*Risk and Return

*Expected ReturnThe future is uncertain. Investors do not know with certainty whether the economy will be growing rapidly or be in recession.Investors do not know what rate of return their investments will yield.Therefore, they base their decisions on their expectations concerning the future.The expected rate of return on a stock represents the mean of a probability distribution of possible future returns on the stock.

*Expected ReturnThe table below provides a probability distribution for the returns on stocks A and BState Probability Return On Return On Stock A Stock B 1 20% 5% 50% 2 30% 10% 30% 3 30% 15% 10% 4 20% 20% -10%The state represents the state of the economy one period in the future i.e. state 1 could represent a recession and state 2 a growth economy.The probability reflects how likely it is that the state will occur. The sum of the probabilities must equal 100%.The last two columns present the returns or outcomes for stocks A and B that will occur in each of the four states.

*Expected ReturnGiven a probability distribution of returns, the expected return can be calculated using the following equation: N E[R] = S (piRi) i=1Where:E[R] = the expected return on the stock N = the number of statespi = the probability of state iRi = the return on the stock in state i.

*Expected ReturnIn this example, the expected return for stock A would be calculated as follows:

E[R]A = .2(5%) + .3(10%) + .3(15%) + .2(20%) = 12.5%E[R]B = .2(50%) + .3(30%) + .3(10%) + .2(-10%) = 20%

So we see that Stock B offers a higher expected return than Stock A.However, that is only part of the story; we haven't considered risk.

*Measures of RiskRisk reflects the chance that the actual return on an investment may be different than the expected return.One way to measure risk is to calculate the variance and standard deviation of the distribution of returns. We will once again use a probability distribution in our calculations.The distribution used earlier is provided again for ease of use.

*Measures of RiskProbability Distribution:

State Probability Return On Return On Stock A Stock B 1 20% 5% 50% 2 30% 10% 30% 3 30% 15% 10% 4 20% 20% -10%E[R]A = 12.5%E[R]B = 20%

*Measures of RiskGiven an asset's expected return, its variance can be calculated using the following equation: NVar(R) = s2 = S pi(Ri E[R])2 i=1

Where:N = the number of states pi = the probability of state i Ri = the return on the stock in state iE[R] = the expected return on the stock

*Measures of RiskThe standard deviation is calculated as the positive square root of the variance:

SD(R) = s = s2 = (s2)1/2 = (s2)0.5

*Measures of RiskThe variance and standard deviation for stock A is calculated as follows:

s2A = .2(.05 -.125)2 + .3(.1 -.125)2 + .3(.15 -.125)2 + .2(.2 -.125)2 = .2625sA = (.002625)0.5 = .0512 = 5.12%

*Measures of Risk

s2B = .2(.50 -.20)2 + .3(.30 -.20)2 + .3(.10 -.20)2 + .2(-.10 - .20)2 = .042

sB = (.042)0.5 = .2049 = 20.49%

Although Stock B offers a higher expected return than Stock A, it also is riskier since its variance and standard deviation are greater than Stock A's.This, however, is still only part of the picture because most investors choose to hold securities as part of a diversified portfolio.

*Portfolio Risk and ReturnMost investors do not hold stocks in isolation.Instead, they choose to hold a portfolio of several stocks.When this is the case, a portion of an individual stock's risk can be eliminated, i.e., diversified away.From our previous calculations, we know that:the expected return on Stock A is 12.5%the expected return on Stock B is 20%the variance on Stock A is .00263the variance on Stock B is .04200the standard deviation on Stock A is 5.12%the standard deviation on Stock B is 20.49%

*Portfolio Risk and ReturnThe Expected Return on a Portfolio is computed as the weighted average of the expected returns on the stocks which comprise the portfolio.The weights reflect the proportion of the portfolio invested in the stocks.This can be expressed as follows: NE[Rp] = S wiE[Ri] i=1Where:E[Rp] = the expected return on the portfolioN = the number of stocks in the portfoliowi = the proportion of the portfolio invested in stock i E[Ri] = the expected return on stock i

*Portfolio Risk and ReturnFor a portfolio consisting of two assets, the above equation can be expressed as: E[Rp] = w1E[R1] + w2E[R2]

If we have an equally weighted portfolio of stock A and stock B (50% in each stock), then the expected return of the portfolio is: E[Rp] = .50(.125) + .50(.20) = 16.25%

*Portfolio Risk and ReturnThe variance/standard deviation of a portfolio reflects not only the variance/standard deviation of the stocks that make up the portfolio but also how the returns on the stocks which comprise the portfolio vary together.Two measures of how the returns on a pair of stocks vary together are the covariance and the correlation coefficient.Covariance is a measure that combines the variance of a stocks returns with the tendency of those returns to move up or down at the same time other stocks move up or down.Since it is difficult to interpret the magnitude of the covariance terms, a related statistic, the correlation coefficient, is often used to measure the degree of co-movement between two variables. The correlation coefficient simply standardizes the covariance.

*Portfolio Risk and ReturnThe Covariance between the returns on two stocks can be calculated as follows: NCov(RA,RB) = sA,B = S pi(RAi - E[RA])(RBi - E[RB]) i=1Where:sA,B = the covariance between the returns on stocks A and B N = the number of states pi = the probability of state i RAi = the return on stock A in state i E[RA] = the expected return on stock A RBi = the return on stock B in state iE[RB] = the expected return on stock B

*Portfolio Risk and ReturnThe Correlation Coefficient between the returns on two stocks can be calculated as follows: sA,B Cov(RA,RB)Corr(RA,RB) = rA,B = sAsB = SD(RA)SD(RB)

Where:rA,B=the correlation coefficient between the returns on stocks A and BsA,B=the covariance between the returns on stocks A and B, sA=the standard deviation on stock A, and sB=the standard deviation on stock B

*Portfolio Risk and ReturnThe covariance between stock A and stock B is as follows:

sA,B = .2(.05-.125)(.5-.2) + .3(.1-.125)(.3-.2) + .3(.15-.125)(.1-.2) +.2(.2-.125)(-.1-.2) = -.0105

The correlation coefficient between stock A and stock B is as follows: -.0105rA,B = (.0512)(.2049) = -1.00

*Portfolio Risk and ReturnUsing either the correlation coefficient or the covariance, the Variance on a Two-Asset Portfolio can be calculated as follows:

s2p = (wA)2s2A + (wB)2s2B + 2wAwBrA,B sAsB ORs2p = (wA)2s2A + (wB)2s2B + 2wAwB sA,B

The Standard Deviation of the Portfolio equals the positive square root of the the variance.

*Portfolio Risk and ReturnLets calculate the variance and standard deviation of a portfolio comprised of 75% stock A and 25% stock B:

s2p =(.75)2(.0512)2+(.25)2(.2049)2+2(.75)(.25)(-1)(.0512)(.2049)= .00016

sp = .00016 = .0128 = 1.28%

Notice that the portfolio formed by investing 75% in Stock A and 25% in Stock B has a lower variance and standard deviation than either Stocks A or B and the portfolio has a higher expected return than Stock A.This is the purpose of diversification; by forming portfolios, some of the risk inherent in the individual stocks can be eliminated.

*Capital Asset Pricing Model (CAPM) If investors are mainly concerned with the risk of their portfolio rather than the risk of the individual securities in the portfolio, how should the risk of an individual stock be measured?In important tool is the CAPM.CAPM concludes that the relevant risk of an individual stock is its contribution to the risk of a well-diversified portfolio.CAPM specifiesa linear relationship between risk and required return. The equation used for CAPM is as follows: Ki = Krf + bi(Km - Krf)Where:Ki = the required return for the individual securityKrf = the risk-free rate of returnbi = the beta of the individual securityKm = the expected return on the market portfolio(Km - Krf) is called the market risk premiumThis equation can be used to find any of the variables listed above, given the rest of the variables are known.

*CAPM ExampleFind the required return on a stock given that the risk-free rate is 8%, the expected return on the market portfolio is 12%, and the beta of the stock is 2.

Ki = Krf + bi(Km - Krf)Ki = 8% + 2(12% - 8%)Ki = 16% Note that you can then compare the required rate of return to the expected rate of return. You would only invest in stocks where the expected rate of return exceeded the required rate of return.

*Another CAPM ExampleFind the beta on a stock given that its expected return is 12%, the risk-free rate is 4%, and the expected return on the market portfolio is 10%.

12% = 4% + bi(10% - 4%)bi = 12% - 4% 10% - 4% bi = 1.33 Note that beta measures the stocks volatility (or risk) relative to the market.