CHAPTER 8: Risk and Rates of Return Updated: September 20, 2011
All Financial Assets Produce CFs
Risk of Asset Depends on Risk of CFs
Stand-alone Risk of Asset’s CFs
Portfolio Risk of CFs
Diversifiable and Market Risk
Risk & return: CAPM / SML
The rate of return on an investment can be calculated as follows:
(Amount received – Amount invested) Return = ________________________
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%.
What is investment risk?
Two types of 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.
Risk = Dispersion of Returns around mean, or expected mean: variance or standard deviation
A listing of all possible outcomes, and the probability of each occurrence.
Can be shown graphically.
Expected Rate of Return
Return (%) 100 15 0 -70
Selected Realized Returns, 1926 – 2004
Small-company stocks 17.5% 33.1%
Large-company stocks 12.4 20.3
L-T corporate bonds 6.2 8.6
L-T government bonds 5.8 9.3
U.S. Treasury bills 3.8 3.1
Source: Based on Stocks, Bonds, Bills, and Inflation: (Valuation
Edition) 2005 Yearbook (Chicago: Ibbotson Associates, 2005), p28.
Economy Prob. T-Bill HT Coll USR MKT.
Recession 0.1 5.5% -27.0% 27.0% 6.0% -17.0%
Below avg 0.2 5.5% -7.0% 13.0% -14.0% -3.0%
Average 0.4 5.5% 15.0% 0.0% 3.0% 10.0%
Above avg 0.2 5.5% 30.0% -11.0% 41.0% 25.0%
Boom 0.1 5.5% 45.0% -21.0% 26.0% 38.0%
Why is the T-bill return independent of the economy? Do T-bills promise a completely risk-free return?
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 rate risk.
T-bills are risk-free in the default sense of the word.
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.
Calculating the expected return
12.4% (0.1) (45%)
(0.2) (30%) (0.4) (15%)
(0.2) (-7%) (0.1) (-27%) r
P r r
return of rate expected r
Summary of expected returns Expected return HT 12.4% Market 10.5% USR 9.8% T-bill 5.5% Coll. 1.0%
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?
Calculating standard deviation
1i i P)r(rσ
Standard deviation for each investment
(0.1)5.5) - (5.5
(0.2)5.5) - (5.5 (0.4)5.5) - (5.5
(0.2)5.5) - (5.5 (0.1)5.5) - (5.5
P )r (r
Comparing standard deviations
Prob. T - bill
0 5.5 9.8 12.4 Rate of Return (%)
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.
Comparing risk and return
Security Expected return, r
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%
* Seem out of place.
Coefficient of Variation (CV)
A standardized measure of dispersion about the expected value, that shows the risk per unit of return.
deviation Standard CV
Risk rankings, by coefficient of variation
CV T-bill 0.0 HT 1.6 Coll. 13.2 USR 1.9 Market 1.4
Collections has the highest degree of risk per unit of return.
HT, despite having the highest standard deviation of returns, has a relatively average CV.
Illustrating the CV as a measure of relative risk
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.
Rate of Return (%)
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.
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.
Standard deviation is a little more tricky and requires that a new probability distribution for the portfolio returns be devised.
Calculating portfolio expected return
6.7% (1.0%) 0.5 (12.4%) 0.5 r
:average weighted a is r
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% 0.0% 7.5%
Above avg 0.2 30.0% -11.0% 9.5%
Boom 0.1 45.0% -21.0% 12.0%
6.7% (12.0%) 0.10 (9.5%) 0.20
(7.5%) 0.40 (3.0%) 0.20 (0.0%) 0.10 rp ^
Calculating portfolio standard deviation and CV
6.7) - (12.0 0.10
6.7) - (9.5 0.20
6.7) - (7.5 0.40
6.7) - (3.0 0.20
6.7) - (0.0 0.10
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 HT and Coll.’s σ (16.6%).
Therefore, the portfolio provides the average return of component stocks, but lower than the average risk.
Why? Negative correlation between stocks.
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.
Returns distribution for two perfectly negatively correlated stocks (ρ = -1.0)
See p. 260
25 25 25
Returns distribution for two perfectly positively correlated stocks (ρ = 1.0)
See p. 261
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 40 stocks), and for large stock portfolios, σp tends to converge to 20%.
Illustrating diversification effects of a stock portfolio
# Stocks in Portfolio 10 20 30 40 2,000+
Stand-Alone Risk, p
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.
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.
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 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.
Measures a stock’s market risk, and 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.
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.
Most stocks have betas in the range of 0.5 to 1.5.
Beta = ΔY/ΔX or ΔKi /ΔKm
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.
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.
Illustrating the calculation of beta
_ -5 0 5 10 15 20
ri = -2.59 + 1.44 rM ^ ^
Year rM ri
1 15% 18%
2 -5 -10
3 12 16
Beta coefficients for HT, Coll, and T-Bills
-20 0 20 40
HT: b = 1.30
T-bills: b = 0
Coll: b = -0.87
Comparing expected returns and beta coefficients
Security Expected Return Beta HT 12.4% 1.32 Market 10.5 1.00 USR 9.8 0.88 T-Bills 5.5 0.00 Coll. 1.0 -0.87 Riskier securities have higher returns, so the rank order is OK.
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 = 5.0%. The market (or equity) risk premium is
RPM = (kM – kRF )= 10.5% – 5.5% = 5%.
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 and investors’ degree of risk aversion.
Varies from year to year, but most estimates suggest that it ranges between 4% and 8% per year.
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) = 9.90% rT-bill = 5.5% + (5.0%)(0.00) = 5.50% rColl = 5.5% + (5.0%)(-0.87) = 1.15%
Expected vs. Required returns
r) r( Overvalued 1.2 1.0 Coll.
r) r( uedFairly val 5.5 5.5 bills-T
r) r( Overvalued 9.9 9.8 USR
r) r( uedFairly val 10.5 10.5 Market
r) r( dUndervalue 12.1% 12.4% HT
Illustrating the Security Market Line
. . Coll.
rM = 10.5
rRF = 5.5
-1 0 1 2
SML: ri = 5.5% + (5.0%) bi
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) + 0.5 (-0.87)
bP = 0.225
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%) + 0.5 (1.15%)
rP = 6.63%
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)
rP = 6.63%
Factors that change the SML
What if investors raise inflation expectations by 3%, what would happen to the SML?
ri (%) SML2
0 0.5 1.0 1.5
D I = 3%
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
0 0.5 1.0 1.5
D RPM = 3%
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.
More thoughts on the CAPM
Investors seem to be concerned with both market risk and 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.