Chapter 08 Risk and Rate of Return Fundamental of Financial Management By:S.Zakir Abbas Zaidi

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 Systematic risk, also known as “undiversifiable risk,” “volatility” or “market risk,” affects the overall market, not just a particular stock or industry.

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%.

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 Use the following example to illustrate the mathematical nature of expected returns: Consider a game where you toss a fair coin: If it is Heads then student A pays student B $1. If it is Tails then student B pays student A $1. Most students will remember from their statistics that the expected value is $0 (=.5(1) + .5(-1)). That means that if the game is played over and over then each student should expect to break-even. However, if the game is only played once, then one student will win $1 and one will lose $1.

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% What is the probability of a recession? 0.2 If the risk-free rate is 6.15%, what is the risk premium? Stock C: 9.9 – 6.15 = 3.75% Stock T: 17.7 – 6.15 = 11.55%

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 It’s important to point out that these formulas are for forecast data, and thus use probabilities to weight possible future outcomes, unlike the formulas in chapter 10 that were for historical data (divided by n-1),as past outcomes are all equally likely. Remind the students that standard deviation is the square root of the variance

Example: Variance and Standard Deviation Consider the previous example. What are the variance and standard deviation for each stock? Stock C 2 = .3(.15-.099)2 + .5(.1-.099)2 + .2(.02-.099)2 = .002029  = .045 Stock T 2 = .3(.25-.177)2 + .5(.2-.177)2 + .2(.01-.177)2 = .007441  = .0863

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

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

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? E(R) = .25(.15) + .5(.08) + .15(.04) + .1(-.03) = .0805 Variance = .25(.15-.0805)2 + .5(.08-.0805)2 + .15(.04-.0805)2 + .1(-.03-.0805)2 = .00267475 Standard Deviation = .051717985

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 StockC on the basis of this criterion. E(R) = .25(.15) + .5(.08) + .15(.04) + .1(-.03) = .0805 Variance = .25(.15-.0805)2 + .5(.08-.0805)2 + .15(.04-.0805)2 + .1(-.03-.0805)2 = .00267475 Standard Deviation = .051717985

The Scale Problem: an Example 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

The Expected Return of a Portfolio  

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 DCLK: 2/15 = .133 KO: 3/15 = .2 INTC: 4/15 = .267 KEI: 6/15 = .4 Show the students that the sum of the weights = 1

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

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%

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:

An Example: Perfect Pos. Correlation

An Example: Perfect Neg. Correlation

An Example: Zero Correlation

Determining Covariance and Correlation (cont'd) 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. N

Determining Covariance and Correlation (cont'd) 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.

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. The instructor’s manual provides information on how to compute the portfolio variance using the correlation or covariance between assets.

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 -24% -17% 4.13% 4 3% 1% 0.001 0.000 -5% -4% 0.21% 5 20% 15% 0.040 0.023 1.18%   8% 0.1222 0.0614 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. Metric: A geometric function that describes the distances between pairs of points in a space

Calculation of beta  

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