Time Value of Money

Future ValuePresent Value

AnnuitiesDifferent compounding Periods

Adjusting for frequent compoundingEffective Annual Rate (EAR)

Chapter 5

5-1

The Concept of TVM 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

Application of TVM 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

Time Lines

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

CF0 CF1 CF3CF2

0 1 2 3I%

Important Terminology 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

Different Time Lines

5-6

100 100100

0 1 2 3I%

2. 3-year $100 ordinary annuity100

0 1 2I%

1. $100 lump sum (single payments) due in 2 years

100 100100

0 1 2 3I%

3. 3-year $100 annuity due

Annuity: A series of equal

payments at fixed intervals

for a specified # of periods

Different Time Lines 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

5-7

100 -50 75

0 1 2 3I%

-50

4. Uneven cash flow stream (payments are not equal)

Different Time Lines

5-8

100 100 100

0 1 2I%

4. Perpetuities (annuity that has payments that go forever)

∞

How Compounding and Discounting Works

Compounding interest rates is when interest is earned on interest.

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

115.76105 110.25

0 1 2 35%

100= 100(1.05)Interest = $5Amount = $100

= 100(1.05)2

Interest = $5.25Amount= $105

= 100(1.05)3

Interest = $5.5125Amount= $110.25

A Graphic view of the compounding process (FV) (lump sum)

+ relation between FV and interest rates

+ relation between FV and N

5-10

A Graphic view of the discounting process (PV) (lump sum)

(-) relation between PV and interest rates

(-) relation between PV and N

5-11

Go to Spreadsheet

7-12

Different Compounding Periods

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

Different Compounding Periods 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

compounding5-14

Different Compounding Periods

We can go on compounding every hour, minute, and second continuous compounding

5-15

87.164$)(100)( )5(10.05 eePVFV IN

Continuous Discounting

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:

5-16

1690.1000$)(649,1

)()(

)10(05.0

10

e

eFVe

FVPV IN

IN

How to adjust for frequent compounding?

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

5-17

Annually: FV3 = $100(1.10)3 = $133.10

0 1 2 310%

100 133.10

How to adjust for frequent compounding?

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.

5-18Semiannually: FV6 = $100(1.05)6 = $134.01

0 1 2 35% 4 5 6

134.01

1 2 30

100

Differences in FVs when compounding is frequent

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

Will the PV of a lump sum be larger or smaller if compounded more often, holding the stated I% constant? 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

Classification of Interest Rates

7-21

Classifications of Interest Rates 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

Example

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 and Effective InterestNominal Effective Annual Rate when compounded

Rate 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%

When is each rate used?

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

26

Notes on EAR and APR(nominal rate)

5-26

M

M 1 EAR)(1

nomI

1 -

1EAR) (1 M (ARP)

nomI M

27

Example 2

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?

11.39% or

.11386551511/12.12)(112APR