Stand-alone risk

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Lecture Eight Portfolio Management

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

2
What is investment risk?
Investment risk pertains to the probability of
earning less than the expected 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
Investment Alternatives(Given in the problem)
5
Why is the T-bill return independent of the
economy?
Will return the promised 8 regardless of the
economy.
6
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.
7
Do the returns of HT and Coll. move with or
counter to the economy?

• HT With. Positive correlation. Typical.
• Coll Countercyclical. Negative correlation.
  Unusual.

8
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.
9

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?
10
Whats the standard deviationof returns for each
alternative?
Standard deviation.
.
11
.
sT-bills 0.0.
sColl 13.4. sUSR 18.8. sM 15.3.
sHT 20.0.
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Prob.
T-bill
USR
HT
0
8
13.8
17.4
Rate of Return ()
13

• 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.

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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.
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Coefficient of Variation (CV)
Standardized measure of dispersion about the
expected value
Std dev s
CV .

Mean
k
Shows risk per unit of return.
16
B
A
0
sA sB , but A is riskier because
larger probability of losses.
s
CVA gt CVB.

k
17
Portfolio Risk and Return
Assume a two-stock portfolio with 50,000 in HT
and 50,000 in Collections.

Calculate kp and sp.
18
Portfolio Return, kp


kp is a weighted average
n


kp S wikw.
i 1

kp 0.5(17.4) 0.5(1.7) 9.6.



kp is between kHT and kCOLL.
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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.
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• 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.

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General statements about risk

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

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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
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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
25
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.

26
Prob.
Large
2
1
0
15
Even with large N, sp 20
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sp ()
Company Specific Risk
35
Stand-Alone Risk, sp
20 0
Market Risk
10 20 30 40 2,000
Stocks in Portfolio
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• As more stocks are added, each new stock has a
  smaller risk-reducing impact.
• sp falls very slowly after about 40 stocks are
  included. The lower limit for sp is about 20
  sM .

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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. Firm-specific risk is that part
of a securitys stand-alone risk which can be
eliminated by proper diversification.
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• By forming portfolios, we can eliminate about
  half the riskiness of individual stocks (35 vs.
  20).

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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?
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• 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.

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• 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.

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• 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.

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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.

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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
.
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Find beta

• By Eye. Plot points, draw in regression line,
  set slope as b Rise/Run. The rise is the
  difference in ki , the run is the difference in
  kM . For example, how much does ki increase or
  decrease when kM increases from 0 to 10?

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• Calculator. Enter data points, and calculator
  does least squares regression ki a bkM
  -2.59 1.44kM. r corr. coefficient 0.997.
• In the real world, we would use weekly or monthly
  returns, with at least a year of data, and would
  always use a computer or calculator.

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• 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.

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Can a beta be negative?
Answer Yes, if ri,m is negative. Then in a
beta graph the regression line will slope
downward.
41
_
b 1.29
ki
HT
40 20
b 0
T-Bills
_
kM
-20 0 20 40
-20
Coll.
b -0.86
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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.

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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.
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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



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SML ki 8 (15 - 8) bi .
ki ()
SML
.
HT
.
.
kM 15 kRF 8
USR
.
T-bills
.
Coll.
Risk, bi
-1 0 1 2
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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.
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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.
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If investors raise inflationexpectations by 3,
whatwould 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 2.0
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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 MRP 3
8
Original situation
Risk, bi
1.0
53
Has the CAPM been verified through empirical
tests?

• Not completely. Those statistical tests have
  problems which 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 ?

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• 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.