Chapter 6

Discounted Cash Flow Valuation

McGraw-Hill/Irwin Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved.

Key Concepts and Skills

• Be able to compute the future value of multiple cash flows

• Be able to compute the present value of multiple cash flows

• Be able to compute loan payments• Be able to find the interest rate on a loan• Understand how interest rates are quoted• Understand how loans are amortized or paid off

6C-2

Chapter Outline

• Future and Present Values of Multiple Cash Flows

• Valuing Level Cash Flows: Annuities and Perpetuities

• Comparing Rates: The Effect of Compounding• Loan Types and Loan Amortization

6C-3

Multiple Cash Flows – FV Example 1

• Suppose you invest $500 in a mutual fund today and $600 in one year. If the fund pays 9% annually, how much will you have in two years?

– FV2 = 500(1.09)2 + 600(1.09)1 = 594.05 + 654.00 = 1,248.05

6C-4

Multiple Cash Flows – FV Example 1 Continued

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

• First way:– FV5 = 500(1.09)5 + 600(1.09)4 = 769.31 + 846.95 = 1,616.26

• Second way – use value at year 2:– FV5 = FV2 (1.09)3 = 1,248.05(1.09)3 =1,616.26

6C-5

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%?

– FV5 = 100 (1.08)4 + 300(1.08)2 = 136.05 + 349.92 = 485.97

6C-6

Multiple Cash Flows – PV Example

• You are considering an investment that will pay you $1,000 in one year, $2,000 in two years and $3,000 in three years. If you want to earn 10% on your money, how much would you be willing to pay?

– PV = 1,000 / (1.10)1 + 2,000 / (1.10)2 +3,000 / (1.10)3 = 4,815.93

6C-7

Timeline

0 1 2 3

1,000 2,000 3,000909.09

1,652.89

2,253.94

4,815.93

6C-8

Decisions Based on Discounted Cash Flows

• 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 take the investment?

– PV = -100 + 40 / (1.15)1 + 75 / (1.15)2 = -8.51

– No, the investment costs more than what you would be willing to pay.

6C-9

Saving For Retirement

• You are offered the opportunity to put some money away for 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%?

– PV = 25,000 / (1.12)40 + 25,000 / (1.12)41 +25,000 / (1.12)42 + 25,000 / (1.12)43 +25,000 / (1.12)44 = 1,084.71

6C-10

Saving For Retirement Timeline

0 1 2 … 39 40 41 42 43 44

0 0 0 … 0 25K 25K 25K 25K 25K

Notice that the year 0 cash flow = 0 (0 CF0)

The cash flows in years 1 – 39 are 0 (0 CF1; 39 N)

The cash flows in years 40 – 44 are 25,000 (25,000 CF2; 5 N)

6C-11

Annuities and Perpetuities Defined

• Annuity – finite 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

• Perpetuity – infinite series of equal payments

6C-12

Annuities and Perpetuities – Basic Formulas

Perpetuity with 1st payment at end of the period

PV = C / r

Annuities with 1st payment at end of the period

r

1r)(1CFV

rr)(1

11

CPV

t

t

6C-13

Annuities and the Calculator

• You can use the PMT key on the calculator for the equal payment

• The sign convention still holds

• Ordinary annuity versus annuity due– You can switch your calculator between the two types by

using the 2nd BGN 2nd Set on the TI BA-II Plus – If you see “BGN” or “Begin” in the display of your

calculator, it is set for an annuity due– Most problems are ordinary annuities

6C-14

Annuity – Sweepstakes Example

• Suppose you win the $10 million sweepstakes. The money is paid in equal annual end-of-year installments of $333,333.33 over 30 years. If the appropriate discount rate is 5%, how much is the sweepstakes actually worth today?

Using financial calculator: 30 N; 5 I/Y; 333,333.33 PMT; CPT PV = 5,124,150.29

6C-15

29.150.124,50.05

)05.0(11

1333,333.33

rr)(1

11

CPV30t

Buying a House

• You are ready to buy a house, and you have $20,000 to cover both the down payment and commission costs. Commission costs are estimated to be 4% of the loan value. You have an annual salary of $36,000, and the bank is willing to allow your monthly mortgage payment to be equal to 28% of your monthly income. The interest rate on the loan is 6% per year with monthly compounding (.5% per month) for a 30-year fixed rate loan. How much money will the bank lend you? How much can you offer for the house?

6C-16

Buying a House - Continued

• Bank loan– Monthly income = 36,000 / 12 = 3,000– Maximum mortgage payment = .28(3,000) = 840

• Total Price– Commission costs = .04(140,105) = 5,604– Down payment = 20,000 – 5,604 = 14,396– Total Price = 140,105 + 14,396 = 154,501

6C-17

140,1050.06/120.06/12)(1

11

840r

r)(11

1CPV

12(30)t

Finding 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 wish to make monthly installment payments over a 4-year period, what is your monthly payment?

6C-18

Finding the Number of Payments – Another Example

• Suppose you borrow $2,000 at 5%, and you are going to make annual payments of $734.42. How long will you take to pay off the loan?

6C-19

3t2,0000.05

0.05)(1

11

734.42r

r)(1

11

CPVtt

Finding 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?

6C-20

9%r10,000r/12

r/12)(11

1207.58

r/12r/12)(11

1CPV

6012t

Future Values for Annuities

• Suppose you begin saving for your retirement by depositing $2,000 per year in a savings account. If the interest rate is 7.5%, how much will you have in 40 years?

6C-21

454,513.040.075

10.075)(12,000

r

1r)(1CFV

4 0t

Annuity Due

• You are saving for a new house and you put $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?

6C-22

35,061.12(1.08)0.08

10.08)(110,000r)(1

r

1r)(1CFV

3t

Annuity Due Timeline

0 1 2 3

10000 10000 10000

32,464

35,016.12

6C-23

Table 6.2

6C-24

Growing Annuity

A growing stream of cash flows with a fixed maturity

t

t

r

gC

r

gC

r

CP V

)1(

)1(

)1(

)1(

)1(

1

2

t

r

g

gr

CPV

)1(

)1(1

6C-25

Growing Annuity: Example

A retirement plan offers to pay $20,000 per year for 40 years and increase the annual payment by three-percent each year. What is the present value at retirement if the discount rate is 10 percent?

57.121,265$10.1

03.11

03.10.

000,20$4 0

PV

6C-26

Growing Perpetuity

A growing stream of cash flows that lasts forever

3

2

2 )1(

)1(

)1(

)1(

)1( r

gC

r

gC

r

CP V

gr

CPV

6C-27

Growing Perpetuity - Example

The expected dividend next year is $1.30, and dividends are expected to grow at 5% forever. If the discount rate is 10%, what is the value of this promised dividend stream?

0 0.2 6$0 5.1 0.

3 0.1$

P V

6C-28

Effective Annual Rate (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 the EAR and use that for comparison.

6C-29

Annual Percentage Rate (APR)

• 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

• You should NEVER divide the effective rate by the number of periods per year – it will NOT give you the period rate.

6C-30

Computing APRs

• What is the APR if the monthly rate is 0.5%?– 0.5(12) = 6%

• What is the APR if the semiannual rate is 0.5%?– 0.5(2) = 1%

• What is the monthly rate if the APR is 12% with monthly compounding?– 12 / 12 = 1%

6C-31

Things to Remember

• 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, you have to use monthly periods for lump sums, or adjust the interest rate appropriately if you have payments other than monthly.

6C-32

Computing EARs - Example

• Suppose you can earn 1% per month on $1 invested today.– What is the APR? 1(12) = 12%– How much are you effectively earning?

• FV = 1(1.01)12 = 1.1268• Rate = (1.1268 – 1) / 1 = .1268 = 12.68%

• Suppose you put it in another account and earn 3% per quarter.– What is the APR? 3(4) = 12%– How much are you effectively earning?

• FV = 1(1.03)4 = 1.1255• Rate = (1.1255 – 1) / 1 = .1255 = 12.55%

6C-33

EAR - Formula

1 m

A P R 1 EA R

m

Remember that the APR is the quoted rate, and m is the number of compounding periods per year

6C-34

Making Decisions using 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)365 – 1 = 5.39%

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

• Which account should you choose and why?

6C-35

Making Decisions using EAR Continued

• Let’s verify the choice. Suppose you invest $100 in each account. How much will you have in each account in one year?

– First Account: FV = 100 (1 + .0525/365)365 = 105.39– Second Account:

FV = 100 (1 + .053/2)2 = 105.37

• You have more money in the first account.

6C-36

Computing APRs from EARs If you have an effective rate, how can you

compute the APR? Rearrange the EAR equation and you get:

1 - E A R ) (1 m A P R m

1

6C-37

APR - Example

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?

1 1 .3 9 %o r

8 6 5 5 1 5 21 1 3.1)1 2.1(1 2 1 2/1 A P R

6C-38

Computing Payments with APRs

Suppose you want to buy a new computer system and the store is willing to allow you to make monthly payments. The entire computer system costs $3,500. The loan period is for 2 years, and the interest rate is 16.9% with monthly compounding. What is your monthly payment?

6C-39

1 7 2 .8 8C0 .1 6 9 /1 2

0 .1 6 9 /1 2 )(1

11

Cr

r)(1

11

C3 ,5 0 02 4t

Future Value with Monthly Compounding

Suppose you deposit $50 a month into an account that has an APR of 9%, based on monthly compounding. How much will you have in the account in 35 years?

6C-40

147,089.220.09/12

10.09/12)(150

r

1r)(1CFV

35(12)t

Present Value with Daily Compounding

You need $15,000 in 3 years for a new car. If you can deposit money into an account that pays an APR of 5.5% based on daily compounding, how much would you need to deposit?

FV = PV(1 + r)t

15,000 = PV (1+0.055/365)3(365)

PV = 12,718.56

6C-41

Continuous Compounding

• Sometimes investments or loans are figured based on continuous compounding

EAR = eq – 1

– The e is a special function on the calculator normally denoted by ex

• Example: What is the effective annual rate of 7% compounded continuously?

EAR = e.07 – 1 = .0725 or 7.25%

6C-42

Pure Discount Loans - Example

• Treasury bills are excellent examples of pure discount loans. The principal amount is repaid at some 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?

FV = PV(1 + r)t

10,000 = PV (1+0.07)PV = 9,345.79

6C-43

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 1 – 4: Interest payments of .07(10,000) = 700• Year 5: Interest + principal = 10,700

• This cash flow stream is similar to the cash flows on corporate bonds, and we will talk about them in greater detail later.

6C-44

7,451.474C50,0000.08

0.08)(11

1C

rr)(1

11

CPV10t

Amortized Loan with Fixed Payment - Example

• Each payment covers the interest expense plus reduces principal

• Consider the same 10 year loan with annual payments. The interest rate is 8%, and the principal amount is $50,000.– What is the annual payment? This is an ordinary

annuity.

.

6C-45

Microsoft Excel 97-2003 Worksheet

Amortized Loan with Fixed Payment – Another Example

• Consider a 4 year loan with annual payments. The interest rate is 8%, and the principal amount is $5,000.– What is the annual payment? This is an ordinary

annuity.

6C-46

APR may differ from EAR because of the differentbasis of interest calculation

– flat basis– annual rest basis  – reducing balance basis

Another Reason Why APR May Differ from EAR

A borrower approached a bank for a loan. The bank quoted a rate of 12% but did not tell him the basis of interest calculation. Below are the details of the loan. What is the EAR if flat basis applies? If annual rest basis applies? If reducing balance basis applies?

Loan=$100,000Repay over 3 years in 36 equal installmentsEAR =?

Example

• For interest calculations, the principal of the loan is not reduced by the installment payments

• Interest is charged on the full sum of the principal over 3 yrs.

Monthly payments= [100,000 + (100,000 X 0.12 X 3)] / 36= 3777.78

Flat Basis

EAR =(1+ APR /12)12 - 1 =23.39%

The flat rate of 12% is equivalent to EAR of 23.39%.

2 1 .2 %r1 0 0 ,0 0 0r/1 2

r/1 2 )(1

11

3 ,7 7 7 .7 8r

r)(1

11

CP V1 2 (3 )t

Flat Basis Continued

• For interest calculation, the principal of the loan will only be reduced by the installment amount at the end of every year.

• Interest is charged on balance at beginning of the year.

  Monthly PMT= 41,634.90 / 12

= 3469.58

41,634.90C100,0000.12

0.12)(11

1C

rr)(1

11

CPV3t

Annual Rest Basis

• Based on the monthly payment of 3,469.58, what is the effective rate charged?

APR = 15.06%EAR = (1+APR/12)12 - 1 

= 16.1468%

15.06%r100,000r/12

r/12)(11

13,469.58

rr)(1

11

CPV3(12)t

Annual Rest Basis (Continued)

• For interest rate calculations, the principal of the loan will be reduced as and when the installment payments are made.

 APR = 12%EAR = (1+APR/12)12 - 1 

=12.6825%

This is the best basis for the borrower.

3,321.43C100,0000.12/12

0.12/12)(11

13,777.78

rr)(1

11

CPV12(3)t

Reducing Balance Basis

End of Chapter

6C-54