T5.1 Chapter Outline Chapter 5 Introduction to Valuation: The Time Value of Money Chapter Organization 5.1Future Value and Compounding 5.2Present Value

T5.1 Chapter Outline

Chapter 5Introduction to Valuation: The Time Value of Money

Chapter Organization

5.1 Future Value and Compounding

5.2 Present Value and Discounting

5.3 More on Present and Future Values

5.4 Summary and Conclusions

Irwin/McGraw-Hill The McGraw-Hill Companies, Inc. 2000

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Irwin/McGraw-Hill ©The McGraw-Hill Companies, Inc. 2000

Consider the time line below:

PV is the Present Value, that is, the value today.

FV is the Future Value, or the value at a future date.

The number of time periods between the Present Value and the Future Value is represented by “t”.

The rate of interest is called “r”.

All time value questions involve the four values above: PV, FV, r, and t. Given three of them, it is always possible to calculate the fourth.

T5.2 Time Value Terminology

. . .0 1 2 3 t

PV FV


Notice that 1. $110 = $100 (1 + .10) 2. $121 = $110 (1 + .10) = $100 1.10 1.10 =

$100 1.102

3. $133.10 = $121 (1 + .10) = $100 1.10 1.10 1.10

= $100 ________

T5.3 Future Value for a Lump Sum


Notice that 1. $110 = $100 (1 + .10) 2. $121 = $110 (1 + .10) = $100 1.10 1.10 =

$100 1.102

3. $133.10 = $121 (1 + .10) = $100 1.10 1.10 1.10

= $100 (1.10)3

In general, the future value, FVt, of $1 invested today at r% for t periods is

FVt = $1 (1 + r)t

The expression (1 + r)t is the future value interest factor.

T5.3 Future Value for a Lump Sum


T5.4 Chapter 5 Quick Quiz - Part 1 of 5

Q. Deposit $5,000 today in an account paying 12%. How much will you have in 6 years? How much is simple interest? How much is compound interest?

A. Multiply the $5000 by the future value interest

factor:

$5000 (1 + r )t = $5000 ___________

= $5000 1.9738227

= $9869.11

At 12%, the simple interest is .12 $5000 = $_____ per year. After 6 years, this is 6 $600 = $_____ ; the difference between compound and simple interest is thus $_____ - $3600 = $_____



Q. Deposit $5,000 today in an account paying 12%. How much will you have in 6 years? How much is simple interest? How much is compound interest?

A. Multiply the $5000 by the future value interest

factor:

$5000 (1 + r )t = $5000 (1.12)6

= $5000 1.9738227

= $9869.11

At 12%, the simple interest is .12 $5000 = $600 per year. After 6 years, this is 6 $600 = $3600; the difference between compound and simple interest is thus $4869.11 - $3600 = $1269.11


T5.5 Interest on Interest Illustration

Q. You have just won a $1 million jackpot in the state lottery. You can

buy a ten year certificate of deposit which pays 6% compounded

annually. Alternatively, you can give the $1 million to your brother-in-law,

who promises to pay you 6% simple interest annually over the ten year

period. Which alternative will provide you with more money at the

end of ten years?


T5.5 Interest on Interest Illustration

Q. You have just won a $1 million jackpot in the state lottery. You can

buy a ten year certificate of deposit which pays 6% compounded

annually. Alternatively, you can give the $1 million to your brother-in-law,

who promises to pay you 6% simple interest annually over the ten year

period. Which alternative will provide you with more money at the

end of ten years?

A. The future value of the CD is $1 million x (1.06)10 = $1,790,847.70.

The future value of the investment with your brother-in-law, on the

other hand, is $1 million + $1 million (.06)(10) = $1,600,000.

Compounding (or interest on interest), results in incremental

wealth of nearly $191,000. (Of course we haven’t even begun to address

the risk of handing your brother-in-law $1 million!)


T5.6 Future Value of $100 at 10 Percent (Table 5.1)

Beginning Simple Compound Total Ending

Year Amount Interest Interest Interest Earned Amount

1 $100.00 $10.00 $ 0.00 $10.00 $110.00

2 110.00 10.00 1.00 11.00 121.00

3 121.00 10.00 2.10 12.10 133.10

4 133.10 10.00 3.31 13.31 146.41

5 146.41 10.00 4.64 14.64 161.05

Totals $50.00 $ 11.05 $ 61.05



Want to be a millionaire? No problem! Suppose you are currently 21 years old, and can earn 10 percent on your money. How much must you invest today in order to accumulate $1 million by the time you reach age 65?


T5.7 Chapter 5 Quick Quiz - Part 2 of 5 (concluded)

First define the variables:

FV = $1 million r = 10 percent

t = 65 - 21 = 44 years PV = ?

Set this up as a future value equation and solve for the present value:

$1 million = PV (1.10)44

PV = $1 million/(1.10) 44 = $15,091.

Of course, we’ve ignored taxes and other complications, but stay tuned - right now you need to figure out where to get $15,000!


Q. Suppose you need $20,000 in three years to pay your college tuition. If you can earn 8% on your money, how much do you need today?

A. Here we know the future value is $20,000, the rate (8%), and the number of periods (3). What is the unknown present amount (i.e., the present value)? From before:

FVt = PV (1 + r )t

$20,000 = PV __________

Rearranging:

PV = $20,000/(1.08)3

= $ ________

T5.8 Present Value for a Lump Sum


Q. Suppose you need $20,000 in three years to pay your college tuition. If you can earn 8% on your money, how much do you need today?

A. Here we know the future value is $20,000, the rate (8%), and the number of periods (3). What is the unknown present amount (i.e., the present value)? From before:

FVt = PV x (1 + r )t

$20,000 = PV x (1.08)3

Rearranging:

PV = $20,000/(1.08)3

= $15,876.64

The PV of a $1 to be received in t periods when the rate is r is

PV = $1/(1 + r )t

T5.8 Present Value for a Lump Sum


T5.9 Present Value of $1 for Different Periods and Rates (Figure 5.3)


T5.10 Example: Finding the Rate

Benjamin Franklin died on April 17, 1790. In his will, he gave 1,000 pounds sterling to Massachusetts and the city of Boston. He gave a like amount to Pennsylvania and the city of Philadelphia. The money was paid to Franklin when he held political office, but he believed that politicians should not be paid for their service(!).

Franklin originally specified that the money should be paid out 100 years after his death and used to train young people. Later, however, after some legal wrangling, it was agreed that the money would be paid out 200 years after Franklin’s death in 1990. By that time, the Pennsylvania bequest had grown to about $2 million; the Massachusetts bequest had grown to $4.5 million. The money was used to fund the Franklin Institutes in Boston and Philadelphia.

Assuming that 1,000 pounds sterling was equivalent to 1,000 dollars, what rate did the two states earn? (Note: the dollar didn’t become the official U.S. currency until 1792.)


T5.10 Example: Finding the Rate (continued)

Q. Assuming that 1,000 pounds sterling was equivalent to 1,000 dollars, what rate did the two states earn?

A. For Pennsylvania, the future value is $________ and the present value is $______ . There are 200 years involved, so we need to solve for r in the following:

________ = _____________/(1 + r )200

(1 + r )200 = ________

Solving for r, the Pennsylvania money grew at about 3.87% per year. The Massachusetts money did better; check that the rate of return in this case was 4.3%. Small differences can add up!


T5.10 Example: Finding the Rate (concluded)

Q. Assuming that 1,000 pounds sterling was equivalent to 1,000 dollars, what rate did the two states earn?

A. For Pennsylvania, the future value is $ 2 million and the present value is $ 1,000. There are 200 years involved, so we need to solve for r in the following:

$ 1,000 = $ 2 million/(1 + r )200

(1 + r )200 = 2,000.00

Solving for r, the Pennsylvania money grew at about 3.87% per year. The Massachusetts money did

better; check that the rate of return in this case was 4.3%. Small differences can add up!


T5.11 The Rule of 72

The “Rule of 72” is a handy rule of thumb that states the following:

If you earn r % per year, your money will double in about 72/r % years.

So, for example, if you invest at 6%, your money will double in 12 years.

Why do we say “about?” Because at higher-than-normal rates, the rule breaks down.

What if r = 72%? FVIF(72,1) = 1.72, not 2.00

And if r = 36%? FVIF(36,2) = 1.8496

The lesson? The Rule of 72 is a useful rule of thumb, but it is only a rule of thumb!


T5.12 Chapter 5 Quick Quiz - Part 4 of 5 Suppose you deposit $5000 today in an account paying r

percent per year. If you will get $10,000 in 10 years, what rate of return are you being offered?

Set this up as present value equation:

FV = $10,000 PV = $ 5,000 t = 10 years

PV = FVt/(1 + r )t

$5000 = $10,000/(1 + r)10

Now solve for r:

(1 + r)10 = $10,000/$5,000 = 2.00

r = (2.00)1/10 - 1 = .0718 = 7.18 percent


T5.13 Example: The (Really) Long-Run Return on Common Stocks

According to Stocks for the Long Run, by Jeremy Siegel, the average annual

compound rate of return on common stocks was 8.4% over the period from

1802-1997. Suppose a distant ancestor of yours had invested $1000 in a

diversified common stock portfolio in 1802. Assuming the portfolio remained

untouched, how large would that portfolio be at the end of 1997? (Hint: if you

owned this portfolio, you would never have to work for the rest of your life!)

Common stock values increased by 28.59% in 1998 (as proxied by the growth of

the S&P 500). How much would the above portfolio be worth at the end of 1998?


T5.13 Example: The (Really) Long-Run Return on Common Stocks

According to Stocks for the Long Run, by Jeremy Siegel, the average annual

return on common stocks was 8.4% over the period from 1802-1997. Suppose a

distant ancestor of yours had invested $1000 in a diversified common stock

portfolio in 1802. Assuming the portfolio remained untouched, how large would

that portfolio be at the end of 1997? (Hint: if you owned this portfolio, you would

never have to work for the rest of your life!)

Common stock values increased by 28.59% in 1998 (as proxied by the growth of

the S&P 500). How much would the above portfolio be worth at the end of 1998?

t = 195 years, r = 8.4%, and FVIF(8.4,195) = 6,771,892.09695

So the value of the portfolio would be: $6,771,892,096.95!

The 1998 value would be $6,771,892,096.95 (1 + .2859) = $8,707,976,047.47!


T5.14 Summary of Time Value Calculations (Table 5.4)

I. Symbols:

PV = Present value, what future cash flows are worth today

FVt = Future value, what cash flows are worth in the future

r = Interest rate, rate of return, or discount rate per period

t = number of periods

C = cash amount

II. Future value of C invested at r percent per period for t periods:

FVt = C (1 + r )t

The term (1 + r )t is called the future value factor.


T5.14 Summary of Time Value Calculations (Table 5.4) (concluded)

III. Present value of C to be received in t periods at r percent per period:

PV = C/(1 + r )t

The term 1/(1 + r )t is called the present value factor.

IV. The basic present value equation giving the relationship between present and future value is:

PV = FVt/(1 + r )t



Now let’s see what you remember!

1. Which of the following statements is/are true?

Given r and t greater than zero, future value interest factors FVIF(r,t ) are always greater than 1.00.

Given r and t greater than zero, present value interest factors PVIF(r,t ) are always less than 1.00.

2. True or False: For given levels of r and t, PVIF(r,t ) is the reciprocal of FVIF(r,t ).

3. All else equal, the higher the discount rate, the (lower/higher) the present value of a set of cash flows.


T5.15 Chapter 5 Quick Quiz - Part 5 of 5 (concluded)

1. Both statements are true. If you use time value tables, use this information to be sure that you are looking at the correct table.

2. This statement is also true. PVIF(r,t ) = 1/FVIF(r,t ).

3. The answer is lower - discounting cash flows at higher rates results in lower present values. And compounding cash flows at higher rates results in higher future values.


T5.16 Solution to Problem 5.6

Assume the total cost of a college education will be $200,000 when your child enters college in 18 years. You have $15,000 to invest. What annual rate of interest must you earn on your investment to cover the cost of your child’s college education?

Present value = $15,000

Future value = $200,000

t = 18 r = ?

Solution: Set this up as a future value problem.

$200,000 = $15,000 FVIF(r,18)

FVIF(r,18) = $200,000 / $15,000 = 13.333 . . .

Solving for r gives 15.48%.



Imprudential, Inc. has an unfunded pension liability of $425 million that must be paid in 20 years. To assess the value of the firm’s stock, financial analysts want to discount this liability back to the present. If the relevant discount rate is 8 percent, what is the present value of this liability?

Future value = FV = $425 million

t = 20 r = 8 percent Present value = ?

Solution: Set this up as a present value problem.

PV = $425 million PVIF(8,20)

PV = $91,182,988.15 or about $91.18 million


Chapter 6Discounted Cash Flow Valuation


6.1 Future and Present Values of Multiple Cash Flows

6.2 Valuing Level Cash Flows: Annuities and Perpetuities

6.3 Comparing Rates: The Effect of Compounding

6.4 Loan Types and Loan Amortization





T6.2 Future Value Calculated (Fig. 6.3-6.4)

Future value calculated by compounding forward one period at a time

Future value calculated by compounding each cash flow separately


T6.3 Present Value Calculated (Fig 6.5-6.6)Present value

calculated by

discounting each

cash flow separately

Present value

calculated by

discounting back one

period at a time


T6.4 Chapter 6 Quick Quiz: Part 1 of 4

Example: Finding C

Q. You want to buy a Mazda Miata to go cruising. It costs $25,000. With a 10% down payment, the bank will loan you the rest at 12% per year (1% per month) for 60 months. What will your monthly payment

be?

A. You will borrow ___ $25,000 = $______ . This is the amount today, so it’s the ___________ . The rate is ___ , and there are __ periods:

$ ______ = C { ____________}/.01 = C {1 - .55045}/.01 = C 44.955

C = $22,500/44.955 C = $________


T6.4 Chapter 6 Quick Quiz: Part 1 of 4 (concluded)

Example: Finding C

Q. You want to buy a Mazda Miata to go cruising. It costs $25,000. With a 10% down payment, the bank will loan you the rest at 12% per year (1% per month) for 60 months. What will your monthly payment

be?

A. You will borrow .90 $25,000 = $22,500 . This is the amount today, so it’s the present value. The rate is 1%, and there are 60 periods:

$ 22,500 = C {1 - (1/(1.01)60}/.01 = C {1 - .55045}/.01 = C 44.955

C = $22,500/44.955 C = $500.50 per month


T6.5 Annuities and Perpetuities -- Basic Formulas

Annuity Present Value

PV = C {1 - [1/(1 + r )t]}/r

Annuity Future Value

FVt = C {[(1 + r )t - 1]/r}

Perpetuity Present Value

PV = C/r

The formulas above are the basis of many of the calculations in Corporate Finance. It will be worthwhile to keep them in mind!


T6.6 Examples: Annuity Present Value


Suppose you need $20,000 each year for the next three years to make your tuition payments.

Assume you need the first $20,000 in exactly one year. Suppose you can place your money in a savings account yielding 8% compounded annually. How much do you need to have in the account today?

(Note: Ignore taxes, and keep in mind that you don’t want any funds to be left in the account after the third withdrawal, nor do you want to run short of money.)


T6.6 Examples: Annuity Present Value (continued)

Annuity Present Value - Solution

Here we know the periodic cash flows are $20,000 each. Using the most basic approach:

PV = $20,000/1.08 + $20,000/1.082 + $20,000/1.083

= $18,518.52 + $_______ + $15,876.65 = $51,541.94

Here’s a shortcut method for solving the problem using the annuity present value factor:

PV = $20,000 [____________]/__________ = $20,000 2.577097 = $________________




Here we know the periodic cash flows are $20,000 each. Using the most basic approach:

PV = $20,000/1.08 + $20,000/1.082 + $20,000/1.083

= $18,518.52 + $17,146.77 + $15,876.65 = $51,541.94


PV = $20,000 [1 - 1/(1.08)3]/.08 = $20,000 2.577097 = $51,541.94




Let’s continue our tuition problem.

Assume the same facts apply, but that you can only earn 4% compounded annually. Now how much do you need to have in the account today?


T6.6 Examples: Annuity Present Value (concluded)


Again we know the periodic cash flows are $20,000 each. Using the basic approach:

PV = $20,000/1.04 + $20,000/1.042 + $20,000/1.043

= $19,230.77 + $18,491.12 + $17,779.93 = $55,501.82


PV = $20,000 [1 - 1/(1.04)3]/.04 = $20,000 2.775091 = $55,501.82


T6.7 Chapter 6 Quick Quiz -- Part 2 of 4

Q. Suppose you owe $2000 on a Visa card, and the interest rate is 2% per month. If you make the minimum monthly payments of $50, how long will it take you to pay off the debt? (Assume you quit charging stuff immediately!)

Example 1: Finding t



Q. Suppose you owe $2000 on a Visa card, and the interest rate is 2% per month. If you make the minimum monthly payments of $50, how long will it take you to pay off the debt? (Assume you quit charging stuff immediately!)

Example 1: Finding t

A. A long time: $2000 = $50 {1 - 1/(1.02)t}/.02 .80 = 1 - 1/1.02t 1.02t = 5.0 t = 81.3 months, or about 6.78 years!



Example 2: Finding C

Previously we determined that a 21-year old could accumulate $1 million by age 65 by investing $15,091 today and letting it earn interest (at 10%compounded annually) for 44 years.

Now, rather than plunking down $15,091 in one chunk, suppose she would rather invest smaller amounts annually to accumulate the million. If the first deposit is made in one year, and deposits will continue through age 65, how large must they be?

Set this up as a FV problem:

$1,000,000 = C [(1.10)44 - 1]/.10

C = $1,000,000/652.6408 = $1,532.24

Becoming a millionaire just got easier!


T6.8 Example: Annuity Future Value

Previously we found that, if one begins saving at age 21, accumulating $1 million by age 65 requires saving only $1,532.24 per year.

Unfortunately, most people don’t start saving for retirement that early in life. (Many don’t start at all!)

Suppose Bill just turned 40 and has decided it’s time to get serious about saving. Assuming that he wishes to accumulate $1 million by age 65, he can earn 10% compounded annually, and will begin making equal annual deposits in one year and make the last one at age 65, how much must each deposit be?

Setup: $1 million = C [(1.10)25 - 1]/.10

Solve for C: C = $1 million/98.34706 = $10,168.07

By waiting, Bill has to set aside over six times as much money each year!



Consider Bill’s retirement plans one more time.

Again assume he just turned 40, but, recognizing that he has a lot of time to make up for, he decides to invest in some high-risk ventures that may yield 20% annually. (Or he may lose his money completely!) Anyway, assuming that Bill still wishes to accumulate $1 million by age 65, and will begin making equal annual deposits in one year and make the last one at age 65, now how much must each deposit be?

Setup: $1 million = C [(1.20)25 - 1]/.20

Solve for C: C = $1 million/471.98108 = $2,118.73

So Bill can catch up, but only if he can earn a much higher return (which will probably require taking a lot more risk!).


T6.10 Summary of Annuity and Perpetuity Calculations (Table 6.2)

I. Symbols

PV = Present value, what future cash flows bring today

FVt = Future value, what cash flows are worth in the future

r = Interest rate, rate of return, or discount rate per period

t = Number of time periods

C = Cash amount

II. FV of C per period for t periods at r percent per period:

FVt = C {[(1 + r )t - 1]/r}

III. PV of C per period for t periods at r percent per period:

PV = C {1 - [1/(1 + r )t]}/r

IV. PV of a perpetuity of C per period:

PV = C/r


T6.11 Example: Perpetuity Calculations

Suppose we expect to receive $1000 at the end of each of the next 5 years. Our opportunity rate is 6%. What is the value today of this set of cash flows?

PV = $1000 {1 - 1/(1.06)5}/.06

= $1000 {1 - .74726}/.06

= $1000 4.212364

= $4212.36

Now suppose the cash flow is $1000 per year forever. This is called a perpetuity. And the PV is easy to calculate:

PV = C/r = $1000/.06 = $16,666.66…

So, payments in years 6 thru have a total PV of $12,454.30!



Consider the following questions.

The present value of a perpetual cash flow stream has a finite value (as long as the discount rate, r, is greater than 0). Here’s a question for you: How can an infinite number of cash payments have a finite value?

Here’s an example related to the question above. Suppose you are considering the purchase of a perpetual bond. The issuer of the bond promises to pay the holder $100 per year forever. If your opportunity rate is 10%, what is the most you would pay for the bond today?

One more question: Assume you are offered a bond identical to the one described above, but with a life of 50 years. What is the difference in value between the 50-year bond and the perpetual bond?


T6.12 Solution to Chapter 6 Quick Quiz -- Part 4 of 4

An infinite number of cash payments has a finite present value is because the present values of the cash flows in the distant future become infinitesimally small.

The value today of the perpetual bond = $100/.10 = $1,000.

Using Table A.3, the value of the 50-year bond equals

$100 9.9148 = $991.48

So what is the present value of payments 51 through infinity (also an infinite stream)?

Since the perpetual bond has a PV of $1,000 and the otherwise identical 50-year bond has a PV of $991.48, the value today of payments 51 through infinity must be

$1,000 - 991.48 = $8.52 (!)


T6.13 Compounding Periods, EARs, and APRs

Compounding Number of times Effective

periodcompounded annual rate

Year 1 10.00000%

Quarter 4 10.38129

Month 12 10.47131

Week 5210.50648

Day 36510.51558

Hour 8,76010.51703

Minute 525,600 10.51709


T6.13 Compounding Periods, EARs, and APRs (continued)

EARs and APRs

Q. If a rate is quoted at 16%, compounded semiannually, then the actual rate is 8% per six months. Is 8% per six months the same as 16% per year?

A. If you invest $1000 for one year at 16%, then you’ll have $1160 at the end of the year. If you invest at 8% per period for two periods, you’ll have

FV = $1000 (1.08)2

= $1000 1.1664

= $1166.40,

or $6.40 more. Why? What rate per year is the same as 8% per six months?


T6.13 Compounding Periods, EARs, and APRs (concluded)

The Effective Annual Rate (EAR) is _____%. The “16% compounded semiannually” is the quoted or stated rate, not the effective rate.

By law, in consumer lending, the rate that must be quoted on a loan agreement is equal to the rate per period multiplied by the number of periods. This rate is called the _________________ (____).

Q. A bank charges 1% per month on car loans. What is the APR? What is the EAR?

A. The APR is __ __ = ___%. The EAR is:

EAR = _________ - 1 = 1.126825 - 1 = 12.6825%


T6.13 Compounding Periods, EARs, and APRs (concluded)

The Effective Annual Rate (EAR) is 16.64%. The “16% compounded semiannually” is the quoted or stated rate, not the effective rate.

By law, in consumer lending, the rate that must be quoted on a loan agreement is equal to the rate per period multiplied by the number of periods. This rate is called the Annual Percentage Rate (APR).

Q. A bank charges 1% per month on car loans. What is the APR? What is the EAR?

A. The APR is 1% 12 = 12%. The EAR is:

EAR = (1.01)12 - 1 = 1.126825 - 1 = 12.6825%

The APR is thus a quoted rate, not an effective rate!


T6.14 Example: Amortization Schedule - Fixed Principal

Beginning Total Interest Principal EndingYear Balance Payment Paid Paid Balance

1 $5,000 $1,450 $450 $1,000 $4,000

2 4,000 1,360 360 1,000 3,000

3 3,000 1,270 270 1,000 2,000

4 2,000 1,180 180 1,000 1,000

5 1,000 1,090 90 1,000 0

Totals $6,350 $1,350 $5,000


T6.15 Example: Amortization Schedule - Fixed Payments

Beginning Total Interest Principal EndingYear Balance Payment Paid Paid Balance

1 $5,000.00 $1,285.46 $ 450.00 $ 835.46 $4,164.54

2 4,164.54 1,285.46 374.81 910.65 3,253.88

3 3,253.88 1,285.46 292.85 992.61 2,261.27

4 2,261.27 1,285.46 203.51 1,081.95 1,179.32

5 1,179.32 1,285.46 106.14 1,179.32 0.00

Totals $6,427.30 $1,427.31 $5,000.00


How to lie, cheat, and steal with interest rates:

RIPOV RETAILINGGoing out for business sale!

$1,000 instant credit!

12% simple interest!

Three years to pay!

Low, low monthly payments!


Assume you buy $1,000 worth of furniture from this store and agree to the above credit terms. What is the APR of this loan? The EAR?


T6.16 Solution to Chapter 6 Quick Quiz -- Part 4 of 4 (concluded)

Your payment is calculated as:

1. Borrow $1,000 today at 12% per year for three years, you will owe $1,000 + $1000(.12)(3) = $1,360.

2. To make it easy on you, make 36 low, low payments of $1,360/36 = $37.78.

3. Is this a 12% loan?

$1,000 = $37.78 x (1 - 1/(1 + r )36)/r

r = 1.767% per month

APR = 12(1.767%) = 21.204% EAR = 1.0176712 - 1 = 23.39% (!)



Seinfeld’s Life Insurance Co. is trying to sell you an investment policy that will pay you and your heirs $1,000 per year forever. If the required return on this investment is 12 percent, how much will you pay for the policy?

The present value of a perpetuity equals C/r. So, the most a rational buyer would pay for the promised cash flows is

C/r = $1,000/.12 = $8,333.33

Notice: $8,333.33 is the amount which, invested at 12%, would throw off cash flows of $1,000 per year forever. (That is, $8,333.33 .12 = $1,000.)



In the previous problem, Seinfeld’s Life Insurance Co. is trying to sell you an investment policy that will pay you and your heirs $1,000 per year forever. Seinfeld told you the policy costs $10,000. At what interest rate would this be a fair deal?

Again, the present value of a perpetuity equals C/r. Now solve the following equation:

$10,000 = C/r = $1,000/r

r = .10 = 10.00%

Notice: If your opportunity rate is less than 10.00%, this is a good deal for you; but if you can earn more than 10.00%, you can do better by investing the $10,000 yourself!



Congratulations! You’ve just won the $20 million first prize in the Subscriptions R Us Sweepstakes. Unfortunately, the sweepstakes will actually give you the $20 million in $500,000 annual installments over the next 40 years, beginning next year. If your appropriate discount rate is 12 percent per year, how much money did you really win?

“How much money did you really win?” translates to, “What is the value today of your winnings?” So, this is a present value problem.

PV = $ 500,000 [1 - 1/(1.12)40]/.12

= $ 500,000 [1 - .0107468]/.12

= $ 500,000 8.243776

= $4,121,888.34 (Not quite $20 million, eh?)


Chapter 9Net Present Value and Other Investment Criteria


9.1 Net Present Value

9.2 The Payback Rule

9.3 The Discounted Payback

9.4 The Average Accounting Return

9.5 The Internal Rate of Return

9.6 The Profitability Index

9.7 The Practice of Capital Budgeting





T9.2 NPV Illustrated

Assume you have the following information on Project X:

Initial outlay -$1,100 Required return = 10%

Annual cash revenues and expenses are as follows:

Year Revenues Expenses

1 $1,000 $500

2 2,000 1,000

Draw a time line and compute the NPV of project X.


T9.2 NPV Illustrated (concluded)

0 1 2

Initial outlay($1,100)

Revenues $1,000Expenses 500

Cash flow $500

Revenues $2,000Expenses 1,000

Cash flow $1,000

– $1,100.00

+454.55

+826.45

+$181.00

1$500 x 1.10

1$1,000 x 1.10

2

NPV


T9.3 Underpinnings of the NPV Rule

Why does the NPV rule work? And what does “work” mean? Look at it this way:

A “firm” is created when securityholders supply the funds to acquire assets that will be used to produce and sell a good or a service;

The market value of the firm is based on the present value of the cash flows it is expected to generate;

Additional investments are “good” if the present value of the incremental expected cash flows exceeds their cost;

Thus, “good” projects are those which increase firm value - or, put another way, good projects are those projects that have positive NPVs!

Moral of the story: Invest only in projects with positive NPVs.


T9.4 Payback Rule Illustrated

Initial outlay -$1,000

Year Cash flow

1 $200

2 400 3 600

Accumulated

Year Cash flow

1 $200 2 600 3 1,200

Payback period = 2 2/3 years


T9.5 Discounted Payback Illustrated

Initial outlay -$1,000

R = 10%

PV of Year Cash flow Cash flow

1 $ 200 $ 182 2 400 331 3 700 526 4 300 205

Accumulated Year discounted cash flow

1 $ 182 2 513 3 1,039 4 1,244

Discounted payback period is just under 3 years


T9.6 Ordinary and Discounted Payback (Table 9.3)

Cash Flow Accumulated Cash Flow

Year Undiscounted Discounted Undiscounted Discounted

1 $100 $89 $100 $89

2 100 79 200 168

3 100 70 300 238

4 100 62 400 300

5 100 55 500 355


T9.7 Average Accounting Return Illustrated

Average net income:

Year

1 2 3

Sales $440 $240 $160

Costs 220 120 80

Gross profit 220 120 80

Depreciation 80 80 80

Earnings before taxes 140 40 0

Taxes (25%) 35 10 0

Net income $105 $30 $0

Average net income = ($105 + 30 + 0)/3 = $45


T9.7 Average Accounting Return Illustrated (concluded)

Average book value:

Initial investment = $240

Average investment = ($240 + 0)/2 = $120

Average accounting return (AAR):

Average net income $45

AAR = = = 37.5% Average book value $120


T9.8 Internal Rate of Return Illustrated

Initial outlay = -$200

Year Cash flow

1 $ 50 2 100 3 150

Find the IRR such that NPV = 0

50 100 150

0 = -200 + + + (1+IRR)1 (1+IRR)2 (1+IRR)3

50 100 150

200 = + + (1+IRR)1 (1+IRR)2 (1+IRR)3


T9.8 Internal Rate of Return Illustrated (concluded)

Trial and Error

Discount rates NPV

0% $100

5% 68

10% 41

15% 18

20% -2

IRR is just under 20% -- about 19.44%


Year Cash flow

0 – $275 1 100 2 100 3 100 4 100

T9.9 Net Present Value Profile

Discount rate

2% 6% 10%

14% 18%

120

100

80

60

40

20

Net present value

0

– 20

– 40

22%

IRR


Assume you are considering a project for which the cash flows are as follows:

Year Cash flows

0 -$252

1 1,431

2 -3,035

3 2,850

4 -1,000

T9.10 Multiple Rates of Return


T9.10 Multiple Rates of Return (continued)

What’s the IRR? Find the rate at which the computed NPV = 0:

at 25.00%: NPV = _______

at 33.33%: NPV = _______

at 42.86%: NPV = _______

at 66.67%: NPV = _______


T9.10 Multiple Rates of Return (continued)

What’s the IRR? Find the rate at which the computed NPV = 0:

at 25.00%: NPV = 0

at 33.33%: NPV = 0

at 42.86%: NPV = 0

at 66.67%: NPV = 0

Two questions: 1. What’s going on here? 2. How many IRRs can there

be?


T9.10 Multiple Rates of Return (concluded)

$0.06

$0.04

$0.02

$0.00

($0.02)

NPV

($0.04)

($0.06)

($0.08)

0.2 0.28 0.36 0.44 0.52 0.6 0.68

IRR = 1/4

IRR = 1/3

IRR = 3/7

IRR = 2/3

Discount rate


T9.11 IRR, NPV, and Mutually Exclusive Projects

Discount rate

2% 6% 10%

14% 18%

60

40200

– 20– 40

Net present value

– 60

– 80

– 100

22%

IRR A IRR B

0

140

12010080

160

Year

0 1 2 3 4

Project A: – $350 50 100 150 200

Project B: – $250 125 100 75 50

26%

Crossover Point


T9.12 Profitability Index Illustrated

Now let’s go back to the initial example - we assumed the following information on Project X:

Initial outlay -$1,100Required return = 10%

Annual cash benefits:

Year Cash flows

1 $ 500

2 1,000 What’s the Profitability Index (PI)?


T9.12 Profitability Index Illustrated (concluded)

Previously we found that the NPV of Project X is equal to:

($454.55 + 826.45) - 1,100 = $1,281.00 - 1,100 = $181.00.

The PI = PV inflows/PV outlay = $1,281.00/1,100 = 1.1645.

This is a good project according to the PI rule. Can you explain why?

It’s a good project because the present value of the inflows exceeds the outlay.


T9.13 Summary of Investment Criteria

I. Discounted cash flow criteria

A. Net present value (NPV). The NPV of an investment is the difference between its market value and its cost. The NPV rule is to take a project if its NPV is positive. NPV has no serious flaws; it is the preferred decision criterion.

B. Internal rate of return (IRR). The IRR is the discount rate that makes the estimated NPV of an investment equal to zero. The IRR rule is to take a project when its IRR exceeds the required return. When project cash flows are not conventional, there may be no IRR or there may be more than one.

C. Profitability index (PI). The PI, also called the benefit-cost ratio, is the ratio of present value to cost. The profitability index rule is to take an investment if the index exceeds 1.0. The PI measures the present value per dollar invested.


T9.13 Summary of Investment Criteria (concluded)

II. Payback criteria

A. Payback period. The payback period is the length of time until the sum of an investment’s cash flows equals its cost. The payback period rule is to take a project if its payback period is less than some prespecified cutoff.

B. Discounted payback period. The discounted payback period is the length of time until the sum of an investment’s discounted cash flows equals its cost. The discounted payback period rule is to take an investment if the discounted payback is less than some prespecified cutoff.

III. Accounting criterion

A. Average accounting return (AAR). The AAR is a measure of accounting profit relative to book value. The AAR rule is to take an investment if its AAR exceeds a benchmark.


T9.14 Chapter 9 Quick Quiz

1. Which of the capital budgeting techniques do account for both the time value of money and risk?

2. The change in firm value associated with investment in a project is measured by the project’s _____________ .

a. Payback period

b. Discounted payback period

c. Net present value

d. Internal rate of return

3. Why might one use several evaluation techniques to assess a given project?


T9.14 Chapter 9 Quick Quiz

1. Which of the capital budgeting techniques do account for both the time value of money and risk?

Discounted payback period, NPV, IRR, and PI

2. The change in firm value associated with investment in a project is measured by the project’s Net present value.

3. Why might one use several evaluation techniques to assess a given project?

To measure different aspects of the project; e.g., the payback period measures liquidity, the NPV measures the change in firm value, and the IRR measures the rate of return on the initial outlay.



Offshore Drilling Products, Inc. imposes a payback cutoff of 3 years for its international investment projects. If the company has the following two projects available, should they accept either of them?

Year Cash Flows A Cash Flows B

0 -$30,000 -$45,000

1 15,000 5,000

2 10,000 10,000

3 10,000 20,000

4 5,000 250,000


T9.15 Solution to Problem 9.3 (concluded)

Project A:

Payback period = 1 + 1 + ($30,000 - 25,000)/10,000

= 2.50 years

Project B:

Payback period = 1 + 1 + 1 + ($45,000 - 35,000)/$250,000

= 3.04 years

Project A’s payback period is 2.50 years and project B’s payback period is 3.04 years. Since the maximum acceptable payback period is 3 years, the firm should accept project A and reject project B.



A firm evaluates all of its projects by applying the IRR rule. If the required return is 18 percent, should the firm accept the following project?

Year Cash Flow

0 -$30,000

1 25,000

2 0

3 15,000


T9.16 Solution of Problem 9.7 (concluded)

To find the IRR, set the NPV equal to 0 and solve for the discount rate:

NPV = 0 = -$30,000 + $25,000/(1 + IRR)1 + $0/(1 + IRR) 2

+$15,000/(1 + IRR)3

At 18 percent, the computed NPV is ____.

So the IRR must be (greater/less) than 18 percent. How did you know?


T9.16 Solution of Problem 9.7 (concluded)

To find the IRR, set the NPV equal to 0 and solve for the discount rate:

NPV = 0 = -$30,000 + $25,000/(1 + IRR)1 + $0/(1 + IRR)2 +$15,000/(1 + IRR)3

At 18 percent, the computed NPV is $316.

So the IRR must be greater than 18 percent. We know this because the computed NPV is positive.

By trial-and-error, we find that the IRR is 18.78 percent.

Documents

T5.1 Chapter Outline Chapter 5 Introduction to Valuation: The Time Value of Money Chapter Organization 5.1Future Value and Compounding 5.2Present Value