule




LAS shows how a loan is paid off over time and breaks down each payment into the interest and principal components
Example: We will illustrate this using Rob
Steinberg’s 4-year $10,000 loan which calls for annual payments of $3,432.05. Recall that the interest rate on this loan is 14% per year.

4-33

LAS with Equal Payments
PMT =

PV
10,000
=
= 3,432
PVIFAi,n
2.913

PMT
{1 - (1 + 0.14)- 4 }
0.14
PMT = $3,432
$10,000 =

Year
1

Beginning
Total
Interest
Balance
Payment
Paid
$10,000.00 $3,432.05 $1,400.00

Principal
Paid
$2,032.05

Ending
Balance
$7,967

2

$7,967.95

$3,432.05 $1,115.51

$2,316.54

$5,651.41

3

$5,651.42

$3,432.05

$791.20

$2,640.85

$3,010.56

4

$3,010.57

$3,432.05

$421.48

$3,010.57

$0

Totals

$13,728.2 $3,728.20

$10,000
4-34

17

LAS with Equal Principal Payments

Year
1

Beginning
Balance
$10,000.00

Total
Payment
$3,900

Interest
Paid
$1,400

Principal
Paid
$2,500

Ending
Balance
$7,500

2

$7,500

$3,550

$1,050

$2,500

$5,000

3

$5,000

$3,200

$700

$2,500

$2,500

4

$2,500

$2,850

$350

$2,500

$0

$13,500

$3,500

$10,000

Totals

4-35

Deferred Annuity Example


The PV of the deferred annuity is the difference in the PVs of two annuities


Example: An annuity’s first CF is expected to occur
3 years from today. There are 4 CFs in this annuity, with each CF being $500. At an interest rate of 10% per year, find the annuity’s PV.

4-36

18

Deferred Annuity
$500
0

$500

$500

$500

1

2

3

4

5

6

$500

$500

$500

$500

$500

$500

1

2

3

4

5

6

$500

$500

1

2

3

4

5

6

Equals
0
Minus

0

PV of annuity: 2,177.63 – $867.77 = $1,309.86
PVImm 

500
{1 - (1  0.10)-6 }
0.10

–

PVImm 

500
{1 - (1  0.10)-2 }
0.10
4-37

Perpetuity


A special type of annuity with an infinite life, providing continual annual cash flow.


PV = PP ÷ I
Example: What is the PV of a $500 perpetuity discounted back to the present at 8%?
PV= $500/0.08= $6,250

4-38

19

Solving for an Unknown Interest Rate


The First Commerce
Bank offers a Certificate of Deposit (CD) that pays you $5,000 in four years.
The CD can be purchased today for
$3,477.87. Assuming you hold the CD to maturity, what annual interest rate is the bank paying on this CD?

PV =

FV
(1 + r )n

$3,477.87 =

$5,000
(1 + r ) 4

r = 9.5%

4-39

Classification of Interest Rates




Nominal rate (INOM)– also called the quoted or stated rate, is an annual rate that ignores compounding effects
Periodic rate (IPER)– amount of interest charged each period, e.g. monthly or quarterly




IPER = INOM/m, where m is the number of compounding periods per year

Effective (or equivalent) annual rate– also called APY= EAR = EFF% is the annual rate of interest actually being earned, accounting for compounding 4-40

20

Classification of Interest Rates


If the annual interest rate is 10% which is compounded monthly, the APY= EAR= EFF% will be:
EFF% = ( 1 + INOM/m )m – 1
= ( 1 + 0.10/12 )1×2 – 1 = 10.47%



Thus, an investor should be indifferent between receiving 10.47% annual interest and receiving
10% interest, compounded monthly

4-41

Can the APY Ever Be Equal to the
Nominal Rate?





YES, but only if annual compounding is used, i.e., if m = 1.
If m > 1, APY (EIR%) will always be greater than the nominal rate.

4-42

21

Why to Consider Effective Rates of
Return?


Investments with 10% nominal rate but different compounding intervals provide different effective returns
EARANNUAL

10.00%

EARQUARTERLY

10.38%

EARMONTHLY

10.47%

EARDAILY (365)

10.52%

4-43

Annual Percentage Yield
 Boffo Bank is advertising deposit rates of
6.5% compounded quarterly.

(a) what is the equivalent interest rate compounded annually?
(b) what is the equivalent rate compounded monthly? (c) what is the equivalent rate compounded daily? 4-44

22

Annual Percentage Yield

0.065 1×4
)
–1
4
⇒ APY = 1.0666 – 1 = 0.0666
⇒ APY = 6.66%(App
.)
(a) APY = (1 +

4-45

Annual Percentage Yield
0.065 1×4 r 1×12
)
– 1 = (1 +
)
–1
4
12 r 12
⇒ (1.01625 4 = (1 +
)
)
12
4 r 12
⇒ (1.01625 12 = (1 +
)
) 12
12
r
⇒ 1.0053876= 1 +
12
r
⇒
= (1.0053876 – 1)
12
⇒ r = 12(1.0053876 – 1) = 0.06465
⇒ r = 6.465%( App
.)
(b) APY = (1 +

4-46

23

Annual Percentage Yield
0.065 1×4 r 1×365
) – 1 = (1 +
)
–1
4
365 r 365
⇒ 1.0666 = (1 +
)
365
1
r 365
365
⇒ (1.0666)
= (1 +
) 365
365
⇒ r = 365(1.00017667– 1) = 0.064483
⇒ r = 6.4483%(App.)
(c) APY = (1 +

4-47

24