Stand-alone risk

1
CHAPTER 6 Risk and Rates of Return

• Stand-alone risk
• Portfolio risk
• Risk return CAPM/SML

2
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.
3
Probability distribution
Firm X
Firm Y
Rate of return ()
100
15
0
-70
Expected Rate of Return
4
Annual Total Returns,1926-1998
Average Standard Return Deviation Distribution
Small-companystocks 17.4
33.8 Large-companystocks
13.2 20.3 Long-termcorporate bonds
6.1 8.6 Long-termgovernment
5.7 9.2 Intermediate-termgovernme
nt 5.5 5.7 U.S.
Treasurybills 3.8
3.2 Inflation 3.2
4.5
5
Investment Alternatives(Given in the problem)
Economy
Prob.
T-Bill
HT
Coll
USR
MP
Recession 0.1 8.0 -22.0 28.0 10.0 -13.0 Below
avg. 0.2 8.0 -2.0 14.7 -10.0 1.0 Average 0.4 8.0
20.0 0.0 7.0 15.0 Above avg. 0.2 8.0 35.0 -10.0
45.0 29.0 Boom 0.1 8.0 50.0 -20.0 30.0 43.0 1.0
6
Why is the T-bill return independent of the
economy?
Will return the promised 8 regardless of the
economy.
7
Do T-bills promise a completelyrisk-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.
8
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.

9
Calculate the expected rate of return on each
alternative

k expected rate of return.

kHT (-22)0.1 (-2)0.20 (20)0.40
(35)0.20 (50)0.1 17.4.
10

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?
11
Whats the standard deviationof returns for each
alternative?
? Standard deviation. ?
?
12
1/2
(8.0 8.0)20.1 (8.0 8.0)20.2 (8.0
8.0)20.4 (8.0 8.0)20.2 (8.0 8.0)20.1
s

T
-
bills
sT-bills 0.0.
sColl 13.4. sUSR 18.8. sM 15.3.
sHT 20.0.
13
Prob.
T-bill
USR
HT
0
8
13.8
17.4
Rate of Return ()
14

• 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.

15
Expected Returns vs. Risk
Expected
Risk, s
Security
return
HT 17.4 20.0 Market 15.0 15.3 USR
13.8 18.8 T-bills 8.0 0.0 Coll.
1.7 13.4
Seems misplaced.
16
Coefficient of Variation (CV)
Standardized measure of dispersion about the
expected value
Std dev s
CV .

Mean
k
Shows risk per unit of return.
17
B
A
0
sA sB , but A is riskier because
larger probability of losses.
s
CVA gt CVB.

k
18
Portfolio Risk and Return
Assume a two-stock portfolio with 50,000 in HT
and 50,000 in Collections.

Calculate kp and sp.
19
Portfolio Return, kp


kp is a weighted average
n


kp S wiki.
i 1

kp 0.5(17.4) 0.5(1.7) 9.6.



kp is between kHT and kCOLL.
20
Alternative Method
Estimated Return
Economy
Prob.
HT
Coll.
Port.
Recession 0.10 -22.0 28.0 3.0 Below avg.
0.20 -2.0 14.7 6.4 Average 0.40 20.0 0.0
10.0 Above avg. 0.20 35.0 -10.0 12.5 Boom
0.10 50.0 -20.0 15.0

kp (3.0)0.10 (6.4)0.20 (10.0)0.40
(12.5)0.20 (15.0)0.10 9.6.
21
1
2
/
é
ù
ê
ú
(3.0 9.6)20.10 (6.4 9.6)20.20 (10.0
9.6)20.40 (12.5 9.6)20.20 (15.0 9.6)20.10
ê
ú
ê
ú
ê
ú


?p
3.3.
ê
ú
ê
ú
ê
ú
ê
ú
ê
ú
ë
û
3.3
CVp 0.34.
9.6
22

• 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.

23
General statements about risk

• Most stocks are positively correlated. rk,m
  0.65.
• s 35 for an average stock.
• Combining stocks generally lowers risk.

24
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
0
0
0
.
.
.
.
-10
-10
-10
25
Returns Distributions for Two Perfectly
Positively Correlated Stocks (r 1.0) and for
Portfolio MM
Stock M
Portfolio MM
Stock M
25
15
0
-10
26
What would happen to theriskiness of an average
1-stockportfolio as more randomlyselected
stocks were added?

• sp would decrease because the added stocks would
  not be perfectly correlated but kp would remain
  relatively constant.

27
Prob.
Large
2
1
0
15
Even with large N, sp 20
28
sp ()
Company Specific Risk
35
Stand-Alone Risk, sp
20 0
Market Risk
10 20 30 40 2,000
Stocks in Portfolio
29

• 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 .

30
Stand-alone Market Firm-specific

risk risk risk
Market risk is that part of a securitys
stand-alone risk that cannot be eliminated by
diversification, and is measured by
beta. Firm-specific risk is that part of a
securitys stand-alone risk that can be
eliminated by proper diversification.
31

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

32
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?
33

• NO!
• Stand-alone risk as measured by a stocks 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.

34

• 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.

35
Beta measures a stocks market risk. It shows a
stocks volatility relative to the market.

• Beta shows how risky a stock is if the stock is
  held in a well-diversified portfolio.

36
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.

37
Illustration of beta calculation
Regression line ki -2.59 1.44 kM


.
20 15 10 5
.
Year kM ki 1 15 18 2 -5 -10 3 12 16
_
-5 0 5 10 15 20
kM
-5 -10
.
38

• If beta 1.0, average stock.
• If beta gt 1.0, stock riskier than average.
• If beta lt 1.0, stock less risky than average.
• Most stocks have betas in the range of 0.5 to 1.5.

39
List of Beta Coefficients
Stock Beta
Merrill Lynch
2.00 America Online
1.70 General Electric
1.20 Microsoft Corp.
1.10 Coca-Cola
1.05 IBM
1.05 Procter Gamble 0.85 Heinz

0.80 Energen Corp.
0.80 Empire District Electric 0.45
40
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.
41
_
b 1.29
ki
HT
40 20
b 0
T-Bills
_
kM
-20 0 20 40
-20
Coll.
b -0.86
42
Expected Risk Security Return (Beta)
HT 17.4 1.29 Market 15.0 1.00 USR 13.8
0.68 T-bills 8.0 0.00 Coll. 1.7 -0.86
Riskier securities have higher returns, so the
rank order is OK.
43
Use the SML to calculate therequired 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) 8.0. kColl 8.0
(7)(-0.86) 2.0.
45
Expected vs. Required Returns

k
k
HT 17.4 17.0 Undervalued k gt k Market
15.0 15.0 Fairly valued USR 13.8 12.8
Undervalued k gt k T-bills 8.0 8.0 Fairly
valued Coll. 1.7 2.0 Overvalued k lt k



46
SML ki 8 (15 8) bi .
ki ()
SML
.
HT
.
.
kM 15 kRF 8
USR
.
T-bills
.
Coll.
Risk, bi
-1 0 1 2
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 ()
D I 3
New SML
SML2
SML1
18 15 11 8
Original situation
0 0.5 1.0 1.5 Risk, bi
51
If inflation did not changebut risk aversion
increasedenough to cause the marketrisk premium
to increase by3 percentage points, whatwould
happen to the SML?
52
After increase in risk aversion
Required Rate of Return ()
SML2
kM 18 kM 15
SML1
18 15
D RPM 3
8
Original situation
Risk, bi
1.0
53
Has the CAPM been verified through empirical
tests?

• Not completely. Those statistical tests have
  problems that make verification almost impossible.

54

• 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 companys historical data may
  not reflect investors expectations about future
  riskiness.