Risk and Return: The Basics  Stand-alone risk  Portfolio risk  Risk and return: CAPM/SML

What is investment risk?  Investment risk pertains to the probability of earning less than the expected return.  The greater the chance of low or negative returns, the riskier the investment.

Probability distribution Expected Rate of Return Rate of return (%) Firm X Firm Y

Investment Alternatives Economy Prob.T-BillABC Mkt Port. Recession %-22.0% 28.0% 10.0%-13.0% Below avg Average Above avg Boom

Why is the T-bill return independent of the economy? Will return the promised 8% regardless of the state of the economy.

Do T-bills promise a completely risk-free return? No, T-bills are still exposed to the risk of inflation. However, not much unexpected inflation is likely to occur over a relatively short period.

Do the returns of A and B move with or counter to the economy?  A: With. Positive correlation. Typical.  B: Countercyclical. Negative correlation. Unusual.

k =  k i P i. Calculate the expected rate of return on each alternative k = Expected rate of return k A = (-22%) (-2%) (20%) (35%) (50%)0.10 = 17.4%. ^ ^ ^ i = 1 n

A appears to be the best, but is it really? ^ k A 17.4% Market 15.0 C 13.8 T-bill 8.0 B. 1.7

What’s the standard deviation of returns for each alternative?  = Standard deviation.

 T-bills = 0.0%.  A = 20.0%.  B =13.4%.  C =18.8%.  M =15.3%.. 1/2     T-bills =  

Prob. Rate of Return (%) T-bill C A

 Standard deviation (  i ) measures stand-alone risk.  The larger the  i, the higher the probability that actual returns will be far below the expected return.

Coefficient of Variation (CV) Standardized measure of dispersion about the expected value: Shows risk per unit of return. CV = =. Std dev  ^ k Mean

0 A B  A =  B, but A is riskier because larger probability of losses. = CV A > CV B.  ^ k

Portfolio Risk and Return Assume a two-stock portfolio with $50,000 in A and $50,000 in B. Calculate k p and  p. ^

Portfolio Return, k p k p is a weighted average: k p = 0.5(17.4%) + 0.5(1.7%) = 9.6%. k p is between k A and k B. ^ ^ ^ ^ ^^ ^^ k p =   w i k w  n i = 1

Alternative Method k p = (3.0%) (6.4%) (10.0%) (12.5%) (15.0%)0.10 = 9.6%. ^ Estimated Return EconomyProb.ABPort. Recession % 28.0% 3.0% Below avg Average Above avg Boom

= 3.3%.       p = /     CV p = = % 9.6%

Returns Distribution for Two Perfectly Negatively Correlated Stocks (r = -1.0) and for Portfolio WM Stock WStock MPortfolio WM

Returns Distributions for Two Perfectly Positively Correlated Stocks (r = +1.0) and for Portfolio MM’ Stock M Stock M’ Portfolio MM’

What would happen to the riskiness of an average 1- stock portfolio as more randomly selected stocks were added?   p would decrease because the added stocks would not be perfectly correlated but k p would remain relatively constant. ^

Large 0 15 Prob. 2 1

# Stocks in Portfolio Company Specific Risk Market Risk Stand-Alone Risk,  p  p (%)

 As more stocks are added, each new stock has a smaller risk- reducing impact.   p falls very slowly after about 40 stocks are included. The lower limit for  p is about  M = 18%.

Stand-alone Market Firm-specific Market risk is that part of a security’s stand-alone risk that cannot be eliminated by diversification. Firm-specific risk is that part of a security’s stand-alone risk which can be eliminated by proper diversification. risk risk risk = +

By forming portfolios, we can eliminate about half the riskiness of individual stocks (35% vs. 18%).

If you chose to hold a one-stock portfolio and thus are exposed to more risk than diversified investors, would you be compensated for all the risk you bear?

 NO!  Stand-alone risk as measured by a stock’s  or CV is not important to a well-diversified investor.  Rational, risk averse investors are concerned with portfolio risk, and here the relevant risk of an individual stock is its contribution to the riskiness of a portfolio.

 There can only be one price, hence market return, for a given security. Therefore, no compensation can be earned for the additional risk of a one-stock portfolio.

CAPM( Capital Asset Pricing Model) Conclusion: zThe relevant riskiness of an individual stock is its contribution to the riskiness of well-diversified portfolio. zCAPM links risk and required rate of return

 Beta measures a stock’s market risk. It shows a stock’s volatility relative to the market.  Beta shows how risky a stock is if the stock is held in a well-diversified portfolio. The concept of beta, “b”

Yeark M k i 115% 18% kiki _ kMkM _ Illustration of beta calculations: Regression line: k i = k M ^^

Find beta  “By Eye.”  “By Eye.” Plot points, draw in regression line, get slope as b = Rise/Run. The “rise” is the difference in k i, the “run” is the difference in k M. For example, how much does k i increase or decrease when k M increases from 0% to 10%?

 Calculator. Enter data points, and calculator does least squares regression: k i = a + bk M = k M. r = corr. coefficient =  In the real world, we would use weekly or monthly returns, with at least a year of data, and would always use a computer or calculator.

 If beta = 1.0, average risk.  If beta > 1.0, stock riskier than average.  If beta < 1.0, stock less risky than average.  Most stocks have betas in the range of 0.5 to 1.5.

Can a beta be negative? Yes, in theory, if a stock’s returns are negatively correlated with the market. Then in a “beta graph” the regression line will slope downward. In the “real world,” negative beta stocks do not exist.

A T-Bills b = 0 kiki _ kMkM _ b = 1.29 B b = -0.86

Use the SML to calculate the required returns.  Assume k RF = 8%.  Note that k M = k M is 15%. (From market portfolio.)  RP M = k M - k RF = 15% - 8% = 7%. SML: k i = k RF + (k M - k RF )b i. ^

Required Rates of Return k A = 8.0% + (15.0% - 8.0%)(1.29) = 8.0% + (7%)(1.29) = 8.0% + 9.0%= 17.0%. k M = 8.0% + (7%)(1.00)= 15.0%. k C = 8.0% + (7%)(0.68)= 12.8%. k T-bill = 8.0% + (7%)(0.00)= 8.0%. k B = 8.0% + (7%)(-0.86)= 2.0%.

Expected vs. Required Returns ^ ^ ^ ^ A 17.4% 17.0% Undervalued: k > k Market Fairly valued C Undervalued: k > k T-bills Fairly valued B Overvalued: k < k k k

.. B. A T-bills. C SML k M = 15 k RF = SML: k i = 8% + (15% - 8%) b i. k i (%) Risk, b i

Calculate beta for a portfolio with 50% A and 50% B b p = Weighted average = 0.5(b A ) + 0.5(b B ) = 0.5(1.29) + 0.5(-0.86) = 0.22.

The required return on the A/B portfolio is: k p =Weighted average k =0.5(17%) + 0.5(2%) =9.5%. Or use SML: k p =k RF + (k M - k RF ) b p =8.0% + (15.0% - 8.0%)(0.22) =8.0% + 7%(0.22)=9.5%.

If investors raise inflation expectations by 3 percentage points, what would happen to the SML?

SML 1 Original situation Required Rate of Return k (%) SML New SML  I = 3%

If inflation did not change but risk aversion increased enough to cause the market risk premium to increase by 3 percentage points, what would happen to the SML?

k M = 18% k M = 15% SML 1 Original situation Required Rate of Return (%) SML 2 After increase in risk aversion Risk, b i  MRP = 3%

Has the CAPM been verified through empirical tests?  Not completely. That statistical tests have problems which make verification almost impossible.

 Investors seem to be concerned with both market risk and total risk. Therefore, the SML may not produce a correct estimate of k i : k i = k RF + (k M - k RF )b + ?