Stand-alone risk
Portfolio risk
Risk & return: CAPM/SML
CHAPTER 6 Risk and Rates of Return
What is investment risk?
Investment risk pertains to the probability of actually earning a low or negative return.
The greater the chance of low or negative returns, the riskier the investment.
Probability distribution
Expected Rate of Return
Rate of
return (%)
100
15
0
-70
Firm X
Firm Y
Annual Total Returns,1926-1998
Average Standard
Return Deviation Distribution
Small-company stocks % %
Large-company stocks
Long-term corporate bonds
Long-term government
Intermediate-term government
. Treasury bills
Inflation
Investment Alternatives
(Given in the problem)
Economy
Prob.
T-Bill
HT
Coll
USR
MP
Recession % % % % %
Below avg.
Average
Above avg.
Boom
Why is the T-bill return independent
of the economy?
Will return the promised 8% regardless 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 HT and Coll. move with or counter to the economy?
HT: Moves with the economy, and has a positive correlation. This is typical.
Coll: Is countercyclical of the economy, and has a negative correlation. This is unusual.
Calculate the expected rate of return on each alternative:
k = expected rate of return.
kHT = (-22%) + (-2%)
+ (20%) + (35%)
+ (50%) = %.
^
^
k
HT
%
Market
USR
T-bill
Coll.
HT appears to be the best, but is it really?
^
What’s the standard deviation
of returns for each alternative?
= Standard deviation.
= =
=
sT-bills = %.
sHT = %.
sColl = %.
sUSR = %.
sM = %.
1/2
s
T
-
bills
=
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( – ) + ( – )
+ ( – ) + ( – )
+ ( – )
Prob.
Rate of Return (%)
T-bill
USR
HT
0
8
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.
Expected Returns vs. Risk
Security
Expected
return
Risk, s
HT % %
Market
USR * *
T-bills
Coll. * *
*Seems misplaced.
Coefficient of Variation (CV)
Standardized measure of dispersion
about the expected value:
Shows risk per unit of return.
CV = = .
Std dev s
^
k
Mean
0
A
B
sA = sB , but A is riskier because larger
probability of losses.
= CVA > CVB.
s
^
k
Portfolio Risk and Return
Assume a two-stock portfolio with $50,000 in HT and $50,000 in Collections.
Calculate kp and sp.
^
Portfolio Return, kp
kp is a weighted average:
kp = (%) + (%) = %.
kp is between kHT and kCOLL.
^
^
^
^
^
^
^
^
kp = S wiki.
n
i = 1
Alternative Method
kp = (%) + (%) + (%)
+ (%) + (%) = %.
^
Estimated Return
Economy
Prob.
HT
Coll.
Port.
Recession % % %
Below avg.
Average
Above avg.
Boom
CVp = = .
%
%
p = = %.
1
2
/
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( – )
+ ( – )
+ ( – )
+ ( – )
+ ( – )
sp = % is much lower than that of either stock (20% and %).
sp = % is lower than average of HT and Coll = %.
\ Portfolio provides average k but lower risk.
Reason: negative correlation.
^
General statements about risk
Most stocks are positively correlated. rk,m » .
s » 35% for an average stock.
Combining stocks generally lowers risk.
Returns Distribution for Two Perfectly Negatively Correlated Stocks (r = ) and for Portfolio WM
25
15
0
-10
-10
-10
0
0
15
15
25
25
Stock W
Stock M
Portfolio WM
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Returns Distributions for Two Perfectly Positively Correlated Stocks (r = +) and for Portfolio MM’
Stock M
0
15
25
-10
Stock M’
0
15
25
-10
Portfolio MM’
0
15
25
-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.
^
Large
0
15
Prob.
2
1
Even with large N, sp » 20%
# Stocks in Portfolio
10 20 30 40 2,000+
Company Specific Risk
Market Risk
20
0
Stand-Alone Risk, sp
sp (%)
35
As more stocks are added, each new stock has a smaller risk-reducing impact.
sp falls very slowly after about 10 stocks are included, and after 40 stocks, there is little, if any, effect. The lower limit for sp is about 20% = sM .
Stand-alone Market Firm-specific
Market risk is that part of a security’s stand-alone risk that cannot be eliminated by diversification, and is measured by beta.
Firm-specific risk is that part of a security’s stand-alone risk that can be eliminated by proper diversification.
risk risk risk
= +
By forming portfolios, we can eliminate about half the riskiness of individual stocks (35% vs. 20%).
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 sp , which is based on market risk.
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.
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.
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.
Year kM ki
1 15% 18%
2 -5 -10
3 12 16
.
.
.
ki
_
kM
_
-5 0 5 10 15 20
20
15
10
5
-5
-10
Illustration of beta calculation:
Regression line:
ki = + kM
^
^
If beta = , average stock.
If beta > , stock riskier than average.
If beta < , stock less risky than average.
Most stocks have betas in the range of to .
List of Beta Coefficients
Stock Beta
Merrill Lynch
America Online
General Electric
Microsoft Corp.
Coca-Cola
IBM
Procter & Gamble
Heinz
Energen Corp.
Empire District Electric
Can a beta be negative?
Answer: Yes, if ri, m is negative. Then in a “beta graph” the regression line will slope downward. Though, a negative beta is highly unlikely.
HT
T-Bills
b = 0
ki
_
kM
_
-20 0 20 40
40
20
-20
b =
Coll.
b =
Riskier securities have higher returns, so the rank order is OK.
HT %
Market
USR
T-bills
Coll.
Expected Risk
Security Return (Beta)
Use the SML to calculate the
required returns.
Assume kRF = 8%.
Note that kM = kM is 15%. (Equil.)
RPM = kM – kRF = 15% – 8% = 7%.
SML: ki = kRF + (kM – kRF)bi .
^
Required Rates of Return
kHT = % + (% – %)()
= % + (7%)()
= % + % = %.
kM = % + (7%)() = %.
kUSR = % + (7%)() = %.
kT-bill = % + (7%)() = %.
kColl = % + (7%)() = %.
Expected vs. Required Returns
HT % % Undervalued:
k > k
Market Fairly valued
USR Undervalued:
k > k
T-bills Fairly valued
Coll. Overvalued:
k < k
^
^
^
^
k
k
.
.
Coll.
.
HT
T-bills
.
USR
SML
kM = 15
kRF = 8
-1 0 1 2
.
SML: ki = 8% + (15% – 8%) bi .
ki (%)
Risk, bi
Calculate beta for a portfolio with 50% HT and 50% Collections
bp= Weighted average
= (bHT) + (bColl)
= () + ()
= .
The required return on the HT/Coll. portfolio is:
kp = Weighted average k
= (17%) + (2%) = %.
Or use SML:
kp= kRF + (kM – kRF) bp
= % + (% – %)()
= % + 7%() = %.
If investors raise inflation expectations by 3%, what would happen to the SML?
SML1
Original situation
Required Rate
of Return k (%)
SML2
0 Risk, bi
18
15
11
8
New SML
D I = 3%
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?
kM = 18%
kM = 15%
SML1
Original situation
Required Rate of Return (%)
SML2
After increase
in risk aversion
Risk, bi
18
15
8
D RPM = 3%
Has the CAPM been verified through empirical tests?
Not completely. Those statistical tests have problems that make verification almost impossible.
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 + ?
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.
