chap05ST

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McGraw-Hill 2004 The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/Irwin

5.1

Chapter Outline

1. Future and Present Values of Multiple Cash

Flows

2. Valuing Level Cash Flows: Annuities and

Perpetuities

3. APR and EAR

4. Different Types of Loans

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5.2

Steps in Time Value of Money

Calculation1. Draw the time line;

2. Identify the problem: is it a single cash flow, annuity,or perpetuity problem?

3. Clarify the question: is it asking for PV, FV, periodicpayment, etc?

4. Find out the right formula or the right keys on thecalculator;

5. Double check how frequently the interest is earned,and make sure the number of period matches with the

period rate.

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5.3

Time Value of Money Calculation

Three groups of formulas

Single cash flow: FV = PV(1 + r)t

Annuity:

Perpetuity: PV = C / r

Constant growth perpetuity: PV=C1/(r-g)

r

rCFV

r

rCPV

t

t

1)1(

)1(

1

1

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5.4

Time Value of Money Calculation

Three sets of calculator keys

Single cash flow: N, I/Y, PV, FV

Annuity: N, I/Y, PV(or FV), PMT

Combination of single cash flow and annuity for

bond evaluation: N, I/Y, PV, PMT, FV

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5.5

1. Multiple Cash Flows

Future Value Example 1

You think you will be able to deposit $4,000 at

the end of each of the next three years in a bank

account paying 8 percent interest. You currently

have $7,000 in the account. How much will you

have in three years? In four years?

0 1 2 3

7000 4000 4000 4000

4

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7/46McGraw-Hill 2004 The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/Irwin

5.6

Multiple Cash Flows

Approaches to find FV in Year 4

0 1 2 3

7000 4000 4000 4000

4

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5.8

Multiple Cash Flows

FV Example 1

Find the value at year 3 of each cash flowand add them together. Today (year 0): FV = 7000(1.08)3= 8,817.98

Year 1: FV = 4,000(1.08)2= 4,665.60

Year 2: FV = 4,000(1.08) = 4,320

Year 3: value = 4,000

Total value in 3 years = 8817.98 + 4665.60 + 4320 +4000 = 21,803.58

Value at year 4 = 21,803.58(1.08) = 23,547.87

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5.9

Multiple Cash Flows

FV Example 2

Suppose you plan to deposit $100 into an account

in one year and $300 into the account in three

years. How much will be in the account in five

years if the interest rate is 8%?

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5.10

Example Timeline

100

0 1 2 3 4 5

300

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5.11

Example Timeline

100

0 1 2 3 4 5

300

136.05

349.92

485.97

FV = 100(1.08)4+ 300(1.08)2

= 136.05 + 349.92 = 485.97

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5.12

Multiple Cash Flows

Present Value Example

You are offered an investment that will pay you

$200 in one year, $400 the next year, $600 the

next year and $800 at the end of the next year.

You can earn 12 percent on very similarinvestments. What is the most you should pay for

this one?

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5.13

Example Timeline

0 1 2 3 4

200 400 600 800178.57

318.88

427.07

508.41

1432.93

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5.14

Multiple Cash Flows

PV Example

Find the PV of each cash flow and add them

Year 1 CF: 200 / (1.12)1= 178.57

Year 2 CF: 400 / (1.12)2= 318.88

Year 3 CF: 600 / (1.12)3= 427.07

Year 4 CF: 800 / (1.12)4= 508.41

Total PV = 178.57 + 318.88 + 427.07 + 508.41 =

1432.93

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5.15

2. Annuities and Perpetuities

Defined

Annuityfinite series of equalpayments that

occur at regular intervals

If the first payment occurs at the end of the period, it

is called an ordinary annuity

If the first payment occurs at the beginning of the

period, it is called an annuity due

Perpetuityinfinite series of equalpayments

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5.16

Annuities and Perpetuities

4000 4000 4000 4000

PV FV

Annuity

0 1 2 3

4000 4000 4000

4

4000

PV

Perpetuity. . .

. . .

4000 4000 40004000

FV-duePV-due

Annuity Due

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5.17

Annuities and PerpetuitiesBasic

Formulas

Ordinary Annuities:

Perpetuity: PV = C / r

r

rCFV

r

rCPV

t

t

1)1(

)1(

11

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5.18

2.1 Ordinary Annuity: Present Value

1 2 3 4 5 . . . t

PV= C/(1+r) + C/(1+r)2+ C/(1+r)3+ + C/(1+r)t

PV= C[1/(1+r) + 1/(1+r)2+ 1/(1+r)3+ + 1/(1+r)t]

11(1 )

tr

PV Cr

C C C C C . . . C

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5.19

Ordinary Annuity: Future Value

0 1 2 3 4 5 . . . t

FV= C(1+r)t-1+ C(1+r)t-2+ C(1+r)t-3+ + C

FV= C[(1+r)t-1+ (1+r)t-2+ (1+r)t-3+ + 1]

(1 ) 1t

rFV Cr

C C C C C . . . C

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5.20

Annuities and the Calculator

You can use the PMT key on the calculator forthe equalpayment

The sign convention still holds

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5.21

Annuity PVExample (5.5)

After carefully going over your budget, you have

determined you can afford to pay $632 per month

towards a new sports car. You call up your local

bank and find out that the going rate is 1 percentper month for 48 months. How much can you

borrow?

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5.22

Annuity PVExample (5.5)

You borrow money TODAY so you need to

compute the present value.

48 N; 1 I/Y; -632 PMT; CPT PV = 23,999.54

($24,000)

Formula:54.999,23

01.

)01.1(

11

63248

PV

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5.23

AnnuityFinding the Payment

Suppose you want to borrow $20,000 for a new car. You

can borrow at 8% per year, compounded monthly (8/12

= .66667% per month). If you take a 4 year loan, what is

your monthly payment? 20,000 = C[11 / 1.006666748] / .0066667

C = 488.26

Calculator:

4(12) = 48 N; 20,000 PV; .66667 I/Y;

CPT PMT = 488.26

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5.24

AnnuityNumber of Payments (5.6)

You ran a little short on your spring break

vacation, so you put $1000 on your credit card.

You can only afford to make the minimum

payment of $20 per month. The interest rate onthe credit card is 1.5 percent per month. How

long will you need to pay off the $1,000.

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5.25

AnnuityNumber of Payments (5.6)

Calculator:

1.5 I/Y; 1000 PV; -20 PMT

CPT N = 93.111 MONTHS = 7.75 years

And this is only if you dont charge anything

more on the card!

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5.26

AnnuityFinding the Rate

Suppose you borrow $10,000 from your parents

to buy a car. You agree to pay $207.58 per

month for 60 months. What is the monthly

interest rate? Sign convention matters!!!

60 N

10,000 PV

-207.58 PMT

CPT I/Y = .75%

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5.27

AnnuityFinding the Rate Without

a Financial Calculator Trial and Error Process

Choose an interest rate and compute the PV of the

payments based on this rate

Compare the computed PV with the actual loan amount If the computed PV > loan amount, then the interest rate

is too low

If the computed PV < loan amount, then the interest rate

is too high

Adjust the rate and repeat the process until the computed

PV and the loan amount are equal

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5.28

Future Values for Annuities

Suppose you begin saving for your retirement by

depositing $2000 per year. If the interest rate is

7.5%, how much will you have in 40 years?

FV = 2000(1.075401)/.075 = 454,513.04

Calculator: (Remember the sign convention!!!)

40 N; 7.5 I/Y; -2000 PMT

CPT FV = 454,513.04

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5.29

2.2 Annuity vs. Annuity Due

0 1 2 3

4000 4000 4000

4

4000

0 1 2 3

4000 4000 4000

4

4000

PV

FV-duePV-due

FV

Annuity

Annuity Due

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5.30

Annuity Due

You are saving for a new house and you put $10,000 peryear in an account paying 8%. The first payment is

made today. How much will you have at the end of 3

years?

2ndBGN, 2ndSet

N=3, PMT=-10,000, I/Y=8

CPT FV=35,061.12

2ndBGN 2ndSet

(Be sure to change it back to

an ordinary annuity)

If in END mode,

N=3, PMT=-10,000,

I/Y=8,

CPT FV=32,464

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5.31

2.3 Perpetuity

Perpetuity formula: PV = C / r

1 2 3 4 5 . . .

PV= C/(1+r) + C/(1+r)2+ C/(1+r)3+

PV= C/r

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5.32

Illustration on Perpetuity Formula

1 2 3 4 5 . . .

PV= C/(1+r) + C/(1+r)2

+ C/(1+r)3

+ PV= C/r

2004 2005 2006 2007

Amount of Cash

$5000 $5000 $5000 . . .

Value of Cash (Interest Rate = 3%)

$166,666.67

C C C C C . . .

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5.33

PerpetuityExample

The preferred stock of Placer Corp. currently

sells for $44.44 per share. The annual dividend of

$4 is fixed. Assuming a constant dividend

forever, what is the rate of return on this stock?

$44.44 = $4 / r

r = 9.0%

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3.AnnualPercentage Rate (APR)

This is the annual rate that is quoted by law

By definition APR = period rate times the

number of periods per year

Example: if quarterly rate=3%, what is APR?

Consequently, to get the period rate we rearrange

the APR equation:

Period rate = APR / number of periods per year

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Effective AnnualRate (EAR)

This is the actual rate paid (or received) after accounting

for compounding that occurs during the year

If you want to compare two alternative investments with

different compounding periods you need to compute theEAR and use that for comparison.

Example: if quarterly rate=3%, what is EAR?

You should NEVER divide the effective rate by the

number of periods per yearit will NOT give you the

period rate

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Calculate EAR

You are looking at two savings accounts. One

pays 5.25%, with daily compounding. The other

pays 5.3% with semiannual compounding.

Which account should you use? First account:

EAR = (1 + .0525/365)3651 = 5.39%

Second account: EAR = (1 + .053/2)21 = 5.37%

Which account should you choose and why?

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Calculate EAR

Lets verify the choice. Suppose you invest $100in each account. How much will you have ineach account in one year?

First Account: Daily rate = .0525 / 365 = .00014383562

FV = 100(1.00014383562)365= 105.39

Second Account:

Semiannual rate = .053/ 2 = .0265 FV = 100(1.0265)2= 105.37

You have more money in the first account.

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4. Different Types of Loans

Amortized

Interest only

Pure discount

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Pure Discount LoansExample 5.11

Treasury bills are excellent examples of purediscount loans. The principal amount is repaid atsome future date, without any periodic interest

payments. If a T-bill promises to repay $10,000 in 12

months and the market interest rate is 7 percent,how much will the bill sell for in the market?

PV = 10,000 / 1.07 = 9345.79

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Interest Only Loan - Example

Consider a 5-year, interest only loan with a 7%

interest rate. The principal amount is $10,000.

Interest is paid annually.

What would the stream of cash flows be?

Years 14: Interest payments of .07(10,000) = 700

Year 5: Interest + principal = 10,700

This cash flow stream is similar to the cash flowson corporate bonds and we will talk about them

in greater detail later.

A i d i h i d

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Amortized Loan with Fixed

Payment - Example Each payment covers the interest expense plus reducesprincipal, such as mortgage payment

Consider a 4 year loan with annual payments. The

interest rate is 8% and the principal amount is $5000. What is the annual payment?

5000 = C[11 / 1.084] / .08; C = 1509.60

Calculator4 N; 8 I/Y; 5000 PV

CPT PMT = -1509.60

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Review Questions

1. Know how to calculate PV and FV of multiple cash

flows.

2. Know how to calculate PV, FV, and PMT of annuities;

Know how to calculate PV and r of perpetuity cashflows.

What is ordinary annuity and what is annuity due?

Can you calculate the FV of perpetuity cash flows?

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Review Questions (cont ..)

3. What is the definition of APR, and EAR? What are thedifferences between APR and EAR?

Know how to compute EAR using APR information.

Which rate should you use to compare alternative

investments?

4. What is a pure discount loan? What is a good exampleof a pure discount loan?

What is an interest only loan? What is a good exampleof an interest only loan?

What is an amortized loan? What is a good example ofan amortized loan?

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