1. Risk and Rates of Return
Chapter 8
Risk and Rates of Return
Stand-Alone Risk Portfolio Risk Risk and Return: CAPM/SML

2. Stock Market Level, , 2000=100

3. Apple, Inc. and S&P 500 Monthly Adjusted Price 2000-2016, 2000=100

4. Apple & S&P 500 Monthly Returns, 2000-2016

5. Variance of Apple vs Variance of S&P500
Standard deviation of Apple capital gain in decade shown is 12.8% a month (not annualized) (arithmetic mean 3.47% a month, geometric mean 2.65% a month)
1.0347^123=65, ^123=25
Standard deviation of S&P 500 return in decade shown is 4.7% (arithmetic mean capital gain mean 0.01%, geometric mean -0.16% a month, meaning we’ve lost money)

6. What is risk?
Risk refer to the chance that some unfavorable event will occur
Risk is the potential that a chosen activity will lead to a loss. It results from the variability and the uncertainty of the outcome.
Many of our activities in life entail risk (e.g. going to school, investing in the stock market)

7. Investment Returns
The rate of return on an investment can be calculated as follows:
Investors like returns & dislike risk
Prefer investment with high expected return & low risk
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%.

8. What is investment risk?
Two types of investment risk:
Stand-alone risk
Portfolio risk
Diversifiable risk: can be eliminated by proper diversification
Market risk component: cannot be eliminated
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.
The expected rate of return need to be high enough to compensate the investor for its perceived risk
No investment will be undertake unless the expected rate of return need to be high enough to compensate the investor for its perceived risk
Systematic Vs. unsystematic (company specific)

9. Probability Distributions
A listing of all possible outcomes, and the probability of each occurrence.
Can be shown graphically.
Expected Rate of Return
Rate of
Return (%)
100 15
-70
Firm X
Firm Y

10. Selected Realized Returns, 1926-2010
Source: Based on Ibbotson Stocks, Bonds, Bills, and Inflation: Classic Yearbook (Chicago: Morningstar, Inc., 2011), p. 32.

11. Hypothetical Investment Alternatives
Economy
Prob.
T-Bills 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%

12. Do T-bills promise a completely risk-free return?
Why is the T-bill return independent of the economy?
T-bills will return the promised 5.5%, regardless of the economy
No, T-bills do not provide a completely risk-free return, as they are still exposed to inflation.
Although, very little unexpected inflation is likely to occur over such a short period of time.
T-bills are also risky in terms of reinvestment risk.
T-bills are risk-free in the default sense of the word.

13. Calculating the Expected Return

14. Summary of Expected Returns
High Tech %
Market %
US Rubber %
T-bills %
Collections %
High Tech has the highest expected return, and appears to be the best investment alternative, but is it really? Have we failed to account for risk?

15. Calculating Standard Deviation
In other words, the lesser the variability, the lesser is the risk
SD is a measure of variability
Using historical returns to measure SD
Problems with historical SD: a) how far we should go back b) is what happened in the past going to repeat itself exactly?
A measure of how far the actual is likely to deviate from the expected return
The smaller the STD DEV, The tighter the probability distribution the smaller the risk is. The lower the risk

16. Standard Deviation for Each Investment
σHT = 20%
σColl = 13.2%
σM = 15.2%
σUSR = 18.8%

17. Comparing Standard Deviations
USR
Prob.
T-bills HT
Rate of Return (%)

18. Comments on Std Dev as a Measure of Risk
Standard deviation (σi) measures total,
Stand-alone, risk.
The larger σi is, the lower the probability that actual returns will be close to expected returns.
Larger σi is associated with a wider probability distribution of returns.
If probability distribution is normal:
The actual return between 1 STD DEV of expected return 68.26% of time

19. Comparing Risk & Return
Security
Expected Return,
Risk, s T -
bills 5
.5%
0.0%
High Tech
12.4
20.0
Collections*
1.0
13.2
US Rubber*
9.8
18.8
Market
10.5 1
5.2

20. Coefficient of Variation (CV)
A standardized measure of dispersion about the expected value, that shows the risk per unit of return
Capture the effect of both risk and return
Better measure for evaluating risk in situation which investments have substantially different expected return
If we had two investments with similar returns but different risk, how we choose? Same risk but different return?
What if it was different return and different risk?

21. Illustrating the CV as a Measure of Relative RiskA B
Rate of Return (%)
Prob.
σA = σB , but A is riskier because of a larger probability of losses In other words, the same amount of risk (as measured by σ) for smaller returns

22. Risk Rankings by Coefficient of VariationCV
T-bills
High Tech 1.6
Collections
US Rubber
Market 1.4
Collections has the highest degree of risk per unit of return.
High Tech, despite having the highest standard deviation of returns, has a relatively average CV

23. Example

24. Investor Attitude Towards Risk
Risk aversion: assumes investors dislike risk & require higher rates of return to encourage them to hold riskier securities
Risk premium: the difference between the return on a risky asset & a riskless asset, which serves as compensation for investors to hold riskier securities
The willingness to take risk varies over time
No investment should under take unless the expected rate of return is high enough to compensate for the perceived risk
Remember the coin flip? I either pay you 500 or a 1000?
Risk premium is the incentive I give you to take the risk (higher salary after coming to school)
Higher risk investments will always demand higher expected returns. What will happen if this was not the case?

25. Portfolio Construction: Risk & Return
A portfolio’s expected return is a weighted average of the returns of the portfolio’s component assets
Risk decline as the number of assets increases
Diversification can reduce risk but cannot eliminated it
Impossible for completely riskless stock portfolio
Standard deviation is a little more tricky
Requires that a new probability distribution for the portfolio returns be constructed.
Not weighted average of the STD DEV of the individual
Smaller & depend on both STD DEV & correlation
Diversification does nothing to reduce risk if the portfolio consists of perfectly positively correlated stocks
Most stocks are held in portfolios

26. Calculating Portfolio Expected Return

27. An Alternative Method for Determining Portfolio Expected Return
Economy
Prob HT
Coll
Port
Recession
0.1
-27.0%
27.0%
0.0%
Below avg
0.2
-7.0%
13.0%
3.0%
Average
0.4
15.0%
7.5%
Above avg
30.0%
-11.0%
9.5%
Boom
45.0%
-21.0%
12.0%

28. Calculating Portfolio Standard Deviation and CV

29. Comments on Portfolio Risk Measures
σp = 3.4% is much lower than the σi of either stock
(σHT = 20.0%; σColl = 13.2%).
σp = 3.4% is lower than the weighted average of High Tech & Collections’ σ (16.6%).
Not the weighted average of individual stocks’ STD Dev
The portfolio provides the average return of component stocks, but lower than the average risk
On average, Portfolio Risk decline as the number of assets increase
Why? Negative correlation between stocks

30. General Comments about Risk
σ  35% for an average stock.
Most stocks are positively (though not perfectly) correlated with the market (i.e., ρ between 0 and 1).
Combining stocks in a portfolio generally lowers risk.
Rational investor choose to hold portfolio not a stock
Diversification is only good when assets or stocks are not perfectly correlated

31. Remmber : Correlation
Correlation : The tendency of two variable to move together
Correlation coefficient (ρ ): a measure of the degree of relationship between two variable

32. Returns Distribution for Two Perfectly Negatively Correlated Stocks (ρ = -1.0)
If you decide to invest in each stock separately you would get 15% but with risk
If you invest in a 50/50 portfolio, you would get the 15% with no risk
Buying the portfolio is the correct way to go for rational investors
What will happen to this portfolio?
Their movements offset each other

33. 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
If correlation is perfect, the portfolio has the same risk. Diversification is only good when assets or stocks are not perfectly correlated

34. Partial Correlation, ρ = +0.35
Most stocks are positively correlated
Past data indicates that stocks are 0.35 on average correlated
Creating portfolio is a good way then to reduce risk
You should buy the portfolio not the individual stocks

35. Creating a Portfolio: Beginning with One Stock & Adding Randomly Selected Stocks to Portfolio
σp decreases as stocks are 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 40 stocks), and for large stock portfolios, σp tends to converge to  20%.

36. Illustrating Diversification Effects of a Stock Portfolio

37. Breaking Down Sources of Risk
Stand-alone risk = Market risk + Diversifiable risk
Market risk: portion of a security’s stand-alone risk that cannot be eliminated through diversification.
Measured by beta.
Diversifiable risk: portion of a security’s stand-alone risk that can be eliminated through proper diversification
What is diversifiable risk? Risk caused by random events affecting firms (e.g. product failure, bankruptcy, lawsuits). They are unique to the firm. But when we diversify we offset this risk because when one firm is doing bad the other is doing well.
Market risk is the risk that affects all firms in the market to some extent (e.g. war, inflation, recession). Because it happens to all firms it is not diversifiable.

38. Failure to Diversify
If an investor chooses to hold a one-stock portfolio (doesn’t diversify), would the investor be compensated for the extra risk they bear?
NO! Stand-alone risk is not important to a well diversified investor
Rational, risk-averse investors are concerned with σp, which is based upon market risk.
There can be only one price (the market return) for a given security.
No compensation should be earned for holding unnecessary, diversifiable risk.

39. Example

40. Capital Asset Pricing Model (CAPM)
Model linking risk & 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 that reflects the stock’s risk after diversification.
ri = rRF + (rM – rRF)bi
Primary conclusion: The relevant riskiness of a stock is its contribution to the riskiness of a well-diversified portfolio
Market portfolio is a portfolio of all stock
High administration fees and commissions would be more than offset the benefit for individual investor
Index fund
Since stand-alone, investors are not compensated for and should not care for if they are holding portfolios. How can we measure the market risk of a firm?
Market risk is also called relevant risk, the only risk that matters.
Therefore, this is the risk that measures how much a stock and the market are connected

41. Beta
Measures a stock’s market risk, & shows a stock’s volatility relative to the market.
Indicates how risky a stock is if the stock is held in a well-diversified portfolio.
Theoretically the correct measure of any stock’s risk
The most relevant measure of stock’s risk
Market risk premium: a measure of additional return over the risk-free rate needed to compensate investor for assuming an average amount of risk
It is the slope of SML & reflect the degree of risk in the economy

42. Comments on Beta
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.
Beta=0, riskless security
Most stocks have betas in the range of 0.5 to 1.5.
Changing the stocks in a portfolio can change its riskiness
What does this beta mean if the market went up or down by 10%?

43. Calculating Betas
Well-diversified investors are primarily concerned with how a stock is expected to move relative to the market in the future.
Without a crystal ball to predict the future, analysts are forced to rely on historical data.
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.

44. Can the beta of a security be negative?
Yes, if the correlation between Stock i & 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

45. Illustrating the Calculation of Beta. ri _ rM 20 15 10 5 -5
-10
Regression line:
ri = rM ^
Year rM ri
1 15% 18%

46. Beta Coefficients for High Tech, Collections & T-Billsri 40 20
-20
HT: b = 1.32
T-bills: b = 0
Coll: b = -0.87
Beta is the slope rM

47. Comparing Expected Returns and Beta Coefficients
Security Expected Return Beta High Tech 12.4% 1.32 Market US Rubber T-Bills Collections Riskier securities have higher expected returns, so the rank order is OK.

48. Calculating Portfolio Beta
Explain the formula
What is w?

49. Example

50. Example

51. The Security Market Line (SML): Calculating Required Rates of Return
SML: ri = rRF + (rM – rRF)bi
ri = rRF + (RPM)bi
Assume the yield curve is flat and that rRF = 5.5% and
RPM = rM  rRF = 10.5%  5.5% = 5.0%.

52. What is the market risk premium?
Additional return over the risk-free rate needed to compensate investors for assuming an average amount of risk
Its size depends on:
The perceived risk of the stock market
Investors’ degree of risk aversion
Varies from year to year, but most estimates suggest that it ranges between 4% & 8% per year

53. Calculating Required Rates of Return

54. Expected vs. Required Returns
High Tech
12.4%
12.1%
Undervalued
Market
10.5
Fairly valued
US Rubber
9.8
9.9
Overvalued
T-bills
5.5
Collections
1.0
1.15

55. Illustrating the Security Market Line.
Coll HT
T-bills
USR
SML
rM = 10.5
rRF = 5.5
SML: ri = 5.5% + (5.0%)bi
ri (%)
Risk, bi .
Vertical & horizontal axis
The intercept is the risk-free rate. It is what you would achieve at a beta = 0
The slope is the risk premium
The slope therefore reflects the degree of risk aversion. Higher slope higher aversion
Both the SML and a stock’s position on the line changes over time because of changes in the interest rate, risk aversion, and a company’s beta
Why would beta for a firm change? Things a firm controls (e.g. capital structure, business risk), industry effects (competition)

56. An Example: Equally-Weighted Two-Stock Portfolio
Create a portfolio with 50% invested in High Tech and 50% invested in Collections.
The beta of a portfolio is the weighted average of each of the stock’s betas.
bP = wHTbHT + wCollbColl
bP = 0.5(1.32) + 0.5(-0.87)
bP = 0.225

57. Calculating Portfolio Required Returns
The required return of a portfolio is the weighted average of each of the stock’s required returns.
rP = wHTrHT + wCollrColl
rP = 0.5(12.10%) + 0.5(1.15%)
rP = 6.625%
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)

58. Example

59. Example

60. Is it possible to construct a portfolio of real- world stocks that has a required rate if return equal to the risk-free rate?

61. Factors That Change the SML
What if investors raise inflation expectations by 3%, what would happen to the SML?
SML1
ri (%)
SML2
13.5
10.5
8.5
5.5
ΔI = 3%
Risk, bi
Risk-free rate = real rate + inflation premium
Inflation premium = expected inflation rate
Inflation premium is a return to compensate investors for loss of purchasing power

62. 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?
SML1
ri (%)
SML2
ΔRPM = 3%
Risk, bi
13.5
10.5
5.5
What if people were not risk averse? Risk did not matter to them? What would the slope of the line be?

63. Factors That Change the SML
As risk aversion increase, so do the this premium & slope of SML
A firm influence its market risk by changing composition of assets or/and use of debt
The greater the average investor aversion of risk:
The steeper the slop of the line
The greater the risk premium for all stocks
The higher the required rate of returns
What if people were not risk averse? Risk did not matter to them? What would the slope of the line be?

64. Factors That Change the SML

65. Factors That Change the SML

66. Example

67. Verifying the CAPM Empirically
The CAPM has not been verified completely.
Statistical tests have problems that make verification almost impossible.
Some argue that there are additional risk factors, other than the market risk premium, that must be considered.
Fama and French’s finding (no relationship between beta and stock returns)
CAPM remains to be very widely used in practice (e.g. analysts’ reports)

68. More Thoughts on the CAPM
Investors seem to be concerned with both market risk & total risk.
Therefore, the SML may not produce a correct estimate of ri.
ri = rRF + (rM – rRF)bi + ???
CAPM/SML concepts are based upon expectations, but betas are calculated using historical data.
A company’s historical data may not reflect investors’ expectations about future riskiness.

69. Some Concluding Thoughts
There is a trade-off between risk and return
Needs to take higher risk To achieve a higher expected return
Diversification is crucial
Real returns are what matters, not nominal returns.
The risk of an investment depends investment horizon
Stocks are very risky on the short-term but their risk is reduced if you hold them over the long-run
While historical data can be helpful, there is no guarantee that the past will repeat itself
Studies have raised concerns about the validity of CAPM
Talk here about stock return forecast. Structural changes.

70. The (in a Sense Fallacious) Risk/Return Pyramid