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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.