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Chapter 5
Future Value
Present Value
Annuities
Different compounding Periods
Adjusting for frequent compounding
Effective Annual Rate (EAR)
5-1

You want to buy a computer and a friend
offers you a $1000.
 Would you prefer use the money now.
 Later (after year for example).

The answer to that question depends on:
 Inflation rate.
 Deferred consumption.
 Forgone investment opportunity
 Uncertainty (Risk)
5-2

There are several application for the TVM from
which both individuals and firms benefit, such as:
 Planning for retirement,
 Valuing businesses or any asset (including stocks and
bonds),
 Setting up loan payment schedules
 Making corporate decisions regarding investing in new
plants and equipments.

The rest of this book and course heavily depends on
your understanding of the concepts of TVM and
your proficiency in doing its calculations.
5-3
0
CF0
I%
1
2
3
CF1
CF2
CF3

Help visualize what is happening in a particular
problem.

Show the timing of cash flows.

Tick marks occur at the end of periods, so Time 0 is
today; Time 1 is the end of the first period (year,
month, etc.) or the beginning of the second period.
5-4

Finding the future value (FV) or compounding):
The amount to which a cash flow or series of cash
flows will grow over a given period of time when
compounded at a given interest rate.

Finding the present value (PV): The value today of
a future cash flow or series of cash flows when
discounted at a given interest.

Compounding : is the process to determine the FV
of a cash flow or series of payments. (multiplying)

Discounting : is the reverse of compounding. The
process of determining the PV of a cash flow or
series of payments (dividing)
5-5
1. $100 lump sum (single payments) due in 2 years
0
I%
1
2
2. 3-year $100 ordinary annuity
0
I%
100
1
2
3
100
100
100
3. 3-year $100 annuity due
0
100
I%
1
2
100
100
3
Annuity:
A series of equal
payments at
fixed intervals
for a specified #
of periods
5-6

Examples of obligations that uses annuities:
 Auto, student, mortgage loans

However, many financial decisions involve non constant
(not equal) payments:
 Dividend on common stocks.
 Investment in capital equipment
4. Uneven cash flow stream (payments are not equal)
0
-50
I%
1
2
3
100
75
-50
5-7
4. Perpetuities (annuity that has payments that go
forever)
0
I%
1
100
2
100
∞
100
5-8

Compounding interest rates is when interest is
earned on interest.
0
5%
100
1
105
= 100(1.05)
Interest = $5
Amount = $100
2
110.25
= 100(1.05)2
Interest = $5.25
Amount= $105
3
115.76
= 100(1.05)3
Interest = $5.5125
Amount= $110.25
 Thus, FV of annuity due > FV of ordinary annuity

Simple interests: interest is not earned on interest
 FV = PV + PV (i)(N) = 100 + 100(0.05)(3) = 115
5-9
+ relation between FV and
interest rates
+ relation between FV and N
5-10
(-) relation between PV and
interest rates
(-) relation between PV and N
5-11
7-12

So far we are assuming that interest is
compounded yearly (annual compounding).

However, there are many situations where
interest is due 2,4, 12, 26, 52, 365 times a
year.
 In general, bonds pay interest semiannually.
 Most mortgages, student, and auto loans require
payments to be monthly.
5-13

A CD that offers a state rate of 10% compounded annually is
different from a CD that offers a state rate of 10% compounded
semiannually.

The 10% is called the nominal rate (INOM), quoted, stated, or
annual percentage rate (APR) since it ignores compounding
effects.
 It is the rate that is stated by banks, credit card companies, and auto,
student, and mortgage loans.

Periodic rate (IPER): amount of interest charged each period,
e.g. annually, monthly, quarterly, daily, and/or continuously.
 IPER = INOM/M, where M is the number of compounding periods per
year.
 M = 4 for quarterly, M = 12 for monthly , and M = continuous
compounding
5-14

We can go on compounding every hour, minute,
and second  continuous compounding
FV5  PV (e )  100(e
IN
0.10( 5 )
)  $164.87
5-15

Thus, if $1,649 is due in 10 years, and if the
appropriate continuous discount rate, is
5%, then the present value of this future
payment is $1,000:
PV10
FV
 IN

 FV (e
)
IN
(e )
 1,649(e 0.05(10) )  $1000.1690
5-16

You have $100 and an investment horizon
of 3 year and have 2 choices:
 CD that offers a state rate of 10% annually
 CD that offers a state rate of 10% semiannually.

The first choice will offer you a FV of
0
10%
1
2
100
3
133.10
Annually: FV3 = $100(1.10)3 = $133.10
5-17

As for the second choice (semiannually
compounding):

There must be 2 main adjustments:
▪ Covert the stated interests to periodic rate
▪ Convert the number of year into number of periods.
0
0
100
5%
1
1
2
3
2
4
5
3
6
134.01
Semiannually: FV6 = $100(1.05)6 = $134.01
5-18

Thus, the FV of a lump sum will be larger if
compounded is more often, holding the
stated I% constant
 Because interest is earned on interest more
often.
 Will the PV of a lump sum be larger
or smaller if compounded more often,
holding the stated I% constant? Why?
5-19

PV of a lump sum will be lower when
interest rate is discounted more frequently.
 This is because interest is discounted sooner
and thus there will be more discounting.

PV of 100 at 10% annually for 3 year is

PV of 100 at 10% semiannually for 3 year is
5-20
7-21

In general, different compounding is used by
different investments.

However, we cannot compare between these
investments until we put them on a common basis.
 We cannot compare a CD that offers 10% annually with
that that offers it semiannually or quarterly
 use the Effective Annual Rate (EAR)

(EAR or EFF%): the annual rate of interest actually
(truly)being earned, accounting for compounding.
5-22

EFF% for 10% semiannual interest
 EFF%= (1 + INOM/M)M – 1
= (1 + 0.10/2)2 – 1 = 10.25%
 Excel: =EFFECT(nominal_rate,npery)
=EFFECT(.10,2)
 Should be indifferent between receiving 10.25%
annual interest and receiving 10% interest,
compounded semiannually.
5-23
Nominal
Rate
Effective Annual Rate when compounded
Yearly
Semiannually
Quarterly
Monthly
Daily
Continuously
1%
1%
1.0025%
1.0038%
1.0046%
1.0050%
1.0050%
2%
2%
2.0100%
2.0151%
2.0184%
2.0201%
2.0201%
3%
3%
3.0225%
3.0339%
3.0416%
3.0453%
3.0455%
4%
4%
4.0400%
4.0604%
4.0742%
4.0808%
4.0811%
5%
5%
5.0625%
5.0945%
5.1162%
5.1267%
5.1271%
6%
6%
6.0900%
6.1364%
6.1678%
6.1831%
6.1837%
8%
8%
8.1600%
8.2432%
8.3000%
8.3278%
8.3287%
10%
10%
10.2500%
10.3813%
10.4713% 10.5156%
10.5171%
15%
15%
15.5625%
15.8650%
16.0755% 16.1798%
16.1834%
25%
25%
26.5625%
27.4429%
28.0732% 28.3916%
28.4025%

INOM: Written into contracts, quoted by banks and brokers.
Not used in calculations or shown on time lines.

IPER: Used in calculations and shown on time lines.

If M = 1  INOM = IPER = EAR = [1+(Inom/1].

EAR: Used to compare returns on investments with
different payments per year. Used in calculations when
annuity payments don’t match compounding periods.
 For example: interest rate of 10% is compounded semiannually,
but payments of annuity are occurring annually.
5-25
 I nom 
(1  EAR)  1 

M 

M
1


M
I
(ARP)  M (1  EAR)
-1


nom
265-26

Suppose you want to earn an effective rate
of 12% and you are looking at an account
that compounds on a monthly basis. What
APR must they pay?

APR  12 (1  .12)
1/12

 1  .113865515
or 11.39%
27