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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.

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Lecture Eight

Portfolio Management

  • Stand-alone risk

  • Portfolio risk

  • Risk & return: CAPM/SML


What is investment risk
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.


Probability distribution

Firm X

Firm Y

Rate of

return (%)

-70

0

15

100

Expected Rate of Return


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

Investment Alternatives(Given in the problem)


Why is the t bill return independent of the economy
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
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.


Do the returns of ht and coll move with or counter to the economy
Do the returns of HT and Coll. move with or counter to the economy?

  • HT: With. Positive correlation. Typical.

  • Coll: Countercyclical. Negative correlation. Unusual.


Calculate the expected rate of return on each alternative
Calculate the expected rate of return on each alternative: economy?

^

k = expected rate of return.

^

kHT = (-22%)0.1 + (-2%)0.20

+ (20%)0.40 + (35%)0.20

+ (50%)0.1 = 17.4%.


^ economy?

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?


What s the standard deviation of returns for each alternative
What’s the standard deviation economy?of returns for each alternative?

= Standard deviation.

.


( economy?

)

(

)

2

2

é

ù

8.0

-

8.0

0

.

1

+

8.0

-

8.0

0

.

2

.5

ê

ú

(

)

(

)

2

2

s

=

+

8.0

-

8.0

0

.

4

+

8.0

-

8.0

0

.

2

ê

ú

T

-

bills

ê

ú

(

)

2

+

8

.

0 -

8

.

0

0

.

1

ë

û

.

sT-bills = 0.0%.

sColl = 13.4%.

sUSR = 18.8%.

sM = 15.3%.

sHT = 20.0%.


Prob. economy?

T-bill

USR

HT

0

8

13.8

17.4

Rate of Return (%)


  • Standard deviation economy? (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
Expected Returns vs. Risk economy?

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.


Coefficient of variation cv
Coefficient of Variation (CV) economy?

Standardized measure of dispersion

about the expected value:

Std dev s

CV = = .

^

Mean

k

Shows risk per unit of return.


B economy?

A

0

sA = sB , but A is riskier because larger

probability of losses.

s

= CVA > CVB.

^

k


Portfolio risk and return
Portfolio Risk and Return economy?

Assume a two-stock portfolio with $50,000 in HT and $50,000 in Collections.

^

Calculate kp and sp.


Portfolio return k p
Portfolio Return, k economy?p

^

^

kp is a weighted average:

n

^

^

kp = Swikw.

i = 1

^

kp = 0.5(17.4%) + 0.5(1.7%) = 9.6%.

^

^

^

kp is between kHT and kCOLL.


Alternative method
Alternative Method economy?

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%.


1 economy?

/

2

2

(

)

é

ù

3.0

-

9.6

0

.

10

ê

ú

ê

ú

2

(

)

+

6

.

4 -

9

.

6

0

.

20

ê

ú

ê

ú

2

(

)

s

=

+

10

.

0 -

9

.

6

0

.

40

= 3.3%.

ê

ú

p

ê

ú

2

(

)

+

12

.

5 -

9

.

6

0

.

20

ê

ú

ê

ú

ê

ú

2

(

)

+

15

.

0 -

9

.

6

0

.

10

ë

û

3.3%

CVp = = 0.34.

9.6%


  • s economy?p = 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.

^


General statements about risk
General statements about risk economy?

  • Most stocks are positively correlated. rk,m» 0.65.

  • s » 35% for an average stock.

  • Combining stocks generally lowers risk.


Returns distribution for two perfectly negatively correlated stocks r 1 0 and for portfolio wm
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


Returns distributions for two perfectly positively correlated stocks r 1 0 and for portfolio mm

25 Stocks (r = -1.0) and for Portfolio WM

25

15

15

0

0

-10

-10

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


What would happen to the Stocks (r = -1.0) and for Portfolio WMriskiness 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.

^


Prob. Stocks (r = -1.0) and for Portfolio WM

Large

2

1

0

15

Even with large N, sp» 20%


s Stocks (r = -1.0) and for Portfolio WMp (%)

Company Specific Risk

35

Stand-Alone Risk, sp

20

0

Market Risk

10 20 30 40 2,000+

# Stocks in Portfolio



Stand alone market firm specific
Stand-alone Market Firm-specific risk-reducing impact.

= +

risk risk risk

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.



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! exposed to more risk than diversified investors, would you be compensated for all

  • Stand-alone risk as measured by a stock’s sor CV is not important to a well-diversified investor.

  • Rational, risk averse investors are concerned with sp , which is based on market risk.




How are betas calculated
How are betas calculated? volatility relative to the market.

  • 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.


_ volatility relative to the market.

ki

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

.


Find beta
Find beta volatility relative to the market.

  • “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%?


  • Calculator volatility relative to the market.. 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.


  • If beta = 1.0, average stock. volatility relative to the market.

  • 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.


Can a beta be negative
Can a beta be negative? volatility relative to the market.

Answer: Yes, if ri,m is negative. Then in a “beta graph” the regression line will slope downward.


_ volatility relative to the market.

b = 1.29

ki

HT

40

20

b = 0

T-Bills

_

kM

-20 0 20 40

-20

Coll.

b = -0.86


Expected Risk volatility relative to the market.

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.


Use the sml to calculate the required returns
Use the SML to calculate the volatility relative to the market.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
Required Rates of Return volatility relative to the market.

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%.


Expected vs required returns
Expected vs. Required Returns volatility relative to the market.

^

k

k

HT 17.4% 17.0% Undervalued:

k > k

Market 15.0 15.0 Fairly valued

USR 13.8 12.8 Undervalued:

k > k

T-bills 8.0 8.0 Fairly valued

Coll. 1.7 2.0 Overvalued:

k < k

^

^

^


SML: k volatility relative to the market.i = 8% + (15% - 8%) bi .

ki (%)

SML

.

HT

.

.

kM = 15

kRF = 8

USR

.

T-bills

.

Coll.

Risk, bi

-1 0 1 2


Calculate beta for a portfolio with 50 ht and 50 collections
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.


The required return on the ht coll portfolio is
The required return on the HT/Coll. portfolio is: Collections

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%.


If investors raise inflation expectations by 3 what would happen to the sml
If investors raise inflation Collectionsexpectations by 3%, whatwould happen to the SML?


Required Rate Collections

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


If inflation did not change Collectionsbut risk aversion increasedenough to cause the marketrisk premium to increase by3 percentage points, whatwould happen to the SML?


After increase Collections

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


Has the capm been verified through empirical tests
Has the CAPM been verified through empirical tests? Collections

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




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