Chapter 08Risk and Rate of Return

By:S.Zakir Abbas Zaidi

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Fundamental of Financial

Management

Key Concepts and Skills

• Know how to calculate expected returns• Understand the impact of diversification• Understand the systematic risk principle• Understand the security market line• Understand the risk-return trade-off

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Simple Returns• The return from holding an investment over some

period – say, a year – is simply any cash payments received due to ownership, plus the change in market price, divided by the beginning price.1 You might, for example, buy for $106 a security that and one year from worth $107 one year later. The return would be ($107 + $106)/$100 = 1%.

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Expected Returns

• Expected returns are based on the probabilities of possible outcomes• In this context, “expected” means “average” if the process is

repeated many times• The “expected” return does not even have to be a possible return

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Example: Expected Returns

• Suppose you have predicted the following returns for stocks C and T in three possible states of nature. What are the expected returns?

• State Probability C T• Boom 0.3 0.15 0.25• Normal 0.5 0.10 0.20• Recession ??? 0.02 0.01

• E(RC) = .3(.15) + .5(.10) + .2(.02) = .099 OR 9.9%

• E(RT) = .3(.25) + .5(.20) + .2(.01) = .177 OR 17.7%

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Variance and Standard Deviation

• Variance and standard deviation still measure the volatility of returns• Using unequal probabilities for the entire range of possibilities• Weighted average of squared deviations

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Example: Variance and Standard Deviation

• Consider the previous example. What are the variance and standard deviation for each stock?

• Stock C2 = .3(.15-.099)2 + .5(.1-.099)2 + .2(.02-.099)2

= .002029 = .045

• Stock T2 = .3(.25-.177)2 + .5(.2-.177)2 + .2(.01-.177)2 = .007441 = .0863

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Use of Standard Deviation Information• So far we have been working with a discrete (noncontinuous) probability distribution, one where a

random variable, like return, can take on only certain values within an interval. In such cases we do not have to calculate the standard deviation in order to determine the probability of specific outcomes. To determine the probability of the actual return in our example being less than zero, we look at the shaded section of Table 5.1 and see that the probability is 0.05 + 0.10 = 15%.

• The procedure is slightly more complex when we deal with a continuous distribution.• Suppose that our return distribution had been approximately normal with an expected return equal

to 9 percent and a standard deviation of 8.38 percent. Let’s say that we wish to find the probability that the actual future return will be less than zero. We first determine how many standard deviations 0 percent is from the mean (9 percent). To do this we take the difference between these two values, which happens to be −9 percent, and divide it by the standard deviation. In this case the result is −0.09/0.0838 = −1.07 standard deviations. (The negative sign reminds us that we are looking to the left of the mean.) In general, we can make use of the formula

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Use of Standard Deviation Information• So far we have been working with a discrete (noncontinuous) probability distribution, one where a

random variable, like return, can take on only certain values within an interval. In such cases we do not have to calculate the standard deviation in order to determine the probability of specific outcomes. To determine the probability of the actual return in our example being less than zero, we look at the shaded section of Table 5.1 and see that the probability is 0.05 + 0.10 = 15%.

• The procedure is slightly more complex when we deal with a continuous distribution.• Suppose that our return distribution had been approximately normal with an expected return equal

to 9 percent and a standard deviation of 8.38 percent. Let’s say that we wish to find the probability that the actual future return will be less than zero. We first determine how many standard deviations 0 percent is from the mean (9 percent). To do this we take the difference between these two values, which happens to be −9 percent, and divide it by the standard deviation. In this case the result is −0.09/0.0838 = −1.07 standard deviations. (The negative sign reminds us that we are looking to the left of the mean.) In general, we can make use of the formula

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Another Example

• Consider the following information:• State Probability Ret. on ABC, Inc.• Boom .25 .15• Normal .50 .08• Slowdown .15 .04• Recession .10 -.03

• What is the expected return?• What is the variance?• What is the standard deviation?

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Coefficient of Variation

• Coefficient of Variation StockC = 0.045/.099 = 0.45

• Coefficient of Variation StockT = 0.0863/0.177= 0.49

• Thus, StockT is more riskier than Stock

C on the basis of this criterion.

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The Scale Problem: an Example

Potential ReturnsProb ABC XYZ10% -12% -24%15% -5% -10%50% 2% 4%15% 9% 18%10% 16% 32%

E(R) 2.0% 4.0%Variance 0.00539 0.02156Std. Dev. 7.34% 14.68%C.V. 3.6708 3.6708

Is XYZ really twice as risky as ABC?

No!

Measuring Stand-Alone Risk

• Standard deviation measures the stand-alone risk of an investment.• The larger the standard deviation, the higher the probability that

returns will be far below the expected return.• Coefficient of variation is an alternative measure of stand-alone risk.

Portfolios

• A portfolio is a collection of assets• An asset’s risk and return are important to how the stock affects the

risk and return of the portfolio• The risk-return trade-off for a portfolio is measured by the portfolio

expected return and standard deviation, just as with individual assets

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The Expected Return of a Portfolio

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Example: Portfolio Weights

• Suppose you have $15,000 to invest and you have purchased securities in the following amounts. What are your portfolio weights in each security?

• $2,000 of DCLK• $3,000 of KO• $4,000 of INTC• $6,000 of KEI

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•DCLK: 2/15 = .133•KO: 3/15 = .2•INTC: 4/15 = .267•KEI: 6/15 = .4

Portfolio Expected Returns• The expected return of a portfolio is the weighted

average of the expected returns of the respective assets in the portfolio

• You can also find the expected return by finding the portfolio return in each possible state and computing the expected value as we did with individual securities

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Example: Expected Portfolio Returns• Consider the portfolio weights computed previously. If the individual

stocks have the following expected returns, what is the expected return for the portfolio?

•DCLK: 19.65%•KO: 8.96%• INTC: 9.67%•KEI: 8.13%

• E(RP) = .133(19.65) + .2(8.96) + .267(9.67) + .4(8.13) = 10.24%

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Determining Covariance and Correlation

• To find the risk of a portfolio, one must know the degree to which the stocks’ returns move together.

The Portfolio Standard Deviation

• The portfolio standard deviation can be thought of as a weighted average of the individual standard deviations plus terms that account for the co-movement of returns

• For a two-security portfolio:

P w w r w w 12

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1 2 1 2 1 22 ,

An Example: Perfect Pos. Correlation

Potential ReturnsState of Economy Probability ABC XYZ 50/50 Portfolio

Recession 25% 2% 2% 2%Moderate Growth 50% 8% 8% 8%Boom 25% 14% 14% 14%Expected Return 8% 8% 8%Standard Deviation 4.24% 4.24% 4.24%Correlation 1.00

P . . . . . . . . . .5 0 0424 5 0 0424 2 100 0 0424 0 0424 0 5 0 5 0 04242 2 2 2

An Example: Perfect Neg. Correlation

Potential ReturnsState of Economy Probability ABC XYZ 50/50 Portfolio

Recession 25% 2% 14% 8%Moderate Growth 50% 8% 8% 8%Boom 25% 14% 2% 8%Expected Return 8% 8% 8%Standard Deviation 4.24% 4.24% 0.00%Correlation -1.00

P . . . . . . . . . .5 0 0424 5 0 0424 2 100 0 0424 0 0424 0 5 0 5 0 002 2 2 2

An Example: Zero Correlation

Potential ReturnsState of Economy Probability ABC XYZ 50/50 Portfolio

Recession 25% 2% 2% 2%Moderate Growth 50% 8% 2% 5%Boom 25% 14% 2% 8%Expected Return 8% 2% 5%Standard Deviation 4.24% 0.00% 2.12%Correlation 0.00

P . . . . . . . . .5 0 0424 5 0 0424 2 0 0 0424 0 0424 0 5 0 5 0 02122 2 2 2

Determining Covariance and Correlation (cont'd)

• Covariance• The expected product of the deviations of two returns from their means

• Covariance between Returns Ri and Rj

• Estimate of the Covariance from Historical Data

• If the covariance is positive, the two returns tend to move together.• If the covariance is negative, the two returns tend to move in opposite

directions.

( , ) [( [ ]) ( [ ])] i j i i j jCov R R E R E R R E R

, ,

1( , ) ( ) ( )

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i j i t i j t jtCov R R R R R R

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Determining Covariance and Correlation (cont'd)

• Correlation• A measure of the common risk shared by stocks that does not depend on their

volatility

• The correlation between two stocks will always be between –1 and +1.

( , )( , )

( ) ( )i j

i ji j

Cov R RCorr R R

SD R SD R

Diversification & Correlation

•The extent to which adding stocks to a portfolio reduces its risk depends on the degree of correlation among the stocks: The smaller the correlation coefficients, the lower the risk in a large portfolio. If we could find a set of stocks whose correlations were zero or negative, all risk could be eliminated. However, in the real world, the correlations among the individual stocks are generally positive but less than -1.0, so some but not all risk can be eliminated.

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Variance, Correlation and Beta from Historical Data

Year X M X^2 M^2 (X-Avg) (M - Avg) (X -Avg).(M -Avg)

1 14% 12% 0.020 0.014 6% 7% 0.41%

2 19% 10% 0.036 0.010 11% 5% 0.53%

3 -16% -12% 0.026 0.014 -24% -17% 4.13%

4 3% 1% 0.001 0.000 -5% -4% 0.21%

5 20% 15% 0.040 0.023 12% 10% 1.18%

8% 5% 0.1222 0.0614 0% 0% 6.45%

SD 0.15 0.109 COV 1.61%

CORR 0.98

Beta 1.35

The Concept of Beta

• A stock’s risk consists of two components, market risk and diversifiable risk

• Diversifiable risk can be eliminated by diversification• Beta Coefficient, b – is a metric that shows the extent to which a

given stock’s returns move up and down with the stock market. Beta thus measures market risk.

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Calculation of beta

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Microsoft Excel Worksheet

Unsystematic risk Vs systematic Risk

Diversifiable Risk• That part of a security’s risk associated with random events; it can be

eliminated by proper diversification. This risk is also known as company specific, or unsystematic, risk.

Market Risk• The risk that remains in a portfolio after diversification has eliminated

all company-specific risk. This risk is also known as nondiversifiable or systematic or beta risk.

Beta measures market Risk

THE RELATIONSHIP BETWEEN RISK AND RATES OF RETURN

Security Market Line(SML) Equation

Betas: Relative Volatility of Stocks H, A, and L

The Security Market Line (SML)

Shift in the SML Caused by an Increase in Expected Inflation

Shift in the SML Caused by Increased Risk Aversion

Beta of a Portfolio

• The beta of a portfolio is a weighted average of its individual securities’ betas:

• For example, if an investor holds a $100,000 portfolio consisting of $33,333.33 invested in each of three stocks, and if each of the stocks has a beta of 0.7, then the portfolio’s beta will be

Exercise

Quiz # 04• Define the following terms using graphs or equations to illustrate your

answers whenever feasible:

Quiz # 04