1. Lecture Eight
Portfolio Management
Stand-alone risk
Portfolio risk
Risk & return: CAPM/SML

2. 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.

3. Probability distribution
Firm X
Firm Y
Rate of
return (%)
-70 15
100
Expected Rate of Return

4. Investment Alternatives (Given in the problem)
Economy
Prob.
T-Bill HT
Coll
USR MP
Recession % -22.0% 28.0% 10.0% -13.0%
Below avg
Average
Above avg
Boom
1.0

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

6. 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.

7. Do the returns of HT and Coll. move with or counter to the economy?
HT: With. Positive correlation. Typical.
Coll: Countercyclical. Negative correlation. Unusual.

8. Calculate the expected rate of return on each alternative:^
k = expected rate of return. ^
kHT = (-22%)0.1 + (-2%)0.20
+ (20%) (35%)0.20
+ (50%)0.1 = 17.4%.

9. ^ k HT
17.4%
Market
15.0
USR
13.8
T-bill
8.0
Coll.
1.7
HT appears to be the best, but is it really?

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

11. ( ) . .5 sT-bills = 0.0%. sColl = 13.4%. sUSR = 18.8%. sM = 15.3%.
8.0 + 2 1 4 8
0 - . é ë ê ù û ú
sT-bills = 0.0%.
sColl = 13.4%.
sUSR = 18.8%.
sM = 15.3%.
sHT = 20.0%.

12. Prob.
T-bill
USR HT 8
13.8
17.4
Rate of Return (%)

13. Standard deviation (si) measures total, or stand-alone, risk.
The larger the si , the lower the probability that actual returns will be close to the expected return.

14. Expected Returns vs. Risk
Security
Risk, s
return
HT 17.4% 20.0%
Market
USR 13.8* 18.8*
T-bills
Coll. 1.7* 13.4*
*Seems misplaced.

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

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

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

18. kp is a weighted average:^
Portfolio Return, kp ^
kp is a weighted average: n ^ ^
kp = S wikw.
i = 1 ^
kp = 0.5(17.4%) + 0.5(1.7%) = 9.6%. ^ ^ ^
kp is between kHT and kCOLL.

19. Alternative Method ^ kp = (3.0%)0.10 + (6.4%)0.20 + (10.0%)0.40
Estimated Return
Economy
Prob. HT
Coll.
Port.
Recession % 28.0% 3.0%
Below avg
Average
Above avg
Boom ^
kp = (3.0%) (6.4%) (10.0%)0.40
+ (12.5%) (15.0%)0.10 = 9.6%.

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

21. sp = 3.3% is much lower than that of either stock (20% and 13.4%).
sp = 3.3% is lower than average of HT and Coll = 16.7%.
\ Portfolio provides average k but lower risk.
Reason: negative correlation. ^

22. General statements about risk
Most stocks are positively correlated. rk,m » 0.65.
s » 35% for an average stock.
Combining stocks generally lowers risk.

23. 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

24. 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

25. 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. ^

26. Prob.
Large 2 1 15
Even with large N, sp » 20%

27. sp (%) # Stocks in Portfolio Company Specific Risk35
Stand-Alone Risk, sp 20
Market Risk
,000+
# Stocks in Portfolio

28. 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 20% = sM .

29. 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.

30. By forming portfolios, we can eliminate about half the riskiness of individual stocks (35% vs. 20%).

31. 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?

32. 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 sp , which is based on market risk.

33. 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.

34. Beta measures a stock’s market risk
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.

35. How are betas calculated?
Run a regression of past returns on Stock i versus returns on the market. Returns = D/P + g.
The slope of the regression line is defined as the beta coefficient.

36. . . . Illustration of beta calculation: _ ki _ kM Regression line:20 15 10 5 . ^ ^ .
Year kM ki
1 15% 18% _ kM -5
-10 .

37. Find beta
“By Eye.” Plot points, draw in regression line, set 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%?

38. Calculator. Enter data points, and calculator does least squares regression: ki = a + bkM = kM. 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.

39. If beta = 1.0, average stock.
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.

40. Can a beta be negative?
Answer: Yes, if ri,m is negative. Then in a “beta graph” the regression line will slope downward.

41. _ ki HT T-Bills _ kM Coll. b = 1.29 b = 0 -20 0 20 40 b = -0.86 40 20 kM
-20
Coll.
b = -0.86

42. Expected Risk
Security Return (Beta)
HT 17.4% 1.29
Market
USR
T-bills
Coll
Riskier securities have higher returns, so the rank order is OK.

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

44. Required Rates of Return
kHT = 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%.
kUSR = 8.0% + (7%)(0.68) = 12.8%.
kT-bill = 8.0% + (7%)(0.00) = %.
kColl = 8.0% + (7%)(-0.86) = %.

45. Expected vs. Required Returns^ k k
HT 17.4% 17.0% Undervalued:
k > k
Market Fairly valued
USR Undervalued:
T-bills Fairly valued
Coll Overvalued:
k < k ^ ^ ^

46. . . . . . SML: ki = 8% + (15% - 8%) bi . ki (%) SML Risk, bi HT
kM = 15
kRF = 8 .
USR
T-bills .
Coll.
Risk, bi

47. Calculate beta for a portfolio with 50% HT and 50% Collections
bp = Weighted average
= 0.5(bHT) + 0.5(bColl)
= 0.5(1.29) + 0.5(-0.86)
= 0.22.

48. The required return on the HT/Coll. 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%.

49. If investors raise inflation expectations by 3%, what would happen to the SML?

50. Required Rate of Return k (%) SML2 SML1 Original situation
D I = 3%
New SML
SML2
SML1 18 15 11 8
Original situation

51. 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?

52. 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

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

54. 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 + ?

55. Also, CAPM/SML concepts are based on expectations, yet betas are calculated using historical data. A company’s historical data may not reflect investors’ expectations about future riskiness.