Time Value of Money

Chapter 5

© 2003 South-Western/Thomson Learning

2

Time is Money

$100 in your hand today is worth more than $100 in one year Money earns interest

• The higher the interest, the faster your money grows

3

Time is Money

Present Value The amount that must be deposited today to

have a future sum at a certain interest rate The discounted value of a sum is its present

value

4

Outline of Approach

Deal with four different types of problems Amount

• Present value• Future value

Annuity• Present value• Future value

5

Outline of Approach

Mathematics For each type of problem an equation will be

presented Time lines

Graphic portrayal of a time value problem

0 1 2

• Helps with complicated problems

6

Amount Problems—Future Value The future value (FV) of an amount

How much a sum of money placed at interest (k) will grow into in some period of time

• If the time period is one year• FV1 = PV + kPV or FV1 = PV(1+k)

• If the time period is two years• FV2 = FV1 + kFV1 or FV2 = PV(1+k)2

• If the time period is generalized to n years• FVn = PV(1+k) n

7

Amount Problems—Future Value

The (1 + k)n depends on Size of k and n

• Can develop a table depicting different values of n and k and the proper value of (1 + k)n

• Can then use a more convenient formula• FVn = PV [FVFk,n]

These values can be looked up in an interest

factor table.

8

Other Issues

Problem-Solving Techniques Three of four variables are given

• We solve for the fourth

The Opportunity Cost Rate The opportunity cost of a resource is the

benefit that would have been available from its next best use

9

Financial Calculators

Work directly with equations How to use a typical financial calculator in time

value Five time value keys

• Use either four or five keys

Some calculators distinguish between inflows and outflows

• If a PV is entered as positive the computed FV is negative

10

The Expression for the Present Value of an Amount

The future and present values factors are reciprocals Either equation can be used to solve any amount

problems

n

n

n n

Interest Factor

FV PV 1+k

Solve for PV

1PV = FV

1 k

k,nk,n

1FVF

PVF Solving for k or n involves

searching a table.

11

Annuity Problems

Annuity A finite series of equal payments separated

by equal time intervals• Ordinary annuities

• Payments occur at the end of the time periods

• Annuity due• Payments occur at the beginning of the time periods

12

The Future Value of an Annuity—Developing a Formula

Future value of an annuity The sum, at its end, of all payments and all

interest if each payment is deposited when received

13

The Future Value of an Annuity—Developing a Formula

Thus, for a 3-year annuity, the formula is

FVFAk,n

0 1 2

0 1 2 n -1

n

nn i

ni=1

FVA = PMT 1+k PMT 1+k PMT 1+k

Generalizing the Expression:

FVA = PMT 1+k PMT 1+k PMT 1+k PMT 1+k

which can be written more conveniently as:

FVA PMT 1+k

Factoring PMT outside the summation, we

n

n i

ni=1

obtain:

FVA PMT 1+k

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The Future Value of an Annuity—Solving Problems

There are four variables in the future value of an annuity equation The future value of the annuity itself The payment The interest rate The number of periods

• Helps to draw a time line

15

The Sinking Fund Problem

Companies borrow money by issuing bonds for lengthy time periods No repayment of principal is made during the

bonds’ lives• Principal is repaid at maturity in a lump sum

• A sinking fund provides cash to pay off a bond’s principal at maturity

• Problem is to determine the periodic deposit to have the needed amount at the bond’s maturity—a future value of an annuity problem

16

Compound Interest and Non-Annual Compounding

Compounding Earning interest on interest

Compounding periods Interest is usually compounded annually,

semiannually, quarterly or monthly• Interest rates are quoted by stating the nominal

rate followed by the compounding period

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The Effective Annual Rate

Effective annual rate (EAR) The annually compounded rate that pays the

same interest as a lower rate compounded more frequently

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The Effective Annual Rate

EAR can be calculated for any compounding period using the following formula:

m

nominalEAR - 1k

1 m

Effect of more frequent compounding is greater at higher interest rates

19

The Effective Annual Rate

The APR and EAR Annual percentage rate (APR)

• Is actually the nominal rate and is less than the EAR

Compounding Periods and the Time Value Formulas Time periods must be compounding periods Interest rate must be the rate for a single

compounding period• For instance, with a quarterly compounding period the

knominal must be divided by 4 and the n must be multiplied by 4

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The Present Value of an Annuity—Developing a Formula

Present value of an annuity Sum of all of the annuity’s payments

• Easier to develop a formula than to do all the calculations individually

2 3

1 2 3

1 2 n

PMT PMT PMTPVA =

1+k 1+k 1+k

which can also be written as:

PVA = PMT 1+k PMT 1+k PMT 1+k

Generalized for any number of periods:

PVA = PMT 1+k PMT 1+k PMT 1+k

Factoring PMT and using summation, we o

n

i

i=1

btain:

PVA PMT 1+k

PVFAk,n

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The Present Value of an Annuity—Solving Problems

There are four variables in the present value of an annuity equation The present value of the annuity itself The payment The interest rate The number of periods

• Problem usually presents 3 of the 4 variables

22

Spreadsheet Solutions

Time value problems can be solved on a spreadsheet such as Microsoft Excel™ or Lotus 1-2-3™

To solve for: FV use =FV(k, n, PMT, PV) PV use =PV(k, n, PMT, FV) K use =RATE(n, PMT, PV, FV) N use =NPER(k, PMT, PV, FV) PMT use =PMT(k, n, PV, FV)

Select the function for the unknown

variable, place the known variables in

the proper order within the

parentheses, and input 0 for the

unknown variable.

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Spreadsheet Solutions

Complications Interest rates in entered as decimals, not

percentages Of the three cash variables (FV, PMT or PV)

• One is always zero• The other two must be of the opposite sign

• Reflects inflows (+) versus outflows (-)

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Amortized Loans

An amortized loan’s principal is paid off regularly over its life Generally structured so that a constant

payment is made periodically• Represents the present value of an annuity

25

Loan Amortization Schedules

Detail the interest and principal in each loan payment

Show the beginning and ending balances of unpaid principal for each period

Need to know Loan amount (PVA) Payment (PMT) Periodic interest rate (k)

26

Mortgage Loans

Mortgage loans (AKA: mortgages) Loans used to buy real estate

Often the largest single financial transaction in an average person’s life Typically an amortized loan over 30 years

• During the early years of the mortgage nearly all the payment goes toward paying interest

• This reverses toward the end of the mortgage

27

Mortgage Loans

Implications of mortgage payment pattern Early mortgage payments provide a large tax

savings which reduces the effective cost of a loan

Halfway through a mortgage’s life half of the loan has not been paid off

Long-term loans like mortgages result in large total interest amounts over the life of the loan

28

The Annuity Due

In an annuity due payments occur at the beginning of each period

The future value of an annuity due Because each payment is received one

period earlier• It spends one period longer in the bank earning

interest

n -1

n

n k,n

FVAd = PMT + PMT 1+k PMT 1+k 1 k

which written with the interest factor becomes:

FVAd PMT FVFA 1 k

29

The Annuity Due

The present value of an annuity due Formula

k,nPVAd PMT PVFA 1 k

Recognizing types of annuity problems Always represent a stream of equal payments Always involve some kind of a transaction at one

end of the stream of payments• End of stream—future value of an annuity• Beginning of stream—present value of an annuity

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Perpetuities

A perpetuity is a stream of regular payments that goes on forever An infinite annuity

Future value of a perpetuity Makes no sense because there is no end point

Present value of a perpetuity A diminishing series of numbers

• Each payment’s present value is smaller than the one before

p

PMTPV

k

31

Continuous Compounding

Compounding periods can be shorter than a day As the time periods become infinitesimally

short, interest is said to be compounded continuously

To determine the future value of a continuously compounded value:

knnFV PV e

32

Multipart Problems

Time value problems are often combined due to complex nature of real situations A time line portrayal can be critical to

keeping things straight

33

Uneven Streams and Imbedded Annuities Many real world problems have sequences of

uneven cash flows These are NOT annuities

• For example, if you were asked to determine the present value of the following stream of cash flows

$100 $200 $300

Must discount each cash flow individually Not really a problem when attempting to determine either a

present or future value• Becomes a problem when attempting to determine an interest rate

34

Calculator Solutions for Uneven Streams

Financial calculators and spreadsheets have the ability to handle uneven streams with a limited number of payments

Generally programmed to find the present value of the streams or the k that will equate a present value to the stream

35

Imbedded Annuities

Sometimes uneven streams of cash flows will have annuities embedded within them We can use the annuity formula to calculate

the present or future value of that portion of the problem