4-0

Discounted Cash Flow Valuation

Chapter 4

Copyright © 2013 by the McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/Irwin

4-1

Be able to compute the future value and/or present value of a single cash flow or series of cash flows

Be able to compute the return on an investment

Be able to use a financial calculator and/or spreadsheet to solve time value problems

Understand perpetuities and annuities

4-2

4.1 Valuation: The One-Period Case

4.2 The Multiperiod Case

4.3 Compounding Periods

4.4 Simplifications

4.5 Loan Amortization

4.6 What Is a Firm Worth?

4-3

If you were to invest $10,000 at 5-percent interest for one year, your investment would grow to $10,500.

$500 would be interest ($10,000 × .05)

$10,000 is the principal repayment ($10,000 × 1)

$10,500 is the total due. It can be calculated as:

$10,500 = $10,000×(1.05)

The total amount due at the end of the investment is called the Future Value (FV).

4-4

In the one-period case, the formula for FV can be written as:

FV = C0×(1 + r)

Where C0 is cash flow today (time zero), and

r is the appropriate interest rate.

Interest rate – “exchange rate” between earlier money and later money

4-5

If you were to be promised $10,000 due in one year when interest rates are 5-percent, your investment would be worth $9,523.81 in today’s dollars.

05.1

000,10$81.523,9$

The amount that a borrower would need to set aside

today to be able to meet the promised payment of

$10,000 in one year is called the Present Value (PV).

Note that $10,000 = $9,523.81×(1.05),

i.e., FV = PV (1+r)

4-6

In the one-period case, the formula for PV can be written as:

r

CPV

1

1

Where C1 is cash flow at date 1, and

r is the appropriate interest rate,

i.e., PV = FV / (1+r)

4-7

How much is the FV of $1 invested at 10 percent for one year?

FV = $1 x (1.1) = $1.10

How much do we need to invest today at 10 percent to get $1 in one year?

PV = ?

Need to solve PV x (1+ r) =$1,

i.e., PV x (1.1) = $1,

i.e., PV = $1/(1+r) = $1/(1.1) = $.909

Suppose you need $105 in one year. If you can earn 5% annually, how much do you need to invest today?

PV = $105/(1.05) = $100

4-8

The Net Present Value (NPV) of an investment is the present value of the expected cash flows, less the cost of the investment.

Suppose an investment that promises to pay $10,000 in one year is offered for sale for $9,500. Your interest rate is 5%. Should you buy?

4-9

81.23$

81.523,9$500,9$

05.1

000,10$500,9$

NPV

NPV

NPV

The present value of the cash inflow is greater

than the cost. In other words, the Net Present

Value is positive, so the investment should be

purchased.

4-10

In the one-period case, the formula for NPV can be written as:

NPV = –Cost + PV

If we had not undertaken the positive NPV project

considered on the last slide, and instead invested our

$9,500 elsewhere at 5 percent, our FV would be less

than the $10,000 the investment promised, and we

would be worse off in FV terms :

$9,500×(1.05) = $9,975 < $10,000

4-11

The general formula for the future value of an investment over many periods can be written as:

FV = C0×(1 + r)T

Where

C0 is cash flow at date 0,

r is the appropriate interest rate, and

T is the number of periods over which the cash is invested.

4-12

Suppose a stock currently pays a dividend of $1.10, which is expected to grow at 40% per year for the next five years.

What will the dividend be in five years?

FV = C0×(1 + r)T

$5.92 = $1.10×(1.40)5

4-13

Notice that the dividend in year five, $5.92, is considerably higher than the sum of the original dividend plus five increases of 40-percent on the original $1.10 dividend:

$5.92 > $1.10 + 5×[$1.10×.40] = $3.30

This is due to compounding.

4-140 1 2 3 4 5

10.1$

3)40.1(10.1$

02.3$

)40.1(10.1$

54.1$

2)40.1(10.1$

16.2$

5)40.1(10.1$

92.5$

4)40.1(10.1$

23.4$

4-15

4-16

4-17

Suppose you invest the $1,000 at 5% for 5 years.

How much would you have?

FV = $1000 x (1.05)5 = $1276.28

Value with simple interest=(.05)x1,000x5=$250, so future value with simple interest = $1,250

The effect of compounding is small for a small

number of periods, but increases as the number of

periods increases.

The effect of compounding also increases with the

interest rate.

4-18

Suppose you had a relative deposit $10 at 5.5%

interest 200 years ago. How much would the

investment be worth today?

FV = $10 x (1.055)200 = $447,189.84

What is the effect of compounding?

Simple interest = (.055)x$10x200 = $110, so future value with simple interest = $120

Compounding added $447,069.84 to the value of the investment!

4-19

Suppose your company expects to increase unit

sales of widgets by 15% per year for the next 5

years. If you currently sell 3 million widgets in one

year, how many widgets do you expect to sell

during the fifth year?

FV = 3,000,000(1.15)5 = 6,034,072

4-20

How much do I have to invest today to have some amount in the future?

Discounting: The process of going from future values (FVs) to Present Values (PVs)

When we talk about discounting, we mean finding the present value of some future amount.

When we talk about the “value” of something, we are talking about the present value unless we specifically indicate that we want the future value.

4-21

General Formula: FV = PV(1 + r)t

Rearrange to solve for PV = FV / (1 + r)t

Example: Suppose you need $115.76 in three years.

If you can earn 5% annually, how much do you need

to invest today?

PV = $115.76/(1.05)3 = 115.76/1.1576 = $100

4-22

How much would an investor have to set aside today in order to have $20,000 five years from now if the current rate is 15%?

0 1 2 3 4 5

$20,000PV

5)15.1(

000,20$53.943,9$

4-23

Make sure that your calculator can display a large

number of decimals.

- [orange key=2nd][FORMAT]9[ENTER]

Make sure that calculator assumes one payment per

period/per year (this is default)

- Press 1 [2nd] [P/Y]

Make sure it is in end mode (this is default, but can

change to BGN mode w/ [2nd][BGN][ENTER]).

4-24

• Remember to clear the registers (orange key + clr work) before

(and after) each problem

Put a negative sign on cash outflows, positive sign

on cash inflows.

- e.g., a loan:

Payments are negative,

FV is negative (outflows to pay off the loan)

PV is positive (loan inflow)

4-25

Texas Instruments BA-II Plus

FV = future value

PV = present value

I/Y = periodic (annual) interest rate Interest will be compounded for number of periods you enter

in P/Y

Interest is entered as a percent, not a decimal

N = number of periods

Remember to clear the registers (CLR WORK) after each problem

Other calculators are similar in format

4-26

Suppose you need $10,000 in one year for the down

payment on a new car. If you can earn 7% annually, how

much do you need to invest today?

PV = $10,000 / (1.07)1 = $9,345.79

Calculator

1 N

7 I/Y

10,000 FV

CPT PV = -9,345.79

4-27

You want to begin saving for your daughter’s college

education and you estimate that she will need $150,000 in 17

years. If you feel confident that you can earn 8% per year,

how much do you need to invest today?

PV = $150,000 / (1.08)17 = $40,540.34

To use the calculator:

17 N

8 I/Y

150000 FV

CPT PV

4-28

Your parents set up a trust fund for you 10 years ago

that is now worth $19,671.51. If the fund earned 7%

per year, how much did your parents invest?

PV = $19,671.51 / (1.07)10 = $10,000

Calculator

10 N

7 I/Y

19671.51 FV

CPT PV

4-29

Your parents set up a trust fund for you and invest

$20,000 today. How much will the fund be worth in

8 years at 6% per year?

FV = $20,000 (1.06)8 = $31,877

Calculator

8 N

6 I/Y

-20000 PV

CPT FV = 31876.96

4-30

For a given interest rate – the longer the time period,

the lower the present value (ceteris paribus: all else

equal)

What is the present value of $500 to be received

in 5 years? 10 years? The discount rate is 10%

5 years: PV = $500 / (1.1)5 = $310.46

10 years: PV = $500 / (1.1)10 = $192.77

4-31

For a given time period – the higher the interest rate, the

smaller the present value (ceteris paribus)

What is the present value of $500 received in 5 years

if the interest rate is 10%? 15%?

Rate = 10%: PV = $500 / (1.1)5 = $310.46

Rate = 15%; PV = $500 / (1.15)5 = $248.59

4-32

4-33

What is the relationship between present value and future value?

Suppose you need $15,000 in 3 years. If you can earn 6% annually, how much do you need to invest today? PV = 15,000/(1.06)3 = 12,594

If you could invest the money at 8%, would you have to invest more or less than at 6%?

A) More, B) Less, C) The Same, D) Can’t Tell

How much? PV at 8% = 15,000/(1.08)3 = 11,907

So, less by 12594 – 11907 = 687

4-34

Often, we will want to know what the implied interest rate is in an investment

- e.g., you have been offered an investment that doubles your money in 10 years. What is the approximate rate of return on the investment?

Rearrange the basic PV equation and solve for r

FV = PV(1 + r)t

r = (FV / PV)1/t – 1

4-35

You are looking at an investment that will pay

$1,200 in 5 years if you invest $1,000 today. What

is the implied rate of interest?

r = ($1,200 / $1,000)1/5 – 1 = .03714 = 3.714%

Calculator – the sign convention matters!!!

5 N

-1000 PV (you pay $1,000 today)

1200 FV (you receive $1,200 in 5 years)

CPT I/Y = 3.714%

4-36

Suppose you are offered an investment that will

allow you to double your money in 6 years. You

have $10,000 to invest. What is the implied rate of

interest?

r = ($20,000 / $10,000)1/6 – 1 = .122462 = 12.25%

Calculator:

6 N

-10000 PV

20000 FV

CPT I/Y = 12.25%

4-37

I would like to retire in 30 years as a millionaire. If I

have $10,000 today, what rate of return I need to

achieve my goal?

$10,000 = $1,000,000/(1 + r)30

(1+r) 30 = 100

r = 16.59%

Calculator: N =30; FV = 1,000,000;

PV = -10,000; CPT I/Y = 16.59%

4-38

What are some situations in which you might want to compute the implied interest rate?

Suppose you are offered the following investment choices: You can invest $500 today and receive $600 in 5 years.

You can invest the $500 in a bank account paying 4% annually.

What is the implied interest rate for the first choice and which investment should you choose?

r = (600 / 500)1/5 – 1 = 3.714% Calculator: N = 5; PV = -500; FV = 600; CPT I/Y = 3.714%

Note that 4% > 3.714% and the FV of depositing the money in a bank account is $608.326 > 600.

4-39

Start with basic equation and solve for t (remember

your logs)

FV = PV(1 + r)t

(1 + r)t = FV / PV

ln (1 + r)t = ln(FV / PV)

tln (1 + r) = ln(FV / PV)

t = ln(FV / PV) / ln(1 + r)

You can use the financial keys on the calculator as

well.

4-40

Recall formula: t = ln(FV / PV) / ln(1 + r)

You want to purchase a new car and you are willing to pay $20,000. If you can invest at 10% per year and you currently have $15,000, how long will it be before you have enough money to pay cash for the car?

t = ln($20,000 / $15,000) / ln(1.1) = 3.02 years

Calculator: -15000 PV, 20000 FV, 10 I/Y,

CPT N = 3.02

4-41

Suppose you want to buy a new house. You

currently have $15,000 and you figure you need to

have a 10% down payment plus an additional 5% in

closing costs on the remaining balance. If the type

of house you want costs about $150,000 and you

can earn 7.5% per year, how long will it be before

you have enough money for the down payment and

closing costs?

4-42

How much do you need to have in the future?

Compute the number of periods

Using the formula

4-43

How much do you need to have in the future?

Down payment = .1($150,000) = $15,000

Closing costs = .05($150,000 – 15,000) = $6,750

Total needed = $15,000 + 6,750 = $21,750

Compute the number of periods

PV = -15,000

FV = 21,750

I/Y = 7.5

CPT N = 5.14 years

Using the formula, t = ln(FV/PV)/ln(1+r)

t = ln($21,750 / $15,000) / ln(1.075) = 5.14 years

4-44

If we deposit $5,000 today in an account paying 10%, how long does it take to grow to $10,000?

TrCFV )1(0 T)10.1(000,5$000,10$

2000,5$

000,10$)10.1( T

)2ln()10.1ln( T

years 27.70953.0

6931.0

)10.1ln(

)2ln(T

4-45

When might you want to compute the number of periods?

Suppose you want to buy some new furniture for your family room. You currently have $500 and the furniture you want costs $600. If you can earn 6%, how long will you have to wait if you don’t add any additional money?

t = ln(600/500) / ln(1.06) = 3.13 years Calculator: PV = -500; FV = 600; I/Y = 6; CPT N

= 3.13 years

4-46

Consider an investment that pays $200 one year from now, with cash flows increasing by $200 per year through year 4. If the interest rate is 12%, what is the present value of this stream of cash flows?

If the issuer offers this investment for $1,500, should you purchase it?

4-47

First, set your calculator to 1 payment per year i.e., 2ND P/Y=1.000

Then, use the cash flow menu:

C02

C01

F02

F01

CF0

1

200

1

$-67.068

-1,500

400

I

NPV

12

C04

C03

F04

F03 1

600

1

800

4-48

0 1 2 3 4

200 400 600 800178.57

318.88

427.07

508.41

1,432.93

Present Value < Cost OR NPV < 0 → Do Not Purchase

4-49

Back to first e.g. we did in class (not on ppt’s)

Cost=$1,200 in t=0, Returns: $100 in t=1, $100 in t=2, $400 in t=3, $500 in t=4, $500 in t=5

If r=3% → NPV = $232.952 (hence, there was a rounding up error in my handwritten calculations), invest

If r=10% → NPV = $-73.953, do not invest; the alternative investment at 10% is better

4-50

Compounding periods

Within-year compounding:

To this point, we have assumed annual interest rates; however, many projects / investments have different periods. For example, bonds typically pay interest semi-annually, and house loans are on a monthly payment schedule.

Continuous compounding:

We could compound semiannually, quarterly, monthly, daily, hourly, each minute or at every infinitesimal instant (i.e., continuously)

Effective Annual Rate (EAR)

4-51

Compounding an investment m times a year for Tyears provides for future value of wealth:

By setting T=1, we get the formula for compounding over one year.

r is the stated annual interest rate without consideration of compounding. Annual Percentage Rate (APR) is the most common synonym.

Tm

m

rCFV

10

4-52

For example, if you invest $50 for 3 years at 12% compounded semi-annually, your investment will grow to:

93.70$)06.1(50$2

12.150$ 6

32

FV

4-53

You are considering two savings accounts. One pays 5.25%, with daily compounding. The other pays 5.3% with semiannual compounding. Which account should you use to deposit $100?

How much will you have in each 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.

4-54

A reasonable question to ask in the above example is “what is the effective annual rate of interest on that investment?”

The Effective Annual Rate (EAR) of interest is

the annual rate that would give us the same

end-of-investment wealth after 3 years:

93.70$)06.1(50$)2

12.1(50$ 632 FV

93.70$)1(50$ 3 EAR

4-55

Thus, EAR = (FV / PV)1/T – 1 (as in an earlier formula we developed for r)

So, investing at 12.36% compounded annually is the same as investing at 12% compounded semi-annually. Thus, EAR > APR due to compounding.

93.70$)1(50$ 3 EARFV

50$

93.70$)1( 3 EAR

1236.150$

93.70$31

EAR

4-56

Find the Effective Annual Rate (EAR) of an18% APR loan that is compounded monthly.

What we have is a loan with a monthly interestrate rate of 18/12 % = 1½ %.

This is equivalent to a loan with an annualinterest rate of 19.56%.

In other words, EAR = [1 + r/m]m – 1

1956.1)015.1(12

18.11 12

12

m

m

r

4-57

Texas Instruments BA-II Plus FV = future value

PV = present value

I/Y = periodic (annual) interest rate Interest will be compounded for number of periods

you enter in P/Y. Thus, P/Y must equal 1 for the I/Y to be the periodic (annual) rate.

Interest is entered as a percent, not a decimal

N = number of periods

Remember to clear the registers (CLR WORK) after each problem

Other calculators are similar in format

4-58

keys: description:

[2nd] [ICONV] Opens interest rate conversion menu

[↓] [EFF=] [CPT] 19.562

Texas Instruments BAII Plus

[↓][NOM=] 18 [ENTER] Sets 18 APR.

[↑] [C/Y=] 12 [ENTER] Sets 12 payments per year

4-59

Compound at every infinitesimal instant.

The general formula for the future value of an investment compounded continuously over many periods can be written as:

FV = C0×erT

(because lim(1 + r/m)m×T = erT when m converges to infinity), where:

C0 is cash flow at date 0, r is the stated annual interest rate, T is the number of years, and e is the base of the natural logarithms. e is approximately equal to 2.718. ex is a key on your calculator.

4-60

The EAR of a continuously compounded investment is:

EAR = er – 1

For example, say APR on a loan (cc) is 10%. Then EAR = e.1 – 1 = .1051709

4-61

Perpetuity

A constant stream of cash flows that lasts forever

Growing perpetuity

A stream of cash flows that grows at a constant rate forever

Annuity

A stream of constant cash flows that lasts for a fixed number of periods

Growing annuity

A stream of cash flows that grows at a constant rate for a fixed number of periods

4-62

A constant stream of cash flows that lasts forever

0

…1

C

2

C

3

C

32 )1()1()1( r

C

r

C

r

CPV

r

CPV

4-63

PV = C/(1+r)+PV/(1+r), i.e., PV(1-(1/1+r)) = C/(1+r),

i.e., PV = C/r

32 )1()1()1( r

C

r

C

r

CPV

4-64

What is the value of a British consol that promises to pay £15 every year for ever?

The interest rate is 10-percent.

0

…

1

£15

2

£15

3

£15

£15010.

£15PV

4-65

A stream of cash flows that grows at a constant rate forever

0

…

1

C

2

C×(1+g)

3

C ×(1+g)2

3

2

2 )1(

)1(

)1(

)1(

)1( r

gC

r

gC

r

CPV

gr

CPV

4-66

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

…

1

$1.30

2

$1.30×(1.05)

3

$1.30 ×(1.05)2

00.26$05.10.

30.1$

PV

4-67

A constant stream of cash flows with a fixed maturity, i.e., a stream of constant cash flows that lasts for a fixed number of periods

0 1

C

2

C

3

C

Tr

C

r

C

r

C

r

CPV

)1()1()1()1( 32

Trr

CPV

)1(

11

T

C

4-68

If you can afford a $400 monthly car payment, how much car can you afford if interest rates are 7% on 36-month loans?

0 1

$400

2

$400

3

$400

59.954,12$)1207.1(

11

12/07.

400$36

PV

36

$400

4-69

Or use calculator

Press 2ND , P/Y

Enter 12 (for 12 periods per year)

Press ENTER

Press 2ND , QUIT

Enter 36, press N

Enter -400, press PMT

Enter 0, press FV

Enter 7, press I/Y

Press CPT, PV

4-70

A stream of cash flows that grows at a constant rate for a fixed number of periods

0 1

C

T

T

r

gC

r

gC

r

CPV

)1(

)1(

)1(

)1(

)1(

1

2

T

r

g

gr

CPV

)1(

11

2

C×(1+g)

3

C ×(1+g)2

T

C×(1+g)T-1

4-71

A defined-benefit retirement plan offers to pay $20,000 per year for 40 years and increase the annual payment by 3% each year. What is the present value at retirement if the discount rate is 10%?

0 1

$20,000

57.121,265$10.1

03.11

03.10.

000,20$40

PV

2

$20,000×(1.03)

40

$20,000×(1.03)39

4-72

Note that a growing annuity (of C that grows at rate g, interest rate r) has the same present value as an ordinary annuity of C* = C/(1+g) at an interest rate r* = [(1+r)/(1+g)]-1. Intuition: we adjust the interest rate and the cash flow for the growth rate.

Thus, C* = C/(1+g) = $20,000/1.03 = 19,417.476, r* = [(1+r)/(1+g)]-1 = (1.1/1.03)-1 = 6.79612%

(make sure P/Y=1), N=40, PMT=-19,417.476, FV=0, I/Y= 6.79612, PV=265,121.47 (small difference due to rounding up)

4-73

You want to receive $5,000 per month in retirement.

If you can earn .75% per month and you expect to

need the income for 25 years, how much do you

need to have in your account at retirement?

PMT = 5,000; N = 25*12 = 300; I/Y = .75; CPT PV = 595,808.11

4-74

Fred starts saving for retirement at age 25 by saving $100 per month. Joe starts at age 35. Both plan to retire at 65. If their retirement accounts earn 12% per year, how much will Joe have to save per month to have saved the same amount as Fred.

F: 40x12=480 months, J: 30x12=360 months

N=480, PMT=-100, PV=0 (they start with $0), I=12/12=1%, FV=1,176,477.85

N=360, PV=0, I=1%, FV=1,176,477.85, PMT=-336.62

4-75

Principal=original loan amount.

Pure Discount Loans are the simplest form of loan. The borrower receives money today and repays a single lump sum (principal and interest) at a future time.

Interest-Only Loans require an interest payment each period, with full principal due at maturity.

Amortized Loans require repayment of principal over time, in addition to required interest.

4-76

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?

PV = 10,000 / 1.07 = 9,345.79

4-77

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.

4-78

Lender requires the borrower to repay parts of the loan amount over time.

Each payment covers the interest expense; plus, it reduces principal

The process of paying off a loan by making regular principal reductions is called amortizing the loan.

Two types:

Fixed principal payment per period

Fixed payment in total (principal plus interest) per

period

4-79

Consider a $50,000, 10 year loan at 8% interest. The loan agreement requires the firm to pay $5,000 in principal each year plus interest for that year.

Click on the Excel icon to see the amortization table

4-80

Each payment covers the interest expense plus reduces principal

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?

4 N

8 I/Y

5,000 PV

CPT PMT = -1,509.60

Click on the Excel icon to see the amortization table

4-81

An investment is worth the present value of its future cash flows. Since a company is a series of investments,

conceptually, a firm should be worth the present value of the firm’s cash flows.

The tricky part is determining the size, timing, and risk of those cash flows.

4-82

How is the future value of a single cash flow computed?

How is the present value of a series of cash flows computed.

What is the Net Present Value of an investment?

What is an EAR, and how is it computed?

What is a perpetuity? An annuity?