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Chapter 5: Discounted Cash Flow Valuation

& Effective Annual Rates

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FV Multiple Cash Flows

1

1. Suppose you invest $500 in a Fidelity Spartan mutualfund today and $600 in one year. If you expect that the

fund will pay 9% annually, how much will you have in two

years?

How much will you have in 5 years if you make no furtherdeposits?

There are 2 ways to solve the problem.

What are the two ways?

2. Suppose you plan to deposit $100 into an account in oneyear 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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PV Multiple Uneven Cash Flows

2

1. You are considering an investment that will pay you $200in one year, $400 in two years, $600 in three years, and

$800 in four years. If you want to earn 12% on your money,

how much would you be willing to pay?

2. Your broker calls you and tells you that he has this great

investment opportunity. If you invest $100 today, you will

receive $40 in one year and $75 in two years. If you require

a 15% return on investments of this risk, should you takethe investment?

What is the highest price you would be willing to

pay?

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Annuities and Perpetuities

3

Annuityfinite series of equal payments 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 equal payments

Perpetuity: PV = C / r

Annuities:

r

rCAFV

t 1)1(

r

tr

CAPV)1(

11

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AnnuitiesSweepstakes Winnings

4

1. Suppose you win the Publishers Clearinghouse $10 millionsweepstakes. The money is paid in equal annual

installments of $333,333.33 over 30 years beginning one

year from today. If the appropriate discount rate is 5%, how

much is the sweepstakes actually worth today?

a. Suppose there is a lump sum option to receive $5 million

dollars today. Will you choose the installments or lump sum?

2. Assume that you choose the installments and invest each

installment at 5%. How much will you have at the end of 30

years?

a. Suppose instead that you chose the lump sum. How much

will you have at the end of 30 years if you invest at 5%?

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Annuities: Finding the Payment (C) -

5

1. Suppose you want buy a new car that costs $20,000. Youcan borrow at 8% per year, compounded monthly. If you

take a 4 year loan, what is your monthly payment?

What if you wanted to borrow more money, but this is

the maximum payment you can afford. What changescan be made to the loan to make this possible?

2. Subsidized Stafford. You will have a accumulated $30,000

in student loans by the time you graduate in 2010. The

interest rate on the loan is 6.8%. What will be your monthlypayment for a term of 30 years?

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Finding t the number of payments

You will not need to solve for t or r in thisclass

Finding r the interest rate

r can not be solved directly, so you must usea financial calculator, a computer, or trial and

error.

)1ln(

)/(11ln

r

CPVrt

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Annuity DueDefinition and Example

7

An ordinary annuity assumes payments begin one periodfrom today (at t=1)

An annuity due is an annuity with cash flows that begin

today (at t=0)

You are saving for a new house and you put threepayments of $10,000 per year in an account paying 8%.

The first payment is made today. How much will you have

at the end of 3 years?

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Annuity Due Timeline

8

0 1 2 3

10K 10K 10K

10K 10K 10K

OrdinaryAnnuity

AnnuityDue

Notice: Because the payments begin oneperiod earlier, the annuity due cash flowearn an extra period of interest. That is whywe multiply our answer by 1+r.

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Practice Problem

You are going to lease a car and can afford to pay $350per month for a 3 year lease. The interest rate is 7% and

your first payment is to be made today. If you will be

putting down $2000 today, how much are you paying to

lease the car? Alternatively, you can purchase the same

car for $15,000. If you plan to keep the car for only 3

years no matter how you decide to pay for the car, would

you prefer to lease or buy the car?

9

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Perpetuity: Defined and Example

10

Perpetuityan annuity in which the cash flows continueforever

Perpetuity formula: PV = C / r

Most common perpetuities - Preferred Stock which often

pay a dividend in perpetuity A company currently pays preferred shareholders a $0.75

dividend per quarter. If the dividend is expected to

continue forever, and the required rate of return on

stocks of similar risk is 7.5% per year, what is the fair priceof the stock?

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Delayed Cash Flow Stream - Example

11

Suppose that you are offered an investment that will payyou $1000 per year forever beginning 2 years from now.

How much should you pay today if your required return is

15%?

t 0 1 2 3.

1000 1000$

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Delayed Cash Flow Stream

The annuity and perpetuity formulas provide the PV ofthe cash flow stream as of the time period before the first

cash flow is paid.

To find PV of a delayed cash flow stream it is a Two-step

process: Find PV of cash flow stream and then discountPV back to today.

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The Two-Step Process

1. Find the PV of the perpetuity. Cash flow stream begins at t=2 so this gives PV as of t=1

2. Discount the PV of the cash flow stream back to today

13

t 0 1 2 3.

1000 1000$PV=$1000/.15=

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PV Annuity: Saving For Retirement

14

You are offered the opportunity to put some money awayfor retirement. You will receive five annual payments of

$25,000 each beginning in 40 years. How much would you

be willing to invest today if you desire an interest rate of

12%?

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Chapter 5, Part 2: Effective Annual Rates

Comparing Rates: The Effect of Compounding

Is 5% every six months the same as 10% per year?

Which account is better an account that earns 15.5%

compounded quarterly or one that earns 16% compounded

annually?

When number of compounding periods differ you must

determine the Effective Annual Rate (EAR) to compare rates.

The EAR captures the effect of compounding periods

throughout the year.

An Annual Percentage Rate (APR) or quoted rate does not

reflect the number of compounding periods per year.

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EAR - Formula

16

APR is the quoted ratedyou may see Q in place of

APR in this equation

Assume (unless noted otherwise) that the rate given inproblems is the APR (quoted rate).

1

m

APR1EAR

m

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Annual Percentage Rate (APR)

17

This is the ANNUAL rate that is quoted by law

By definition APR = period rate times the number of

periods per year

Consequently, to get the period rate we rearrange the

APR equation: Period rate = APR / number of periods per year

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Number of Periods Per Year

18

Annual - 1

Semiannual2

Quarterly4

Monthly12

Weekly52

Daily - 365

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Computing APRs

19

What is the APR if the monthly rate is .5%?

What is the APR if the semiannual rate is .5%?

What is the monthly rate if the APR is 12% with monthlycompounding?

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Remember

20

You ALWAYS need to make sure that the interest rate and

the time period match.

If you are looking at annual periods, you need an annual rate.

If you are looking at monthly periods, you need a monthly rate.

If you have an APR based on monthly compounding, youhave to use monthly periods for lump sums. (Remember

this when pricing bonds!)

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EARs - Example

21

Suppose you can earn 1% per month on $1 invested

today.

What is the APR?

How much are you effectively earning?

Suppose if you put it in another account, you earn 3% perquarter.

What is the APR?

How much are you effectively earning?

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EARExample 2

22

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?

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APR - Example

24

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?

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Remember

25

Use EAR

When comparing interest rates with different

compounding periods.

Use APR (Quoted rate)

When calculating PV and FV.Or If you use EAR when calculating PV and FV, use EAR (do

not divide by number of periods per year) and use t=the

number of years.

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Homework/Practice Problems

Read Chapter 5

Part I: Do 5.1 to 5.5; 2, 7, 9, 10, 11, 24, 25, 30, 35, 36, 38,

46 (a only), 47, 49

Part II: Do 14, 15, 19, 43