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 Firm X Firm Y Rate of return (%) -70 15 100 Expected Rate of Return

Investment Alternatives Economy Prob. T-Bill A B C Mkt Port. Recession 0.1 8.0% -22.0% 28.0% 10.0% -13.0% Below avg. 0.2 8.0 -2.0 14.7 -10.0 1.0 Average 0.4 8.0 20.0 0.0 7.0 15.0 Above avg. 0.2 8.0 35.0 -10.0 45.0 29.0 Boom 0.1 8.0 50.0 -20.0 30.0 43.0 1.0

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.

Calculate the expected rate of return on each alternative ^ k = Expected rate of return n k = S kiPi. ^ i = 1 ^ kA = (-22%)0.10 + (-2%)0.20 + (20%)0.40 + (35%)0.20 + (50%)0.10 = 17.4%.

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

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

( ) . 1/2 sT-bills = 0.0%. sB = 13.4%. sC = 18.8%. sM = 15.3%. 8.0 + 2 1 4 8 0 - . sT-bills = 0.0%. sB = 13.4%. sC = 18.8%. sM = 15.3%. sA = 20.0%.

Prob. T-bill C A 8 13.8 17.4 Rate of Return (%)

Standard deviation (si) measures stand-alone risk. The larger the si , 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: CV = = . Std dev s ^ Mean k Shows risk per unit of return.

sA = sB , but A is riskier because larger sA = sB , but A is riskier because larger probability of losses. s = CVA > CVB. ^ k

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

Portfolio Return, kp ^ ^ kp is a weighted average: ^ ^ kp = S wikw. ^ kp is between kA and kB.

Alternative Method ^ Estimated Return Economy Prob. A B Port. Recession 0.10 -22.0% 28.0% 3.0% Below avg. 0.20 -2.0 14.7 6.4 Average 0.40 20.0 0.0 10.0 Above avg. 0.20 35.0 -10.0 12.5 Boom 0.10 50.0 -20.0 15.0 ^ kp = (3.0%)0.10 + (6.4%)0.20 + (10.0%)0.40 + (12.5%)0.20 + (15.0%)0.10 = 9.6%.

1 / 2 ( 3.0 - 9.6 ) 2 . 10 + ( 2 6 . 4 - 9 . 6 ) . 20 s ( 2 = + 10 . 0 - 9 . 6 ) . 40 = 3.3%. p + ( 12 . 5 - 9 . 6 ) 2 . 20 2 + ( 15 . 0 - 9 . 6 ) . 10 3.3% CVp = = 0.34. 9.6%

Returns Distribution for Two Perfectly Negatively Correlated Stocks (r = -1.0) and for Portfolio WM Stock W Stock M Portfolio WM . . . . 25 25 25 . . . . . . . 15 15 15 . . . . -10 -10 -10

Returns Distributions for Two Perfectly Positively Correlated Stocks (r = +1.0) and for Portfolio MM’ Stock M Stock M’ Portfolio MM’ 15 25 -10 15 25 -10 25 15 -10

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

Prob. Large 2 1 15

sp (%) # Stocks in Portfolio Company Specific Risk 35 18 Stand-Alone Risk, sp Market Risk 10 20 30 40 2000+ # Stocks in Portfolio

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

Stand-alone Market Firm-specific = + risk risk risk 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.

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

you be compensated for all the risk you bear? 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 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 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: The relevant riskiness of an individual stock is its contribution to the riskiness of well-diversified portfolio. CAPM links risk and required rate of return

The concept of beta, “b” 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.

. . . Illustration of beta calculations: _ ki _ kM Regression line: 20 15 10 5 . ^ ^ . Year kM ki 1 15% 18% 2 -5 -10 3 12 16 _ -5 0 5 10 15 20 kM -5 -10 .

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

Calculator. Enter data points, and calculator does least squares regression: ki = a + bkM = -2.59 + 1.44kM . r = corr. coefficient = 0.997. 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.

_ b = 1.29 ki A 40 20 b = 0 T-Bills _ -20 0 20 40 kM -20 B b = -0.86

Use the SML to calculate the required returns. SML: ki = kRF + (kM - kRF)bi . Assume kRF = 8%. Note that kM = kM is 15%. (From market portfolio.) RPM = kM - kRF = 15% - 8% = 7%. ^

Required Rates of Return kA = 8.0% + (15.0% - 8.0%)(1.29) = 8.0% + (7%)(1.29) = 8.0% + 9.0% = 17.0%. kM = 8.0% + (7%)(1.00) = 15.0%. kC = 8.0% + (7%)(0.68) = 12.8%. kT-bill = 8.0% + (7%)(0.00) = 8.0%. kB = 8.0% + (7%)(-0.86) = 2.0%.

Expected vs. Required Returns ^ k k A 17.4% 17.0% Undervalued: k > k Market 15.0 15.0 Fairly valued C 13.8 12.8 Undervalued: T-bills 8.0 8.0 Fairly valued B 1.7 2.0 Overvalued: k < k ^ ^ ^

. . . . . SML: ki = 8% + (15% - 8%) bi . ki (%) SML Risk, bi A kM = 15 kRF = 8 . C T-bills . B Risk, bi -1 0 1 2

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

The required return on the A/B portfolio is: kp = Weighted average k =0.5(17%) + 0.5(2%) =9.5%. Or use SML: kp= kRF + (kM - kRF) bp =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?

Required Rate of Return k (%) SML2 SML1 Original situation D I = 3% New SML SML2 SML1 18 15 11 8 Original situation 0 0.5 1.0 1.5 2.0

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?

Required Rate of Return (%) After increase in risk aversion Required Rate of Return (%) SML2 kM = 18% kM = 15% 18 15 SML1 D MRP = 3% 8 Original situation Risk, bi 1.0

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 Investors seem to be concerned with both market risk and total risk. Therefore, the SML may not produce a correct estimate of ki: ki = kRF + (kM - kRF)b + ?