last study topics project analysis project interaction – equivalent annual cost – replacement...

Last Study Topics

• Project Analysis• Project Interaction

– Equivalent Annual Cost– Replacement– Project Interaction– Timing– Fluctuating Load Factors

Principles of Corporate

Finance

Sixth Edition

Richard A. Brealey

Stewart C. Myers

Lu Yurong

Chapter 7

McGraw Hill/Irwin

Introduction to Risk, Return, and the Opportunity Cost of

Capital

Today’s Study Topics

• 75 Years of Capital Market History• Measuring Risk• Portfolio Risk

75 Years of Capital Market History

• We, concentrate on a study by Ibbotson Associates that measures the historical performance of five portfolios of securities:

• 1. A portfolio of Treasury bills, i.e., United States government debt securities maturing in less than one year.

• 2. A portfolio of long-term United States government bonds.

• 3. A portfolio of long-term corporate bonds.• 4. A portfolio of the common stocks of small

firms.

75 Years of Capital Market History

• These investments offer different degrees of risk.• Treasury bills are about as safe an investment as

you can make. • There is no risk of default, and their short

maturity means that the prices of Treasury bills are relatively stable.

• In fact, an investor who wishes to lend money for, say, three months can achieve a perfectly certain payoff by purchasing a Treasury bill maturing in three months.

Continue

• By switching to long-term government bonds, the investor acquires an asset whose price fluctuates as interest rates vary.

• An investor who shifts from government to corporate bonds accepts an additional default risk.

• An investor who shifts from corporate bonds to common stocks has a direct share in the risks of the enterprise.

The Value of an Investment of $1 in 1926

Source: Ibbotson Associates

0.1

10

1000

1925 1940 1955 1970 1985 2000

S&PSmall CapCorp BondsLong BondT Bill

Inde

x

Year End

1

6402

2587

64.1

48.9

16.6

Continue

• Portfolio performance coincides with our intuitive risk ranking.

• A dollar invested in the safest investment, Treasury bills, would have grown to just over $16 by 2000, barely enough to keep up with inflation.

• An investment in long-term Treasury bonds would have produced $49, and corporate bonds a pinch more.

Continue

• Common stocks were in a class by themselves.• An investor who placed a dollar in the stocks

of large U.S. firms would have received $2,587.

• The jackpot, however, went to investors in stocks of small firms, who walked away with $6,402 for each dollar invested.

Average Returns• Treasury bills have provided the lowest average return

—3.9 % per year in nominal terms and .8 % in real terms.

• In other words, the average rate of inflation over this period was just over 3 % per year.

0.1

10

1000

1925 1940 1955 1970 1985 2000

S&PSmall CapCorp BondsLong BondT Bill


Inde

x

Year End

1

660

267

6.6

5.0

1.7

Real returns

The Value of an Investment of $1 in 1926

Arithmetic Averages and Compound Annual Returns

• Notice that the average returns shown in Table are arithmetic averages.

• In other words, Ibbotson Associates simply added the 75 annual returns and divided by 75.

• The arithmetic average is higher than the compound annual return over the period.

Case: Big Oil’s Stock

• Suppose that the price of Big Oil’s common stock is $100.

• There is an equal chance that at the end of the year the stock will be worth $90, $110, or $130.

• Therefore, the return could be -10%, +10%, or 30%.

• Expected return would be; – E(R)= (-10% +10% +30% )1/3

– = +10%

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• And Present Value would be;– PV = $110 = $100 1.10

• Now suppose that we observe the returns on Big Oil stock over a large number of years.

• If the odds are unchanged, the return will be;– -10% in a third of the years, +10% in a further

third, and +30% in the remaining years.

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• Arithmetic Average of these years returns would be;– -10+10+30 = +10% (Still the same)

3

• The average compound annual return on Big Oil stock would be;

• (.9 x 1.1 x 1.3)1/3 = .088, or 8.8%,

Continue

• Investors would not be willing to invest in a project that offered an 8.8% expected return if they could get an expected return of 10% in the capital markets.– NPV = -100 + 108.8 = -1.1 1.1

• Moral: If the cost of capital is estimated from historical returns or risk premiums, use arithmetic averages, not compound annual rates of return.

Rates of Return 1926-2000


-60

-40

-20

0

20

40

60

26 30 35 40 45 50 55 60 65 70 75 80 85 90 9520

00

Common Stocks

Long T-Bonds

T-Bills

Year

Perc

enta

ge R

etur

n

Using Historical Evidence to Evaluate Today’s Cost of Capital

• Suppose there is an investment project which you know and has the same risk as any Index. We will say that it has the same degree of risk as the market portfolio.

• What rate should you use to discount this project’s forecasted cash flows?

• expected rate of return on the market portfolio;– The return investors would forgo by investing in the

proposed project. Let us call this market return rm.

Continue

• One way to estimate rm is to assume that the future will be like the past and that today’s investors expect to receive the same “normal” rates of return revealed by the averages calculated earlier.

• Unfortunately, this is not the way to do it; rm is not likely to be stable over time.– Remember that it is the sum of the risk-free

interest rate rf and a premium for risk.

Continue• Take the interest rate on Treasury bills and add

9.1%, the average risk premium shown in Table.• Assume, Treasury bills is about 3.5%, then;

– rm(2001) = rf (2001) + normal risk premium = .035 + .091 = .126, or about 12.5%

Average Market Risk Premia (1900-2000)• Market returns in 15 countries and shows the

average risk premium in each country between 1900 and 2000.

4.35.1 6 6.1 6.1 6.5 6.7 7.1 7.5 8

8.5 9.9 9.9 10 11

0123456789

1011

Den

Bel

Ca

n

Sw

i

Sp

a

UK Ire

Net

h

US

A

Sw

e

Au

s

Ger

Fra

Ja

p It

Risk premium, %

Country

Risk Premium

• Now compare the returns in the United States with those in the other countries.

• There is no evidence here that U.S. investors have been particularly fortunate; the USA was exactly average in terms of the risk premium.

• Danish common stocks came bottom of the league; the average risk premium in Denmark was only 4.3%.

• Top of the form was Italy with a premium of 11.1 %.

Summary

• 75 Years of Capital Market History• Measuring Risk• Risk Premium

MEASURING PORTFOLIO RISK• Make you learn;

– (1) how to measure risk and, – (2) the relationship between risks borne and risk

premiums demanded.

• Variance - Average value of squared deviations from mean. A measure of volatility.

• Standard Deviation - Average value of squared deviations from mean. A measure of volatility.

Measuring Risk

• Variance (rm*) = the expected value of ( rm* – rm)2

• The standard deviation is simply the square root of the variance.

• Here is a very simple example showing how variance and standard deviation are calculated. – Suppose that you are offered the chance to play the

following game. You start by investing $100. Then two coins are flipped. For each head that comes up you get back your starting balance plus 20 percent, and for each tail that comes up you get back your starting balance less 10 percent. Clearly there are four equally likely outcomes

continue

– Head + head: You gain 40 percent.– Head + tail: You gain 10 percent.– Tail + head: You gain 10 percent.– Tail + tail: You lose 20 percent.

• There is a chance of 1 in 4, or .25, that you will make 40 percent; a chance of 2 in 4, or .5, that you will make 10 percent; and a chance of 1 in 4, or .25, that you will lose 20 percent.

Expected Return

– Expected return = (.25 x 40) + (.5 x 10) + (.25 x -20) = 10%

• The variance of the percentage returns is 450.• Standard deviation is the square root of 450,

or 21. • This figure is in the same units as the rate of

return, so we can say that the game’s variability is 21%.

Measuring RiskCoin Toss Game-calculating variance and standard deviation

(1) (2) (3)

Percent Rate of Return Deviation from Mean Squared Deviation

+ 40 + 30 900

+ 10 0 0

+ 10 0 0

- 20 - 30 900

Variance = average of squared deviations = 1800 / 4 = 450

Standard deviation = square of root variance = 450 = 21.2%

Continue

• Think of a second game, the same as the first except that each head means a 35% gain and each tail means a 25% loss. Again, there are four equally likely outcomes:– Head head: You gain 70 percent.– Head tail: You gain 10 percent.– Tail head: You gain 10 percent.– Tail tail: You lose 50 percent.

Continue

• For this game the expected return is 10 percent, the same as that of the first game.

• But its standard deviation is double that of the first game, 42 versus 21 percent.

• By this measure the second game is twice as risky as the first.

Measuring Variability

• You would identify the possible outcomes, assign a probability to each outcome, and grind through the calculations.

• But where do the probabilities come from?• Most financial analysts start by observing past

variability. • Of course, there is no risk in hindsight, but it is

reasonable to assume that portfolios with histories of high variability also have the least predictable future performance.

Continue

• You may find it interesting to compare the coin-tossing game and the stock market as alternative investments. – The stock market generated an average annual return

of 13.0 % with a standard deviation of 20.2%.

Market Variability

• Of course, there is no reason to believe that the market’s variability should stay the same over more than 70 years.

• For example, it is clearly less now than in the Great Depression of the 1930s.

Measuring Risk

1 1 24

1311

13 12 13

3 20123456789

10111213

-50

to -

40

-40

to -

30

-30

to -

20

-20

to -

10

-10

to 0

0 to

10

10 t

o 20

20 t

o 30

30 t

o 40

40 t

o 50

50 t

o 60

Return %

# of Years

Histogram of Annual Stock Market ReturnsHistogram of Annual Stock Market Returns

How Diversification Reduces Risk• Remember that the market portfolio’s standard

deviation was about 13 percent in the 1990s. • Of our individual stocks only Exxon Mobil came close

to this figure.• Amazon.com was about eight times more variable than

the market portfolio.

Measuring Risk

Diversification - Strategy designed to reduce risk by spreading the portfolio across many investments.

Unique Risk - Risk factors affecting only that firm. Also called “diversifiable risk.”

Market Risk - Economy-wide sources of risk that affect the overall stock market. Also called “systematic risk.”

Measuring Risk

Portfolio rate

of return=

fraction of portfolio

in first assetx

rate of return

on first asset

+fraction of portfolio

in second assetx

rate of return

on second asset

((

(())

))

Measuring Risk

05 10 15

Number of Securities

Po

rtfo

lio

sta

nd

ard

dev

iati

on

Measuring Risk

05 10 15

Number of Securities

Po

rtfo

lio

sta

nd

ard

dev

iati

on

Market risk

Uniquerisk

Portfolio Risk

22

22

211221

1221

211221

122121

21

σxσσρxx

σxx2Stock

σσρxx

σxxσx1Stock

2Stock 1Stock

The variance of a two stock portfolio is the sum of these four boxes

Portfolio RiskExample

Suppose you invest 65% of your portfolio in Coca-Cola and 35% in Reebok. The expected dollar return on your CC is 10% x 65% = 6.5% and on Reebok it is 20% x 35% = 7.0%. The expected return on your portfolio is 6.5 + 7.0 = 13.50%. Assume a correlation coefficient of 1.

Portfolio Risk

2222

22

211221

2112212221

21

)5.58()35(.σx5.585.311

35.65.σσρxxReebok

5.585.311

35.65.σσρxx)5.31()65(.σxCola-Coca

ReebokCola-Coca

Example


Portfolio RiskExample


% 31.7 1,006.1 Deviation Standard

1.006,15)1x31.5x58.2(.65x.35x

]x(58.5)[(.35)

]x(31.5)[(.65) Valriance Portfolio22

22

Portfolio Risk

)rx()r(x Return PortfolioExpected 2211

)σσρxx(2σxσxVariance Portfolio 21122122

22

21

21

Portfolio RiskThe shaded boxes contain variance terms; the remainder contain covariance terms.

1

2

3

4

5

6

N

1 2 3 4 5 6 N

STOCK

STOCK

To calculate portfolio variance add up the boxes

Beta and Unique Risk

beta

Expected

return

Expectedmarketreturn

10%10%- +

-10%+10%

stock

Copyright 1996 by The McGraw-Hill Companies, Inc

-10%

1. Total risk = diversifiable risk + market risk2. Market risk is measured by beta, the sensitivity to market changes


Market Portfolio - Portfolio of all assets in the economy. In practice a broad stock market index, such as the S&P Composite, is used to represent the market.

Beta - Sensitivity of a stock’s return to the return on the market portfolio.


2m

imiB


2m

imiB

Covariance with the market

Variance of the market

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