6

Analysis of Risk and Return

©2006 Thomson/South-Western

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Introduction

This chapter develops the risk-return relationship for individual projects (investments) and a portfolio of projects.

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Risk and Return

Risk refers to the potential variability of returns from a project or portfolio of projects.

Returns are cash flows. Risk-free returns are known with

certainty. U.S. Treasury Securities

Check out interest rates on the following URLs

http://www.stls.frb.org/fred/data/irates.html http://www.bloomberg.com/

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Expected Return

A weighted average of the individual possible returns

The set of all possible returns is referred to as the “distribution of returns”

The symbol for expected return, r , is called “r hat.”

r = Sum (all possible returns their probability)

^

^

^

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Standard Deviation

Standard Deviation is an absolute measure of risk.

Z score measures the number of standard

deviations a particular rate of return r is from the

expected value of r.

See table V page T5 and slide 6

Coefficient of variation v is a relative measure of

risk.

Risk is an increasing function of time.

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Calculating the Z Score

Z score =

What’s the probability of a loss on an investment with an expected return of 20 percent and a standard deviation of 7 percent?

(0% – 20%)/17% = –1.18 rounded

From table V = 0.1190 or 11.9 percent probability of a loss

Target score – Expected value Standard deviation

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Coefficient of Variation

The coefficient of variation is a relative measure of risk.

The coefficient of variation is an appropriate measure of total risk when comparing two investment projects of different size.

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Risk-Return Relationship

Required return = Risk-free return + Risk premium

Check out the risk-free rate at this Web site:

http://www.cnnfn.com/markets/rates.html

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What Makes Up the Risk-Free Rate? The risk-free rate of return is the sum of

two components:

Real rate of return

+ expected inflation premium

The inflation premium compensates investors for the loss of purchasing power due to inflation

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Sources of the Risk Premium Maturity risk premium

Default risk premium

Seniority risk premium

Marketability risk premium

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Explaining the Maturity PremiumWhat causes interest rates to change

with the time to maturity?

Expectations theory

Liquidity premium theory

Market segmentation theory

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Business Risk and Financial Risk Business risk refers to the variability

of operating earnings over time.

Financial risk refers to the additional variability in earnings per share resulting from the use of debt financing.

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Conceptual Risk-Return Relationship

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Characteristics of the Securities Comprising the Portfolio

Expected return

Standard deviation,

Correlation coefficient

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Efficient Portfolio

Has the highest possible return for a given

Has the lowest possible for a given expected return

^ rr

aa

c bc b RiskRiska and c are preferred to ba and c are efficient

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Diversification

The Portfolio effect is the risk reduction accompanying diversification.

Systematic

Risk

Unsystematic Diversifiable

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Expected Return on a Portfolio A portfolio has common stock from

companies A and B. Stock from A makes up 75% of the portfolio and has an expected return of 12%. Stock from B makes up 25% of the portfolio and has an expected return of 16%.

rp = 0.75(12%) + 0.25(16%)

= 13.0%

^

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Standard Deviation of Portfolio Return The standard deviations of returns for the

securities for companies A and B are 10%(σa) and 20%(σb), respectively. With a correlation coefficient(ρab) between the returns on the securities equal to +0.50, the standard deviations of return for the same portfolio is:

σp = √(.75)2(10)2 + (.25)2(20)2 + 2(.75)(.25)(+.50)(10)(20)

= 10.90%

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(a) Perfect Positive Correlation for Two

Investments

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(b) Perfect Negative Correlation for Two

Investments

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(c) Zero Correlation for Two Investments

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CAPM: Only Systematic Risk is Relevant Systematic risk caused by

factors affecting the market as a whole

undiversifiable interest rate changes changes in purchasing

power change in business outlook

Unsystematic risk caused by factors unique to the firm

diversifiable strikes government

regulations management’s

capabilities

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Systematic Risk is Measured by Beta, A measure of the volatility of a securities

return compared to the Market Portfolio

j,m

j,m

j Variance

Covarianceβ

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The Characteristic Line

A regression line of periodic rates of return for security j and the Market Index

Search for (stock beta) on this search engine: http://www.altavista.digital.com/

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Beta Measures Slope

Return on Market Index

Return on GM

Slope = β

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SML Shows the Relationship Between r and ß

r SML

rf

^

^

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Required Rate of Return

The required return for any security j may be defined in terms of systematic risk, j, the expected market return, rm, and the expected risk free rate, rf.

)ˆˆ(βˆfmjfj rrrk

^

^

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Risk Premium

(rm – rf)

Slope of security market line

Will increase or decrease with uncertainties about the future economic

outlook

the degree of risk aversion of investors

^

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SML ^

^

^

1.0

Risk Premium = (9% – 6%) = 3%ka = 6% + 1.5(9% – 6%) = 10.5%

a10.5% ra

1.5

^

6% rf

r SML

9% rm

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CAPM Assumptions

Investors hold well-diversified portfolios

Competitive markets Borrow and lend at

the risk-free rate Investors are risk

averse No taxes

Investors are influenced by systematic risk

Freely available information

Investors have homogeneous expectations

No brokerage charges

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Major Problems in the Practical Application of the CAPM Estimating expected future market returns Determining an appropriate rf

Determining the best estimate of Investors don’t totally ignore

unsystematic risk. Betas are frequently unstable over time. Required returns are determined by

macroeconomic factors.

^

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International Investing

Appears to offer diversification benefits Returns from DMCs tend to have high

positive correlations. Returns from MNCs tend to have lower

correlations. Obtains the benefits of international

diversification by investing in MNCs or DMCs operating in other countries

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Risk of Failure is Not Necessarily Captured by Risk Measures Risk of failure especially relevant

For undiversified investor Costs of bankruptcy

Loss of funds when assets are sold at distressed prices

Legal fees and selling costs incurred Opportunity costs of funds unavailable to

investors during bankruptcy proceedings.

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High-Yield Securities

Sometimes called “Junk Bonds” Bonds with credit ratings below

investment-grade securities Have high returns relative to the returns

available from investment-grade securities

Higher returns achieved only by assuming greater risk.

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Ethical Issues

Growth in high-risk junk bonds

Savings and loan industry

Insurance industry