Slides developed by:Pamela L. Hall, Western Washington University

Time Value of Money

Chapter 5

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

Q: How much would $1,000 promised in one year be worth today if the bank paid 5% interest?

A: $952.38. If we deposited $952.38 after one year we would have earned $47.62 ($952.38 × .05) in interest. Thus, our future value would be $952.38 + $47.62 = $1,000.

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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.Q: If we deposited $438 at 6% interest

for five years, how much would we have?

A: FV5 = $438(1.06)5 = $438(1.3382) = $586.13Exa

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Table 5-1: The Future Value Factor for k and nFVFk,n = (1+k)n

5

6%

1.3382

9

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

10

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

11

Financial Calculators

N

I/Y

PMT

FV

PV

Number of time periods

Interest rate (%)

Future Value

Present Value

Payment

Basic Calculator Keys

12

Financial Calculators

N

I/Y

PMT

PV

FV

1

6

0

5000

4,716.98 Answer

Q: What is the present value of $5,000 received in one year if interest rates are 6%?

A: Input the following values on the calculator and compute the PV:

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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.

14

The Expression for the Present Value of an Amount—Example

N

PV

PMT

FV

3

-850

0

983.96

5.0I/Y Answer

Or you could look up the

interest factor of 0.8639 [850 983.96] in the

proper table with n=3.

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3

3

3

$983.96 = $850 (1 + i)

1.1576 = (1 + i)

1.1576 1 + i

1.0499 = 1 + i

0.0499 = i or i = 4.99%

Q: What interest rate will grow $850 into $983.96 in three years?

A: Using the FV equation of FV = PV (1 + i)n, the answer is:

15

The Expression for the Present Value of an Amount—Example

Q: How long does it take money invested at 14% to double?

A: The future value must be twice the present value, so make up some values for present and future values that meet that criterion, such as $1 doubling to $2.

N

PV

PMT

FV

5.29

-1

0

2

14I/Y

Answer

Or you could look up the

interest factor of 2.00 [2 1] in the proper

table with k=14%.

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n

n

$2 = $1 (1.14)

$2 = (1.14)

log 2 = n log 1.14

0.3010 = 0.0569n

5.29 = n

16

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

17

Figure 5.1: Ordinary Annuity

18

Figure 5.2: Annuity Due

19

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

20

Figure 5.4: FV of a Three-Year Ordinary Annuity

21

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

22

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

23

The Future Value of an Annuity—Solving Problems—Example

Q: The Brock Corporation owns the patent to an industrial process and receives license fees of $100,000 a year on a 10-year contract for its use. Management plans to invest each payment until the end of the contract to provide funds for development of a new process at that time. If the invested money is expected to earn 7%, how much will Brock have after the last payment is received?

A: Using the FVAn = PMT[FVFAk,n] equation and looking up the interest factor of the annuity at an n of 10 and a k of 7, we obtain an interest factor of 13.8164. This gives up a FVA10 of $100,000[13.8164] = $1,381,640.

FV

N

PMT

I/Y

1,381,645

10

100000

7

0PV

Answer

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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

25

The Sinking Fund Problem –Example

Q: The Greenville Company issued bonds totaling $15 million for 30 years. The bond agreement specifies that a sinking fund must be maintained after 10 years, which will retire the bonds at maturity. Although no one can accurately predict interest rates, Greenville’s bank has estimated that a yield of 6% on deposited funds is realistic for long-term planning. How much should Greenville plan to deposit each year to be able to retire the bonds with the money put aside?

A: The time period of the annuity is the last 20 years of the bond issue’s life. Input the following keystrokes into your calculator.

PMT

N

FV

I/Y

407,768.35

20

15,000,000

6

0PV

Answer

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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

27

Figure 5.5:The Effect of Compound Interest

28

The Effective Annual Rate

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

same interest as a lower rate compounded more frequently

29

The Effective Annual Rate—Example

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Q: If 12% is compounded monthly, what annually compounded interest rate will get a depositor the same interest?

A: If your initial deposit were $100, you would have $112.68 after one year of 12% interest compounded monthly. Thus, an annually compounded rate of 12.68% [($112.68 $100) – 1] would have to be earned.

30

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

31

Table 5.3

32

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

33

The Effective Annual Rate – Example

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PMT

N

FV

I/Y

431.22

30

15,000

1

0PV

Answer

Q: You want to buy a car costing $15,000 in 2½ years. You plan to save the money by making equal monthly deposits in your bank account, which pays 12% compounded monthly. How much must you deposit each month?

A: This is a future value of an annuity problem with a 1% monthly interest rate and a 30-month time period. Input the following keystrokes into your calculator.

34

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

35

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

36

The Present Value of an Annuity—Solving Problems–Example

Q: The Shipson Company has just sold a large machine to Baltimore Inc. on an installment contract. The contract calls for Baltimore to make payments of $5,000 every six months (semiannually) for 10 years. Shipson would like its cash now and asks its bank to discount the contract and pay it the present (discounted) value. Baltimore is a good credit risk, so the bank is willing to discount the contract at 14% compounded semiannually. How much should Shipson receive?

A: The contract represents an annuity with payments of $5,000. Adjust the interest rate and number of periods for semiannual compounding and solve for the present value of the annuity.

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PV

N

FV

I/Y

52,970.07

20

0

7

5000PMT

Answer

This can also be calculated using the PVA formula of PVA = PMT[PVFAk, n] with an n of 20 and a k of 7%,

resulting in PVA = $5,000[10.594] = $52,970.

37

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.

38

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 (-)

39

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

40

Amortized Loans—Example E

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PMT

N

PV

I/Y

293.75

48

10,000

1.5

0FV

Answer

This can also be calculated using the PVA formula of PVA = PMT[PVFAk, n] with an n of 48 and a k of 1.5%,

resulting in $10,000 = PMT[34.0426] = $293.75.

Q: Suppose you borrow $10,000 over four years at 18% compounded monthly repayable in monthly installments. How much is your loan payment?

A: Adjust your interest rate and number of periods for monthly compounding and input the following keystrokes into your calculator.

41

Amortized Loans—Example

PV

N

FV

I/Y

15,053.75

36

0

1

500PMT

Answer

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This can also be calculated using the PVA formula of PVA = PMT[PVFAk, n] with an n of 36 and a k of 1%,

resulting in PVA = $500[30.1075] = $15,053.75.

Q: Suppose you want to buy a car and can afford to make payments of $500 a month. The bank makes three-year car loans at 12% compounded monthly. How much can you borrow toward a new car?

A: Adjust your k and n for monthly compounding and input the following calculator keystrokes.

42

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)

43

Loan Amortization Schedules—Example

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Q: Develop an amortization schedule for the loan demonstrated in Example 5.12.

Note that the Interest portion of the payment is decreasing

while the Principal portion is increasing.

44

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

45

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

46

Mortgage Loans—Example E

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PMT

N

FV

I/Y

1,015.65

360

0

.5979

150,000PV

Answer$150,944Net interest cost

$64,690Tax Savings @ 30%

$215,634Total Interest

$150,000- Original Loan

$365,634Total payments

360X number of payments

$1,015.65Monthly payment

Q: Calculate the monthly payment for a 30-year 7.175% mortgage of $150,000. Also calculate the total interest paid over the life of the loan.

A: Adjust the n and k for monthly compounding and input the following calculator keystrokes.

47

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

48

The Annuity Due—Example E

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FV

N

PMT

I/Y

3,020,099 x 1.02 = 3,080,501

40

50,000

2

0PV

Answer

NOTE: Advanced calculators allow you to

switch from END (ordinary annuity) to BEGIN (annuity

due) mode.

Q: The Baxter Corporation started making sinking fund deposits of $50,000 per quarter today. Baxter’s bank pays 8% compounded quarterly, and the payments will be made for 10 years. What will the fund be worth at the end of that time?

A: Adjust the k and n for quarterly compounding and input the following calculator keystrokes.

49

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

50

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

51

Perpetuities—Example E

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p

PMT $5PV $250

k 0.02 You may also work this by inputting a

large n into your calculator (to simulate infinity), as shown below.

PV

N

PMT

I/Y

250

999

5

2

0FV

Answer

Q: The Longhorn Corporation issues a security that promises to pay its holder $5 per quarter indefinitely. Money markets are such that investors can earn about 8% compounded quarterly on their money. How much can Longhorn sell this special security for?

A: Convert the k to a quarterly k and plug the values into the equation.

52

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

53

Continuous Compounding—Example

.065 3.5knnFV PV e $5,000 e $5,000 1.2255457 $6,277.29

A: To determine the EAR of 12% compounded continuously, find the future value of $100 compounded continuously in one year, then calculate the annual return.

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kn .12nFV PV e $100 e $100 1.1275 $112.75

$112.75 $100Annual Return = 12.75%

$100

NOTE: Some advanced calculators have a function to solve for the

EAR with continuous compounding.

Q: The First National Bank of Charleston is offering continuously compounded interest on savings deposits. Such an offering is generally more of a promotional feature than anything else. If you deposit $5,000 at 6½% compounded continuously and leave it in the bank for 3½ years, how much will you have? Also, what is the equivalent annual rate (EAR) of 12% compounded continuously?

A: To determine the future value of $5,000, plug the appropriate values into the equation.

54

Table 5.5: Time Value Formulas

55

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

56

Multipart Problems—Example E

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Q:Exeter Inc. has $75,000 invested in securities that earn a return of 16% compounded quarterly. The company is developing a new product that it plans to launch in two years at a cost of $500,000. Exeter’s cash flow is good now but may not be later, so management would like to bank money from now until the launch to be sure of having the $500,000 in hand at that time. The money currently invested in securities can be used to provide part of the launch fund. Exeter’s bank has offered an account that will pay 12% compounded monthly. How much should Exeter deposit with the bank each month to have enough reserved for the product launch?

57

Multipart Problems—Example E

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A: Two things are happening in this problem Exeter is saving money every month (an annuity) and The money invested in securities (an amount) is growing

independently at interest

We have two problems that must be handled sequentially. First, we need to find the future value of $75,000 Then, we subtract that future value from $500,000 to

determine how much extra Exeter needs to save via the annuity

Then, we’ll solve a future value of an annuity problem for the payment

58

Multipart Problems—Example E

xam

ple

To find the future value of the $75,000…

PV

N

PMT

I/Y

102,643

8

0

4

75,000FV

Answer

To find the savings annuity value

PMT

N

PV

I/Y

14,731

24

0

1

397,355FV

Answer

$500,000 - $102,645= $397,355

59

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

60

Uneven Streams and Imbedded Annuities

$100 $200 $300

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A: We’ll start with a guess of 12% and discount each amount separately at that rate.

This value is too low, so we need to select a lower interest rate. Using 11% gives us

$471.77. The answer is between 8% and 9%.

1 k,1 2 k,2 3 k,3

12,1 12,2 12,3

PV = FV PVF FV PVF FV PVF

$100 PVF $200 PVF $300 PVF

$100 .8929 $2

$462.2

00 .7972 $300 .71 8

7

1

Q:Calculate the interest rate at which the present value of the stream of payments shown below is $500.

61

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

62

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