Risk and Return The following two assumptions permeate most of what we do in finance: –Other things equal, investors prefer higher expected returns. –Other things equal, investors prefer less risk. How do we measure risk and expected returns? What has risk been rewarded historically?

Risk and Return (Historically)

Risk and Return (Before the Fact) Numerical Example: Suppose investors have a choice to invest in one of two securities, A or B, under the following conditions. How would investors measure expected returns? State of the EconomyProbabilityMarket PortfolioRisk FreeStock AStock B Recession %6.00%-20.00%30.00% Normal %6.00%10.00%10.00% Expansion %6.00%40.00%-10.00% 1.00 Expected Return10.00% Expected returns are the probability weighted average of the returns on the stock across states of nature. For stock B, the expected rate of return is 10% =.25*30% +.50*(10.0%) +.25*(-10%) What is the expected rate of return for stock A? For the risk-free asset?

Risk and Return (Continued) Numerical Example: How would investors measure risk for stocks A and B? State of the Economy Prob.Market PortfolioRisk FreeStock AStock B Recession %6.00%-20.00%30.00% Normal %6.00%10.00%10.00% Expansion %6.00%40.00%-10.00% 1.00 Expected Return10.00% Standard Deviation14.14% One measure of risk is the standard deviation. In this context, it is the square root of the sum of the probability weighted squared deviations from the expected returns. For stock B, the standard deviation is –14.14% = [.25*(30-10) 2 +.5*(10-10) *(-10-10) 2 ] 1/2 What is the standard deviation for stock A? For the risk-free asset?

Risk and Return (Continued) What does it mean when two stocks have the same expected returns but different standard deviations? In our problem if returns were normally distributed, with a standard deviation for stock A of 21.21% and an expected return of 10%, the probability a negative return for stock A is about 31.5% What is the probability of a negative return for stock B? Z = (0%-10%)/14.14% = > Probability = ? (See p.752) Which stock is riskier? Why? Given this information, if you could only invest in stock A or B, but not both, which would you choose? Why?

Risk and Return (Continued) Numerical Example: Now suppose investors could invest 40% of their wealth in stock A and 60% in stock B. What are the expected return and standard deviation for the portfolio of A and B? State of the EconomyProbabilityMarket PortfolioRisk FreeStock AStock BA & B Recession %6.00%-20.00%30.00%10.00% Normal %6.00%10.00%10.00%10.00% Expansion %6.00%40.00%-10.00%10.00% 1.00 Expected Return14.00%10.00%10.00% Standard Deviation8.49%21.21%14.14% E(R p ) = SD(R p ) =

Risk and Return (Continued) Why is the expected return of the portfolio just a weighted average of the expected returns of stocks A and B, but the standard deviation of the portfolio is smaller than the standard deviation of either A or B? –The magic of correlation!!!! What is correlation?

Risk and Return (Continued)

The formula for the expected return of our two-stock portfolio, in which 40% of our wealth is invested in stock A and 60% is invested in stock B, is The formula for the standard deviation of our portfolio is

Risk and Return (Continued) In our example, the correlation between returns on stocks A and B is -1. What values can correlation coefficients take on? What meaning do you associate with certain correlation coefficients? How, for example, are returns on the risk-free asset correlated with returns on stocks A and B? State of the EconomyProbabilityMarket PortfolioRisk FreeStock AStock BA & B Recession %6.00%-20.00%30.00%10.00% Normal %6.00%10.00%10.00%10.00% Expansion %6.00%40.00%-10.00%10.00% 1.00 Given your choice between an investment in the risk-free asset and the portfolio of stocks A and B, which would you choose? Why?

Risk and Return (Continued)

Why do we care what the correlation is between returns on investment opportunities? –ASSIGNMENT FOR NEXT TIME: Assume you have two stocks in your portfolio. Stock A has an expected return of 10% and a standard deviation of 15%. Stock B has an expected return of 14% and a standard deviation of 20%. –Using the formulas for portfolio expected returns and standard deviations, plot the expected returns and standard deviations using correlations of -1, 0, and 1. Use weights that go from 0.0 to 1.0 on stock A (1.0 to 0.0 on stock B) in increments of.10. Put expected returns on the vertical axis and the standard deviation on the horizontal axis. –Choose the “optimal” weights for stocks A and B under each correlation assumption. –Make overheads of your graphs, one set of overheads per group.

Risk and Return (Continued) ASSIGNMENT CONTINUED: WeightsCorrelation = -1 Correlation = 0Correlation = 1 ABE(Rp) SD(Rp) E(Rp) SD(Rp) E(Rp) SD(Rp)