1. CHAPTER 8 Risk and Rates of Return
Stand-alone risk
Portfolio risk
Risk & return: CAPM / SML

2. Investment returns (Amount received – Amount invested)
The rate of return on an investment can be calculated as follows:
(Amount received – Amount invested)
Return = ________________________
Amount invested
For example, if $1,000 is invested and $1,100 is returned after one year, the rate of return for this investment is:
($1,100 - $1,000) / $1,000 = 10%.

3. What is investment risk?
Investment risk is related to the probability of earning a low or negative actual return.
The greater the chance of lower than expected or negative returns, the riskier the investment.

4. Selected Realized Returns, 1990 – 2007
Average Standard
Return Deviation
Small-company stocks 17.5% 33.1%
Large-company stocks
L-T corporate bonds
Source: Based on Stocks, Bonds, Bills, and Inflation: (Valuation Edition) 2008 Yearbook (Bursa Malaysia, 2008)

5. Investment alternatives & Rate of Returns( %)
Economy
Prob.
T-Bill HT
Coll
USR MP
Recession
0.1
5.5%
-27.0%
27.0%
6.0%
-17.0%
Below avg
0.2
-7.0%
13.0%
-14.0%
-3.0%
Average
0.4
15.0%
0.0%
3.0%
10.0%
Above avg
30.0%
-11.0%
41.0%
25.0%
Boom
45.0%
-21.0%
26.0%
38.0%

6. Investment alternatives & Rate of Returns( %)
T-bills will return the promised 5.5%, regardless of the economy- risk-free return
Do T-bills promise a completely risk-free return?
T-bills do not provide a completely risk-free return, as they are still exposed to inflation..
T-bills are also risky in terms of reinvestment rate risk.
However, T-bills are risk-free in the default sense of the word.

7. “Reinvestment Risk” “reinvest cash inflow at going market rates”CF CF CF CF CF
5.5%
5.5%
5.5%
5.5%
5.5%
“reinvest cash inflow at going market rates”
Thus, if market rates , may experience income reduction

8. “inflation risk” If inflation rate = 8%
- funds deposited loses the purchasing power
- “inflation risk”

9. How do the returns of HT and Coll. behave in relation to the market?
HT – Moves with the economy, and has a positive correlation. This is typical.
Coll. – Is countercyclical with the economy, and has a negative correlation. This is unusual.

10. Calculating the expected return

11. Summary of expected returns
HT %
Market %
USR %
T-bill %
Coll %
HT has the highest expected return, and appears to be the best investment alternative, but is it really? Have we failed to account for risk?

12. Expected returns & Standard deviation
12.4
HT’s expected return

13. Calculating standard deviation

14. 6HT 6HT 20% 1/2 (15-12.4)2(0.4) + (30-12.4)2(0.2) (45-12.4)2(0.1)
( )2(0.1) + ( )2(0.2)
( )2(0.4) + ( )2(0.2)
( )2(0.1)
6HT =
6HT
20% =

15. Expected returns & Standard deviation
-7.6%
12.4%
32.4%

16. Standard deviation for each investment

17. Comparing standard deviations
USR
Prob.
T - bill HT
Rate of Return (%)

18. Comments on standard deviation as a measure of risk
Standard deviation (σi) measures total, or stand-alone, risk.
The larger σi is, the lower the probability that actual returns will be closer to expected returns.
Larger σi is associated with a wider probability distribution of returns.

19. Comparing risk and return
Security
Expected return, r
Risk, σ
T-bills
5.5%
0.0% HT
12.4%
20.0%
Coll
1.0%
13.2%
USR
9.8%
18.8%
Market
10.5%
15.2% ^

20. Investor attitude towards risk
Risk aversion – assumes investors dislike risk and require higher rates of return to encourage them to hold riskier securities.
Risk premium – the difference between the return on a risky asset and a riskless asset, which serves as compensation for investors to hold riskier securities.

21. Portfolio construction: Risk and return
Assume a two-stock portfolio is created with $50,000 invested in both HT and Collections.
A portfolio’s expected return is a weighted average of the returns of the portfolio’s component assets.

22. Calculating portfolio expected return
= 12.4%
= 1.0%

23. An alternative method for determining portfolio expected return
Economy
Prob. HT
Coll
Port.
Recession
0.1
-27.0%
27.0%
0.0%*A
Below avg
0.2
-7.0%
13.0%
3.0%*B
Average
0.4
15.0%
0.0%
7.5%
Above avg
30.0%
-11.0%
9.5%
Boom
45.0%
-21.0%
12.0%
*A: 0.5(-27%) +
0.5(27)
= 0%
*B: 0.5(-7%) +
0.5(13%)
= 3%

24. Calculating portfolio standard deviation

25. Comments on portfolio risk measures
σp = 3.4% is much lower than the σi of either stock (σHT = 20.0%; σColl. = 13.2%).
Therefore, the portfolio provides the average return of component stocks, but lower than the average risk.
Why? Negative correlation between stocks.
Combining stocks in a portfolio generally lowers risk.

26. Returns distribution for two perfectly negatively correlated stocks (ρ = -1.0)25 15
-10
Stock W
Stock M
-10
Portfolio WM 25 25 15 15
-10

27. Returns distribution for two perfectly positively correlated stocks (ρ = 1.0)
Stock M 15 25
-10
Stock M’ 15 25
-10
Portfolio MM’ 15 25
-10

28. Illustrating diversification effects of a stock portfolio
# Stocks in Portfolio
,000+
Diversifiable Risk
Market Risk 20
Stand-Alone Risk, sp
sp (%) 35

29. Creating a portfolio: Beginning with one stock and adding randomly selected stocks to portfolio
σp decreases as stocks added, because they would not be perfectly correlated with the existing portfolio.
Expected return of the portfolio would remain relatively constant.
Eventually the diversification benefits of adding more stocks dissipates (after about 10 stocks), and for large stock portfolios, σp tends to converge to  20%.

30. Breaking down sources of risk
Diversifiable risk – portion of a security’s stand-alone risk that can be eliminated through proper diversification.
Market risk – portion of a security’s stand-alone risk that cannot be eliminated through diversification. Measured by beta.

31. Beta
Measures a stock’s market risk, and shows a stock’s volatility relative to the market.
If beta = 1.0, the security is just as risky as the average stock.
If beta > 1.0, the security is riskier than average.
If beta < 1.0, the security is less risky than average.
Most stocks have betas in the range of 0.5 to 1.5.

32. Calculating betas
Well-diversified investors are primarily concerned with how a stock is expected to move relative to the market in the future.
A typical approach to estimate beta is to run a regression of the security’s past returns against the past returns of the market.
The slope of the regression line is defined as the beta coefficient for the security.

33. Illustrating the calculation of beta. ri _ rM 20 15 10 5 -5
-10
Regression line:
ri = rM ^
Year rM ri
1 15% 18%

34. Comparing expected returns and beta coefficients
Security Expected Return Beta
HT %
Market
USR
T-Bills
Coll
Riskier securities have higher returns, so the rank order is OK.

35. Can the beta of a security be negative?
Yes, if the correlation between Stock i and the market is negative (i.e., ρi,m < 0).
If the correlation is negative, the regression line would slope downward, and the beta would be negative.
However, a negative beta is highly unlikely.

36. Beta coefficients for HT, Coll, and T-Billsri _ kM 40 20
-20
HT: b = 1.30
T-bills: b = 0
Coll: b = -0.87

37. Capital Asset Pricing Model (CAPM)
Model linking risk and required returns. CAPM suggests that there is a Security Market Line (SML) that states that a stock’s required return equals the risk-free return plus a risk premium.
ri = rRF + (rM – rRF) bi

38. The Security Market Line (SML): Calculating required rates of return
SML: ri = rRF + (rM – rRF) bi
ri = rRF + (RPM) bi
Assume the rRF = 5.5% and RPM = 5.0%.

39. What is the market risk premium (RPM)?
Additional return over the risk-free rate needed to compensate investors for assuming an average amount of risk.
Varies from year to year, but most estimates suggest that it ranges between 4% and 8% per year.

40. Calculating required rates of return
rHT = 5.5% + (5.0%)(1.32)
= 5.5% + 6.6% = 12.10%
rM = 5.5% + (5.0%)(1.00) = 10.50%
rUSR = 5.5% + (5.0%)(0.88) = %
rT-bill = 5.5% + (5.0%)(0.00) = %
rColl = 5.5% + (5.0%)(-0.87) = %

41. Illustrating the Security Market Line
SML: ri = 5.5% + (5.0%) bi
ri (%)
SML . HT . .
rM = 10.5
rRF = 5.5 .
USR
T-bills .
Risk, bi
Coll.

42. Expected vs. Required returns

43. An example: Equally-weighted two-stock portfolio
Create a portfolio with 50% invested in HT and 50% invested in Collections.
The beta of a portfolio is the weighted average of each of the stock’s betas.
bP = wHT bHT + wColl bColl
bP = 0.5 (1.32) (-0.87)
bP = 0.225

44. Calculating portfolio required returns
The required return of a portfolio is the weighted average of each of the stock’s required returns.
rP = wHT rHT + wColl rColl
rP = 0.5 (12.10%) (1.2%)
rP = 6.6%
Or, using the portfolio’s beta, CAPM can be used to solve for expected return.
rP = rRF + (RPM) bP
rP = 5.5% + (5.0%) (0.225)

45. Factors that change the SML
What if investors raise inflation expectations by 3%, what would happen to the SML?
ri (%)
SML2
D I = 3%
SML1
13.5
10.5
8.5
5.5
Risk, bi

46. Y = C + mx ri = rf + (Rm - Rf)b RPm SML line
- Inflation is captured by rf
- Inflation - rf
- Risk factor is captured by (Rm – Rf)
- Risk factor - (Rm – Rf)

47. Factors that change the SML
What if investors’ risk aversion increased, causing the market risk premium to increase by 3%, what would happen to the SML?
ri (%)
SML2
D RPM = 3%
SML1
13.5
10.5
5.5
Risk, bi