0

0 20 40 60 80 100 120

Portfolio standard deviation (%)

bearers of bad news. What the investor wants is to reward bearers of accurate news. Investors

should monitor forecasts of portfolio managers on a regular basis to develop a track record of their

forecasting accuracy. Portfolio choices of the more accurate forecasters will, in the long run,

outperform the field.

6. a. Beta, the slope coefficient of the security on the factor: Securities R1“R6 have a positive beta.

These securities move, on average, in the same direction as the market (RM ). R1, R2, R6 have

large betas, so they are “aggressive” in that they carry more systematic risk than R3, R4, R5,

which are “defensive.” R7 and R8 have a negative beta. These are hedge assets that carry

negative systematic risk.

b. Intercept, the expected return when the market is neutral: The estimates show that R1, R4, R8

have a positive intercept, while R2, R3, R5, R6, R7 have negative intercepts. To the extent that one

believes these intercepts will persist, a positive value is preferred.

c. Residual variance, the nonsystematic risk: R2, R3, R7 have a relatively low residual variance.

With sufficient diversification, residual risk eventually will be eliminated, and hence, the

difference in the residual variance is of little economic significance.

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d. Total variance, the sum of systematic and nonsystematic risk: R3 has a low beta and low residual

variance, so its total variance will be low. R1, R6 have high betas and high residual variance, so

their total variance will be high. But R4 has a low beta and high residual variance, while R2 has a

high beta with a low residual variance. In sum, total variance often will misrepresent systematic

risk, which is the part that matters.

7. a. To obtain the characteristic line of XYZ we continue the spreadsheet of Example 6.4 and run a

regression of the excess return of XYZ on the excess return of the market index fund.

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210 Part TWO Portfolio Theory

Summary Output

Regression Statistics

Multiple R 0.363

R Square 0.132

Adjusted R Square 0.023

Standard Error 41.839

Observations 10

Coefficients Standard Error t Stat P-value Lower 95% Upper 95%

Intercept 3.930 14.98 0.262 0.800 30.62 38.48

Market 0.582 0.528 1.103 0.302 0.635 1.798

The regression output shows that the slope coefficient of XYZ is .58 and the intercept is 3.93%,

hence the characteristic line is: RXYZ 3.93 .582RMarket.

b. The beta coefficient of ABC is 1.15, greater than XYZ™s .58, implying that ABC has greater

systematic risk.

c. The regression of XYZ on the market index shows an R-Square of .132. Hence the percent of

unexplained variance (nonsystematic risk) is .868, or 86.8%.

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Fifth Edition

Appendix

THE FALLACY OF TIME

DIVERSIFICATION

RISK POOLING VERSUS RISK SHARING

Suppose the probability of death within a year of a healthy 35-year-old is 5%. If we treat the

event of death as a zero-one random variable, the standard deviation is 21.79%. In a group of

1,000 healthy 35-year-olds, we expect 50 deaths within a year (5% of the sample), with a stan-

dard deviation of 6.89 deaths (.689%). Measured by the mean-standard deviation criterion, the

life insurance business apparently becomes less risky the more policies an insurer can write.

This is the concept of risk pooling. risk pooling

However, the apparent risk reduction from pooling many policies is really a fallacy; risk is Lowering the

not always appropriately measured by standard deviation. The complication in this case is that variance of returns

insuring more people puts more capital at risk. The owners of a small insurance company may by combining

risky projects.

be unwilling to take the increasing risk of ruin that would be incurred by insuring very large

groups of clients.

What really makes the insurance industry tick is risk sharing. When an insurer sells more risk sharing

policies, it can also bring in more partners. Each partner then takes a smaller share of the Lowering the risk per

growing pie, thus obtaining the benefits of diversification without scaling up the amount of invested dollar by

capital put at risk. selling shares to

investors.

To reiterate, the reason that risk pooling alone does not improve the welfare of an investor

(insurer) is that the size of the capital at risk, that is, total risk, is increasing. Risk-averse

investors will shy away from a large degree of risk pooling unless they can share the risk of

a growing pool with other investors, thereby keeping the size of their investment relatively

stable.

Risk sharing is analogous to portfolio investment. You take a fixed budget and by investing

it in many risky assets, that is, investing small proportions in various assets, you lower the risk

without giving up expected returns.

TIME DIVERSIFICATION

A related version of the risk pooling versus sharing misconception is “time diversification.”

Consider the case of Mr. Frier. Planning to retire in five years, he has a five-year horizon. Con-

fronted with the fact that the standard deviation of stock returns exceeds 20% per year,

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212 Part TWO Portfolio Theory

Mr. Frier has become aware of his acute risk aversion and is keeping most of his retirement

portfolio in money market assets.

Recently, Mr. Frier has learned of the large potential gains from diversification. He won-

ders whether investing for as long as five years might not take the standard deviation sting out

of stocks without giving up expected return.

Mr. Mavin, a highly recommended financial adviser, argues that the time factor is all-

important. He cites academic research showing that asset rates of return over successive hold-

ing periods are nearly independent. Therefore, he argues that over a five-year period, returns

in good years and bad years will cancel out, making the average rate of return on the portfolio

over the investment period less risky than would appear from an analysis of single-year

volatility. Because returns in each year are nearly independent, Mr. Mavin tells Mr. Frier that

a five-year investment is equivalent to a portfolio of five equally weighted, independent

assets.

Mr. Frier is convinced and intends to transfer his funds to a stock mutual fund right away.

Is his conviction warranted? Does Mr. Mavin™s time diversification really reduce risk?

It is true that the standard deviation of the average annual rate of return over the five years

will be smaller than the one-year standard deviations, as Mr. Mavin claims. But what about the

volatility of Mr. Frier™s total retirement fund?

Mr. Mavin is wrong: Time diversification does not reduce risk. While it is true that the per-

year average rate of return has a smaller standard deviation for a longer time horizon, it is also

true that the uncertainty compounds over a greater number of years. Unfortunately, this latter

effect dominates; that is, the total T-year return becomes more uncertain the longer the in-

vestment horizon (T years).

Investing for more than one holding period means total risk is growing. This is analogous

to an insurer taking on more insurance policies. The fact that these policies are independent

does not offset the effect of placing more funds at risk. Focus on the standard deviation of the

average rate of return should never obscure the more proper emphasis on the ultimate dollar

value of a portfolio strategy. Indeed, insuring a portfolio to guarantee a minimum T-year re-

turn equal to a money market fund rate will cost you more as T grows longer. This is an em-

pirically verifiable fact that is well anchored in economic theory.

There may in fact be good reasons for the commonly accepted belief that younger investors

with longer investment horizons should invest higher fractions of their portfolios in risky

assets with higher expected returns, such as stocks. For example, if things go wrong, there is

more time to spread out the burden and recover from the loss. But the rationale for these

investors to direct their funds to the stock market should not be that the stock market is less

risky if one™s investment horizon is longer.

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Essentials of Investments, and Arbitrage Pricing Companies, 2003

Fifth Edition Theory

7

CAPITAL ASSET PRICING

AND ARBITRAGE

PRICING THEORY

AFTER STUDYING THIS CHAPTER

YOU SHOULD BE ABLE TO:

> Use the implications of capital market theory to compute

security risk premiums.

> Construct and use the security market line.

> Take advantage of an arbitrage opportunity with a portfolio

that includes mispriced securities.

> Use arbitrage pricing theory with more than one factor to

identify mispriced securities.

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http://www.mhhe.com/edumarketinsight

Related Websites

This site contains information on monthly, weekly, and

http://finance.yahoo.com

daily returns that can be used in estimating beta

http://moneycentral.msn.com/investor/home.asp coefficients.

http://bloomberg.com http://www.efficientfrontier.com

http://www.411stocks.com/ Here you™ll find information on modern portfolio theory

http://www.morningstar.com and portfolio allocation.

You can use these sites to assess beta coefficients for

individual securities and mutual funds.

he capital asset pricing model, almost always referred to as the CAPM, is a

T centerpiece of modern financial economics. It was first proposed by William F.

Sharpe, who was awarded the 1990 Nobel Prize for economics.

The CAPM provides a precise prediction of the relationship we should observe

between the risk of an asset and its expected return. This relationship serves two vital

functions.

First, it provides a benchmark rate of return for evaluating possible investments.

For example, a security analyst might want to know whether the expected return she

forecasts for a stock is more or less than its “fair” return given its risk. Second, the

model helps us make an educated guess as to the expected return on assets that have

not yet been traded in the marketplace. For example, how do we price an initial pub-

lic offering of stock? How will a major new investment project affect the return in-

vestors require on a company™s stock? Although the CAPM does not fully withstand

empirical tests, it is widely used because of the insight it offers and because its accu-

racy suffices for many important applications.

The exploitation of security mispricing to earn risk-free economic profits is called

arbitrage. It typically involves the simultaneous purchase and sale of equivalent se-

curities (often in different markets) in order to profit from discrepancies in their price

relationship.

The most basic principle of capital market theory is that equilibrium market

prices should rule out arbitrage opportunities. If actual security prices allow for arbi-

trage, the resulting opportunities for profitable trading will lead to strong pressure on

security prices that will persist until equilibrium is restored. Only a few investors need

be aware of arbitrage opportunities to bring about a large volume of trades, and

these trades will bring prices back into alignment. Therefore, no-arbitrage restrictions

on security prices are extremely powerful. The implications of no-arbitrage principles

for financial economics were first explored by Modigliani and Miller, both Nobel

Laureates (1985 and 1990).

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Essentials of Investments, and Arbitrage Pricing Companies, 2003

Fifth Edition Theory

216 Part TWO Portfolio Theory

BU TD

TA B L E 7.1

Price per share ($) 39.00 39.00

Share prices and

Shares outstanding 5,000,000 4,000,000

market values

of Bottom Up Market value ($ millions) 195 156

(BU) and Top

Down (TD)

The Arbitrage Pricing Theory (APT) developed by Stephen Ross uses a no-arbitrage argu-

ment to derive the same relationship between expected return and risk as the CAPM. We ex-

plore the risk-return relationship using well-diversified portfolios and discuss the similarities

and differences between the APT and the CAPM.

7.1 DEMAND FOR STOCKS AND EQUILIBRIUM PRICES

So far we have been concerned with efficient diversification, the optimal risky portfolio, and

its risk-return profile. We haven™t had much to say about how expected returns are determined