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Lecture Eight Portfolio Management. Stand-alone risk Portfolio risk Risk & return: CAPM/SML. 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. - PowerPoint PPT Presentation
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Lecture Eight Portfolio Management
Stand-alone riskPortfolio riskRisk & return: CAPM/SML
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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.
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Probability distribution
Expected Rate of Return
Rate ofreturn (%)100150-70
Firm X
Firm Y
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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 1.0Average 0.4 8.0 20.0 0.0 7.0 15.0Above avg. 0.2 8.0 35.0 -10.0 45.0 29.0Boom 0.1 8.0 50.0 -20.0 30.0 43.0
1.0
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Why is the T-bill return independent of the economy?
Will return the promised 8% regardless of the economy.
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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.
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Do the returns of HT and Coll. move with or counter to the economy?
HT: With. Positive correlation. Typical.
Coll: Countercyclical. Negative correlation. Unusual.
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Calculate the expected rate of return on each alternative:
.k = k Pi ii=1
n
k = expected rate of return.
kHT = (-22%)0.1 + (-2%)0.20 + (20%)0.40 + (35%)0.20 + (50%)0.1 = 17.4%.
^
^
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kHT 17.4%Market 15.0USR 13.8T-bill 8.0Coll. 1.7
HT appears to be the best, but is it really?
^
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What’s the standard deviationof returns for each alternative?
= Variance = 2
= (k k) Pi2
ii=1
n
= Standard deviation.
.
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= (k k) Pi2
ii=1
n
T-bills = 0.0%.HT = 20.0%.
Coll = 13.4%.USR = 18.8%. M = 15.3%.
.
.5
T-bills = 8.0- 8.0 + 8.0 - 8.0 8.0 - 8.0 + 8.0 - 8.0
2 2
2 2
2
01 0 20 4 0 2
8 0 - 8 0 01
. .. .
. . .
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Prob.
Rate of Return (%)
T-bill
USR
HT
0 8 13.8 17.4
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Standard deviation (i) measures total, or stand-alone, risk.
The larger the i , the lower the probability that actual returns will be close to the expected return.
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Expected Returns vs. Risk
SecurityExpected
return Risk, HT 17.4% 20.0%Market 15.0 15.3USR 13.8* 18.8*T-bills 8.0 0.0Coll. 1.7* 13.4*
*Seems misplaced.
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Coefficient of Variation (CV)
Standardized measure of dispersionabout the expected value:
Shows risk per unit of return.
CV = = . Std dev
k̂Mean
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0
A B
A = B , but A is riskier because largerprobability of losses.
= CVA > CVB.k̂
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Portfolio Risk and Return
Assume a two-stock portfolio with $50,000 in HT and $50,000 in Collections.
Calculate kp and p.^
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Portfolio Return, kp
kp is a weighted average:
kp = 0.5(17.4%) + 0.5(1.7%) = 9.6%.
kp is between kHT and kCOLL.
^
^
^
^
^ ^
^ ^
kp = wikwn
i = 1
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Alternative Method
kp = (3.0%)0.10 + (6.4%)0.20 + (10.0%)0.40 + (12.5%)0.20 + (15.0%)0.10 = 9.6%.
^
Estimated ReturnEconomy Prob. HT Coll. Port.Recession 0.10 -22.0% 28.0% 3.0%Below avg. 0.20 -2.0 14.7 6.4Average 0.40 20.0 0.0 10.0Above avg. 0.20 35.0 -10.0 12.5Boom 0.10 50.0 -20.0 15.0
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= 3.3%.
p =
3.0 - 9.6 2
2
2
2
2
1 20 10
6 4 - 9 6 0 20
10 0 - 9 6 0 40
12 5 - 9 6 0 20
15 0 - 9 6 0 10
.
. . .
. . .
. . .
. . .
/
CVp = = 0.34. 3.3% 9.6%
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p = 3.3% is much lower than that of either stock (20% and 13.4%).
p = 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.
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
25
15
0
-10 -10 -10
0 0
15 15
25 25
Stock W Stock M Portfolio WM
.. .
. .
..
..
.. . . . .
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Returns Distributions for Two Perfectly Positively Correlated Stocks (r = +1.0) and
for Portfolio MM’
Stock M
0
15
25
-10
Stock M’
0
15
25
-10
Portfolio MM’
0
15
25
-10
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What would happen to theriskiness of an average 1-stock
portfolio as more randomlyselected stocks were added?
p would decrease because the added stocks would not be perfectly correlated but kp would remain relatively constant.^
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Large
0 15
Prob.
2
1
Even with large N, p 20%
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# Stocks in Portfolio10 20 30 40 2,000+
Company Specific Risk
Market Risk20
0
Stand-Alone Risk, p
p (%)35
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As more stocks are added, each new stock has a smaller risk-reducing impact.
p falls very slowly after about 40 stocks are included. The lower limit for p is about 20% = M .
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Stand-alone Market Firm-specific
Market risk is that part of a security’s stand-alone risk that cannot be eliminated by diversification.Firm-specific risk is that part of a security’s stand-alone risk which can be eliminated by proper diversification.
risk risk risk= +
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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
stock’s or CV is not important to a well-diversified investor.
Rational, risk averse investors are concerned with p , 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 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.
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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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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 = -2.59 + 1.44 kM^ ^
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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 > 1.0, stock riskier than
average.If beta < 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.
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HT
T-Bills
b = 0
ki
_
kM
_-20 0 20 40
40
20
-20
b = 1.29
Coll.b = -0.86
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Riskier securities have higher returns, so the rank order is OK.
HT 17.4% 1.29Market 15.0 1.00USR 13.8 0.68T-bills 8.0 0.00Coll. 1.7 -0.86
Expected RiskSecurity Return (Beta)
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Use the SML to calculate therequired returns.
Assume kRF = 8%.Note that kM = kM is 15%. (Equil.)RPM = kM - kRF = 15% - 8% = 7%.
SML: ki = kRF + (kM - kRF)bi .
^
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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 > kMarket 15.0 15.0 Fairly valuedUSR 13.8 12.8 Undervalued:
k > kT-bills 8.0 8.0 Fairly valuedColl. 1.7 2.0 Overvalued:
k < k
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..Coll.
.HT
T-bills
.USR
SML
kM = 15
kRF = 8
-1 0 1 2
.
SML: ki = 8% + (15% - 8%) bi .
ki (%)
Risk, bi
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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%, what
would happen to the SML?
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SML1
Original situation
Required Rate of Return k (%)
SML2
0 0.5 1.0 1.5 2.0
181511 8
New SML I = 3%
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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?
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kM = 18%kM = 15%
SML1
Original situation
Required Rate of
Return (%)SML2
After increasein risk aversion
Risk, bi
18
15
8
1.0
MRP = 3%
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Has the CAPM been verified through empirical tests?
Not completely. Those statistical tests have problems which make verification almost impossible.
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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 company’s historical data may not reflect investors’ expectations about future riskiness.