Risk-Return Corporate Finance

8/3/2019 Risk-Return Corporate Finance

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CHAPTER 6 and 7- Risk and Return

1. Basic return concepts2. Basic risk concepts3. Individual Securities

Expected Return, Variance, and Covariance

1. The Return and Risk for Portfolios2. The Efficient Set for Two Assets3. The Efficient Set for Many Securities4. Diversification: An Example5. Riskless Borrowing and Lending

6. Market Equilibrium7. Relationship between Risk and Expected Return (CAPM)8. Summary and Conclusions


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What are investment returns?

Investment returns measure thefinancial results of an investment.

Returns may be historical orprospective (anticipated).

Returns can be expressed in:

Dollar terms.

Percentage terms.


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What is the return on an investment that

costs $1,000 and is sold after 1 year for$1,100?

Dollar return:

Percentage return:

$ Received - $ Invested$1,100 - $1,000 = $100.

$ Return/$ Invested$100/$1,000 = 0.10 = 10%.


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What is investment risk?

Typically, investment returns are not

known with certainty.Investment risk pertains to the

probability of earning a return lessthan that expected.

The greater the chance of a return farbelow the expected return, thegreater the risk.


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Probability distribution

Rate ofreturn (%)50150-20

Stock X

Stock Y

Which stock is riskier? Why?


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Individual Securities

The characteristics of individual securitiesthat are of interest are the:

Expected ReturnVariance and Standard Deviation

Covariance and Correlation


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Calculate the expected rate of

return on each alternative.

.P=r

n

1=i

iir

r = expected rate of return.


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What is the standard deviation

of returns for each alternative?

( ) .

Variance

deviationStandard

1

2

2

=

=

==

=

n

i

ii Prr


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Standard deviation measures thestand-alone risk of an investment.

The larger the standard deviation, the

higher the probability that returns willbe far below the expected return. Coefficient of variation is an

alternative measure of stand-alone

risk.


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Consider the following two risky asset

world. There is a 1/3 chance of eachstate of the economy and the only assetsare a stock fund and a bond fund.

Rate of Return

Scenario Probability Stock fund Bond fund

Recession 33.3% -7% 17%

Normal 33.3% 12% 7%

Boom 33.3% 28% -3%


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Stock fund Bond Fund

Rate of Squared Rate of Squared

Scenario Return Deviation Return Deviation

Recession-7% 3.24% 17% 1.00%

Normal 12% 0.01% 7% 0.00%

Boom 28% 2.89% -3% 1.00%

Expected return 11.00% 7.00%

Variance 0.0205 0.0067

Standard Deviation14.3% 8.2%


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Recession -7% 3.24% 17% 1.00%

Normal 12% 0.01% 7% 0.00%

Boom 28% 2.89% -3% 1.00%


Variance 0.0205 0.0067

Standard Deviation 14.3% 8.2%


%11)(

%)28(3

1%)12(3

1%)7(3

1)(

=

++=

S

S

rE

rE


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Recession -7% 3.24% 17% 1.00%

Normal 12% 0.01% 7% 0.00%

Boom 28% 2.89% -3% 1.00%


Variance 0.0205 0.0067



%7)(

%)3(3

1%)7(3

1%)17(3

1)(

=

++=

B

B

rE

rE


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Recession -7% 3.24% 17% 1.00%

Normal 12% 0.01% 7% 0.00%

Boom 28% 2.89% -3% 1.00%


Variance 0.0205 0.0067



%24.3%)7%11( 2 =


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Recession-7% 3.24% 17% 1.00%

Normal 12% 0.01% 7% 0.00%

Boom 28% 2.89% -3% 1.00%


Variance 0.0205 0.0067



%89.2%)28%11( 2 =


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Recession -7% 3.24% 17% 1.00%

Normal 12% 0.01% 7% 0.00%Boom 28% 2.89% -3% 1.00%


Variance 0.0205 0.0067



%)89.2%01.0%24.3(3

1%05.2 ++=


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Recession -7% 3.24% 17% 1.00%

Normal 12% 0.01% 7% 0.00%

Boom 28% 2.89% -3% 1.00%


Variance 0.0205 0.0067



0205.0%3.14 =


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Recession -7% 3.24% 17% 1.00%

Normal 12% 0.01% 7% 0.00%

Boom 28% 2.89% -3% 1.00%


Variance 0.0205 0.0067


The Return and Risk for Portfolios

Note that stocks have a higher expected return than bonds

and higher risk. Let us turn now to the risk-return tradeoff

of a portfolio that is 50% invested in bonds and 50%

invested in stocks.


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Rate of Return

Scenario Stock fund Bond fund Portfolio squared deviation

Recession -7% 17% 5.0% 0.160%

Normal 12% 7% 9.5% 0.003%

Boom 28% -3% 12.5% 0.123%

Expected return 11.00% 7.00% 9.0%

Variance 0.0205 0.0067 0.0010

Standard Deviation 14.31% 8.16% 3.08%

The rate of return on the portfolio is a weighted average ofthe returns on the stocks and bonds in the portfolio:

SSBBPrwrwr +=

%)17(%50%)7(%50%5 +=


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Rate of ReturnScenario Stock fund Bond fund Portfolio squared deviation

Recession -7% 17% 5.0% 0.160%

Normal 12% 7% 9.5% 0.003%

Boom 28% -3% 12.5% 0.123%

Expected return 11.00% 7.00% 9.0%Variance 0.0205 0.0067 0.0010



The rate of return on the portfolio is a weighted average of

the returns on the stocks and bonds in the portfolio:

%)7(%50%)12(%50%5.9 +=

SSBBPrwrwr +=


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Recession -7% 17% 5.0% 0.160%

Normal 12% 7% 9.5% 0.003%

Boom 28% -3% 12.5% 0.123%




The rate of return on the portfolio is a weighted average of

the returns on the stocks and bonds in the portfolio:

%)3(%50%)28(%50%5.12 +=

SSBBPrwrwr +=


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Recession -7% 17% 5.0% 0.160%

Normal 12% 7% 9.5% 0.003%

Boom 28% -3% 12.5% 0.123%




The expectedrate of return on the portfolio is a weighted

average of the expectedreturns on the securities in theportfolio.

%)7(%50%)11(%50%9 +=

)()()( SSBBP rEwrEwrE +=


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Rate of Return


Recession -7% 17% 5.0% 0.160%

Normal 12% 7% 9.5% 0.003%

Boom 28% -3% 12.5% 0.123%




The variance of the rate of return on the two risky assets

portfolio isBSSSBB

2

SS

2

BB

2

P ))(w2(w)(w)(w ++=

where BSis the correlation coefficient between the returns

on the stock and bond funds.


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Rate of Return


Recession -7% 17% 5.0% 0.160%

Normal 12% 7% 9.5% 0.003%

Boom 28% -3% 12.5% 0.123%




Observe the decrease in risk that diversification offers.

An equally weighted portfolio (50% in stocks and 50%

in bonds) has less risk than stocks or bonds held in

isolation.


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P o r t f o lo R is k a n d R e t u r n

5 .0 %

6 .0 %

7 .0 %

8 .0 %

9 .0 %

1 0 .0 %

1 1 .0 %

1 2 .0 %

0 .0 % 5 .0 % 1 0 .0 % 1 5 .0 % 2 0 .0 %

P o r t f o l io R is k ( s t a n d a r

PortfolioReturn

% in stocks Risk Return

0% 8.2% 7.0%

5% 7.0% 7.2%

10% 5.9% 7.4%

15% 4.8% 7.6%

20% 3.7% 7.8%

25% 2.6% 8.0%

30% 1.4% 8.2%35% 0.4% 8.4%

40% 0.9% 8.6%

45% 2.0% 8.8%

5 0 .0 0 % 3 .0 8 % 9 .0 0 %

55% 4.2% 9.2%

60% 5.3% 9.4%

65% 6.4% 9.6%70% 7.6% 9.8%

75% 8.7% 10.0%

80% 9.8% 10.2%

85% 10.9% 10.4%

90% 12.1% 10.6%

95% 13.2% 10.8%

100% 14.3% 11.0%

The Efficient Set for Two Assets

We can consider other

portfolio weights besides 50%

in stocks and 50% in bonds

100%

bonds

100%

stocks


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P o r tf o lo R is k a n d R e t u rn C o

5 . 0 %

6 . 0 %

7 . 0 %

8 . 0 %

9 . 0 %

1 0 . 0 %

1 1 . 0 %

1 2 . 0 %

0 . 0 % 2 . 0 % 4 . 0 % 6 . 0 % 8 . 0 % 1 0 . 0 %1 2 . 0 %1 4 . 0 %1 6 . 0 %

P o rt f o lio R is k ( st a n d a rd d

PortfolioReturn

The Efficient Set for Two Assets

We can consider other

portfolio weights besides 50%

in stocks and 50% in bonds

100%

bonds

100%

stocks


0 % 8 .2 % 7 .0 %

5% 7.0% 7.2%

10% 5.9% 7.4%

15% 4.8% 7.6%

20% 3.7% 7.8%

25% 2.6% 8.0%

30% 1.4% 8.2%

35% 0.4% 8.4%

40% 0.9% 8.6%

45% 2.0% 8.8%

50% 3.1% 9.0%

55% 4.2% 9.2%

60% 5.3% 9.4%

65% 6.4% 9.6%

70% 7.6% 9.8%

75% 8.7% 10.0%

80% 9.8% 10.2%

85% 10.9% 10.4%

90% 12.1% 10.6%

95% 13.2% 10.8%

100% 14.3% 11.0%


0% 8.2% 7.0%

5 % 7 .0 % 7 .2 %

10% 5.9% 7.4%

15% 4.8% 7.6%

20% 3.7% 7.8%

25% 2.6% 8.0%

30% 1.4% 8.2%

35% 0.4% 8.4%

40% 0.9% 8.6%

45% 2.0% 8.8%

50% 3.1% 9.0%

55% 4.2% 9.2%

60% 5.3% 9.4%

65% 6.4% 9.6%

70% 7.6% 9.8%

75% 8.7% 10.0%

80% 9.8% 10.2%

85% 10.9% 10.4%

90% 12.1% 10.6%

95% 13.2% 10.8%

100% 14.3% 11.0%


0% 8.2% 7.0%

5% 7.0% 7.2%

1 0 % 5 .9 % 7 .4 %

15% 4.8% 7.6%

20% 3.7% 7.8%

25% 2.6% 8.0%

30% 1.4% 8.2%

35% 0.4% 8.4%

40% 0.9% 8.6%

45% 2.0% 8.8%

50% 3.1% 9.0%

55% 4.2% 9.2%

60% 5.3% 9.4%

65% 6.4% 9.6%

70% 7.6% 9.8%

75% 8.7% 10.0%

80% 9.8% 10.2%

85% 10.9% 10.4%

90% 12.1% 10.6%

95% 13.2% 10.8%

100% 14.3% 11.0%


0% 8.2% 7.0%

5% 7.0% 7.2%

10% 5.9% 7.4%

1 5 % 4 .8 % 7 .6 %

20% 3.7% 7.8%

25% 2.6% 8.0%

30% 1.4% 8.2%

35% 0.4% 8.4%

40% 0.9% 8.6%

45% 2.0% 8.8%

50% 3.1% 9.0%

55% 4.2% 9.2%

60% 5.3% 9.4%

65% 6.4% 9.6%

70% 7.6% 9.8%

75% 8.7% 10.0%

80% 9.8% 10.2%

85% 10.9% 10.4%

90% 12.1% 10.6%

95% 13.2% 10.8%

100% 14.3% 11.0%


0% 8.2% 7.0%

5% 7.0% 7.2%

10% 5.9% 7.4%

15% 4.8% 7.6%

2 0 % 3 .7 % 7 .8 %

25% 2.6% 8.0%

30% 1.4% 8.2%

35% 0.4% 8.4%

40% 0.9% 8.6%

45% 2.0% 8.8%

50% 3.1% 9.0%

55% 4.2% 9.2%

60% 5.3% 9.4%

65% 6.4% 9.6%

70% 7.6% 9.8%

75% 8.7% 10.0%

80% 9.8% 10.2%

85% 10.9% 10.4%

90% 12.1% 10.6%

95% 13.2% 10.8%

100% 14.3% 11.0%


0% 8.2% 7.0%

5% 7.0% 7.2%

10% 5.9% 7.4%

15% 4.8% 7.6%

20% 3.7% 7.8%

2 5 % 2 .6 % 8 .0 %

30% 1.4% 8.2%

35% 0.4% 8.4%

40% 0.9% 8.6%

45% 2.0% 8.8%

50% 3.1% 9.0%

55% 4.2% 9.2%

60% 5.3% 9.4%

65% 6.4% 9.6%

70% 7.6% 9.8%

75% 8.7% 10.0%

80% 9.8% 10.2%

85% 10.9% 10.4%

90% 12.1% 10.6%

95% 13.2% 10.8%

100% 14.3% 11.0%


0% 8.2% 7.0%

5% 7.0% 7.2%

10% 5.9% 7.4%

15% 4.8% 7.6%

20% 3.7% 7.8%

25% 2.6% 8.0%

3 0 % 1 .4 % 8 .2 %

35% 0.4% 8.4%

40% 0.9% 8.6%

45% 2.0% 8.8%

50% 3.1% 9.0%

55% 4.2% 9.2%

60% 5.3% 9.4%

65% 6.4% 9.6%

70% 7.6% 9.8%

75% 8.7% 10.0%

80% 9.8% 10.2%

85% 10.9% 10.4%

90% 12.1% 10.6%

95% 13.2% 10.8%

100% 14.3% 11.0%


0% 8.2% 7.0%

5% 7.0% 7.2%

10% 5.9% 7.4%

15% 4.8% 7.6%

20% 3.7% 7.8%

25% 2.6% 8.0%

30% 1.4% 8.2%

3 5 % 0 .4 % 8 .4 %

40% 0.9% 8.6%

45% 2.0% 8.8%

50% 3.1% 9.0%

55% 4.2% 9.2%

60% 5.3% 9.4%

65% 6.4% 9.6%

70% 7.6% 9.8%

75% 8.7% 10.0%

80% 9.8% 10.2%

85% 10.9% 10.4%

90% 12.1% 10.6%

95% 13.2% 10.8%

100% 14.3% 11.0%


0% 8.2% 7.0%

5% 7.0% 7.2%

10% 5.9% 7.4%

15% 4.8% 7.6%

20% 3.7% 7.8%

25% 2.6% 8.0%

30% 1.4% 8.2%

35% 0.4% 8.4%

4 0 % 0 .9 % 8 .6 %

45% 2.0% 8.8%

50% 3.1% 9.0%

55% 4.2% 9.2%

60% 5.3% 9.4%

65% 6.4% 9.6%

70% 7.6% 9.8%

75% 8.7% 10.0%

80% 9.8% 10.2%

85% 10.9% 10.4%

90% 12.1% 10.6%

95% 13.2% 10.8%

100% 14.3% 11.0%


0% 8.2% 7.0%

5% 7.0% 7.2%

10% 5.9% 7.4%

15% 4.8% 7.6%

20% 3.7% 7.8%

25% 2.6% 8.0%

30% 1.4% 8.2%

35% 0.4% 8.4%

40% 0.9% 8.6%

4 5 % 2 .0 % 8 .8 %

50% 3.1% 9.0%

55% 4.2% 9.2%

60% 5.3% 9.4%

65% 6.4% 9.6%

70% 7.6% 9.8%

75% 8.7% 10.0%

80% 9.8% 10.2%

85% 10.9% 10.4%

90% 12.1% 10.6%

95% 13.2% 10.8%

100% 14.3% 11.0%


0% 8.2% 7.0%

5% 7.0% 7.2%

10% 5.9% 7.4%

15% 4.8% 7.6%

20% 3.7% 7.8%

25% 2.6% 8.0%

30% 1.4% 8.2%

35% 0.4% 8.4%

40% 0.9% 8.6%

45% 2.0% 8.8%

5 0 % 3 .1 % 9 .0 %

55% 4.2% 9.2%

60% 5.3% 9.4%

65% 6.4% 9.6%

70% 7.6% 9.8%

75% 8.7% 10.0%

80% 9.8% 10.2%

85% 10.9% 10.4%

90% 12.1% 10.6%

95% 13.2% 10.8%

100% 14.3% 11.0%


0% 8.2% 7.0%

5% 7.0% 7.2%

10% 5.9% 7.4%

15% 4.8% 7.6%

20% 3.7% 7.8%

25% 2.6% 8.0%

30% 1.4% 8.2%

35% 0.4% 8.4%

40% 0.9% 8.6%

45% 2.0% 8.8%

50% 3.1% 9.0%

5 5 % 4 .2 % 9 .2 %

60% 5.3% 9.4%

65% 6.4% 9.6%

70% 7.6% 9.8%

75% 8.7% 10.0%

80% 9.8% 10.2%

85% 10.9% 10.4%

90% 12.1% 10.6%

95% 13.2% 10.8%

100% 14.3% 11.0%


0% 8.2% 7.0%

5% 7.0% 7.2%

10% 5.9% 7.4%

15% 4.8% 7.6%

20% 3.7% 7.8%

25% 2.6% 8.0%

30% 1.4% 8.2%

35% 0.4% 8.4%

40% 0.9% 8.6%

45% 2.0% 8.8%

50% 3.1% 9.0%

55% 4.2% 9.2%

6 0 % 5 .3 % 9 .4 %

65% 6.4% 9.6%

70% 7.6% 9.8%

75% 8.7% 10.0%

80% 9.8% 10.2%

85% 10.9% 10.4%

90% 12.1% 10.6%

95% 13.2% 10.8%

100% 14.3% 11.0%


0% 8.2% 7.0%

5% 7.0% 7.2%

10% 5.9% 7.4%

15% 4.8% 7.6%

20% 3.7% 7.8%

25% 2.6% 8.0%

30% 1.4% 8.2%

35% 0.4% 8.4%

40% 0.9% 8.6%

45% 2.0% 8.8%

50% 3.1% 9.0%

55% 4.2% 9.2%

60% 5.3% 9.4%

6 5 % 6 .4 % 9 .6 %

70% 7.6% 9.8%

75% 8.7% 10.0%

80% 9.8% 10.2%

85% 10.9% 10.4%

90% 12.1% 10.6%

95% 13.2% 10.8%

100% 14.3% 11.0%


0% 8.2% 7.0%

5% 7.0% 7.2%

10% 5.9% 7.4%

15% 4.8% 7.6%

20% 3.7% 7.8%

25% 2.6% 8.0%

30% 1.4% 8.2%

35% 0.4% 8.4%

40% 0.9% 8.6%

45% 2.0% 8.8%

50% 3.1% 9.0%

55% 4.2% 9.2%

60% 5.3% 9.4%

65% 6.4% 9.6%

7 0 % 7 .6 % 9 .8 %

75% 8.7% 10.0%

80% 9.8% 10.2%

85% 10.9% 10.4%

90% 12.1% 10.6%

95% 13.2% 10.8%

100% 14.3% 11.0%


0% 8.2% 7.0%

5% 7.0% 7.2%

10% 5.9% 7.4%

15% 4.8% 7.6%

20% 3.7% 7.8%

25% 2.6% 8.0%

30% 1.4% 8.2%

35% 0.4% 8.4%

40% 0.9% 8.6%

45% 2.0% 8.8%

50% 3.1% 9.0%

55% 4.2% 9.2%

60% 5.3% 9.4%

65% 6.4% 9.6%

70% 7.6% 9.8%

7 5 % 8 .7 % 1 0 .0 %

80% 9.8% 10.2%

85% 10.9% 10.4%

90% 12.1% 10.6%

95% 13.2% 10.8%

100% 14.3% 11.0%


0% 8.2% 7.0%

5% 7.0% 7.2%

10% 5.9% 7.4%

15% 4.8% 7.6%

20% 3.7% 7.8%

25% 2.6% 8.0%

30% 1.4% 8.2%

35% 0.4% 8.4%

40% 0.9% 8.6%

45% 2.0% 8.8%

50% 3.1% 9.0%

55% 4.2% 9.2%

60% 5.3% 9.4%

65% 6.4% 9.6%

70% 7.6% 9.8%

75% 8.7% 10.0%

8 0 % 9 .8 % 1 0 .2 %

85% 10.9% 10.4%

90% 12.1% 10.6%

95% 13.2% 10.8%

100% 14.3% 11.0%


0% 8.2% 7.0%

5% 7.0% 7.2%

10% 5.9% 7.4%

15% 4.8% 7.6%

20% 3.7% 7.8%

25% 2.6% 8.0%

30% 1.4% 8.2%

35% 0.4% 8.4%

40% 0.9% 8.6%

45% 2.0% 8.8%

50% 3.1% 9.0%

55% 4.2% 9.2%

60% 5.3% 9.4%

65% 6.4% 9.6%

70% 7.6% 9.8%

75% 8.7% 10.0%

80% 9.8% 10.2%

8 5 % 1 0 .9 % 1 0 .4 %

90% 12.1% 10.6%

95% 13.2% 10.8%

100% 14.3% 11.0%


0% 8.2% 7.0%

5% 7.0% 7.2%

10% 5.9% 7.4%

15% 4.8% 7.6%

20% 3.7% 7.8%

25% 2.6% 8.0%

30% 1.4% 8.2%

35% 0.4% 8.4%

40% 0.9% 8.6%

45% 2.0% 8.8%

50% 3.1% 9.0%

55% 4.2% 9.2%

60% 5.3% 9.4%

65% 6.4% 9.6%

70% 7.6% 9.8%

75% 8.7% 10.0%

80% 9.8% 10.2%

85% 10.9% 10.4%

9 0 % 1 2 .1 % 1 0 .6 %

95% 13.2% 10.8%

100% 14.3% 11.0%


0% 8.2% 7.0%

5% 7.0% 7.2%

10% 5.9% 7.4%

15% 4.8% 7.6%

20% 3.7% 7.8%

25% 2.6% 8.0%

30% 1.4% 8.2%

35% 0.4% 8.4%

40% 0.9% 8.6%

45% 2.0% 8.8%

50% 3.1% 9.0%

55% 4.2% 9.2%

60% 5.3% 9.4%

65% 6.4% 9.6%

70% 7.6% 9.8%

75% 8.7% 10.0%

80% 9.8% 10.2%

85% 10.9% 10.4%

90% 12.1% 10.6%

9 5 % 1 3 .2 % 1 0 .8 %

100% 14.3% 11.0%


0% 8.2% 7.0%

5% 7.0% 7.2%

10% 5.9% 7.4%

15% 4.8% 7.6%

20% 3.7% 7.8%

25% 2.6% 8.0%

30% 1.4% 8.2%

35% 0.4% 8.4%

40% 0.9% 8.6%

45% 2.0% 8.8%

50% 3.1% 9.0%

55% 4.2% 9.2%

60% 5.3% 9.4%

65% 6.4% 9.6%

70% 7.6% 9.8%

75% 8.7% 10.0%

80% 9.8% 10.2%

85% 10.9% 10.4%

90% 12.1% 10.6%

95% 13.2% 10.8%

1 0 0 % 1 4 .3 % 1 1 .0 %


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29

Two-Security Portfolios with Various

Correlations

100%

bonds

return

100%

stocks

= 0.2

= 1.0

= -1.0


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30

Portfolio Risk/Return Two Securities:Correlation Effects

Relationship depends on correlationcoefficient

-1.0 < < +1.0

The smaller the correlation, the greater therisk reduction potential

If = +1.0, no risk reduction is possible


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31

What would happen to the

risk of portfolio as more randomlyselected stocks were added?

p would decrease because the added

stocks would not be perfectly correlated,

but rp

would remain relatively constant.


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32

Large

0 15

Prob.

2

1

1 35% ; 2 20%.Return


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33

Portfolio Risk as a Function of the Number of Stocks in thePortfolio

Nondiversifiable risk;

Systematic Risk;Market Risk

Diversifiable Risk;

Nonsystematic Risk;

Firm Specific Risk;

Unique Risk

n

In a large portfolio the variance terms are effectivelydiversified away, but the covariance terms are not.

Thus diversification can eliminate some, but not all of the

risk of individual securities.

Portfolio risk


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Stand-alone Market Diversifiable

Market risk is that part of a securitys

stand-alone risk that cannotbeeliminated by diversification.

Firm-specific, or diversifiable, risk is

that part of a securitys stand-alonerisk that can be eliminated bydiversification.

risk risk risk= +


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No. Rational investors will minimize riskby holding portfolios.

They bear only market risk, so prices andreturns reflect this lower risk.

The one-stock investor bears higher

(stand-alone) risk, so the return is lessthan that required by the risk.

Can an investor holding one stock earn

a return commensurate with its risk?


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Conclusions

As more stocks are added, each newstock has a smaller risk-reducingimpact on the portfolio.

In general p falls very slowly afterabout 40 stocks are included.

By forming well-diversified portfolios,investors can eliminate part of theriskiness of owning a single stock.


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37

The Efficient Set for Many Securities

Consider a world with many risky assets; wecan still identify the opportunity setof risk-return combinations of various portfolios.

return

P

Individual Assets


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38


Given the opportunity setwe can identify

the minimum variance portfolio.

return

P

minimumvarianceportfolio

Individual Assets


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39


The section of the opportunity set above theminimum variance portfolio is the efficient

frontier.

return

P

minimumvarianceportfolio

efficie

ntfro

ntier

Individual Assets


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40

Optimal Risky Portfolio with a Risk-Free Asset

In addition to stocks and bonds, considera world that also has risk-free securitieslike T-bills

100%

bonds

100%

stocks

rf

return


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41

Riskless Borrowing and Lending

Now investors can allocate their moneyacross the T-bills and a balanced mutualfund

100%

bonds

100%

stocks

rf

return

Balanced

fund

CML


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Riskless Borrowing and Lending

With a risk-free asset available andthe efficient frontier identified, wechoose the capital allocation line

with the steepest slope

return

P

efficient frontier

rf

CML


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43

Market Equilibrium

With the capital allocation line identified, all investors choose a

point along the linesome combination of the risk-free assetand the market portfolio M. In a world with homogeneousexpectations, Mis the same for all investors.

return

P

efficient frontier

rf

M

CML


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The Separation Property

The Separation Propertystates that the marketportfolio, M, is the same for all investorstheycan separate their risk aversion from their

choice of the market portfolio.

return

P

efficient frontier

rf

M

CML


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Investor risk aversion is revealed in their choice of whereto stay along the capital allocation linenot in theirchoice of the line.

return

P

efficient frontier

rf

M

CML


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46

Market Equilibrium

Just where the investor chooses along the CapitalAsset Line depends on his risk tolerance. The big

point though is that all investors have the same CML.

100%

bonds

100%

stocks

rf

return

Balanced

fund

CML


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Market Equilibrium

All investors have the same CML becausethey all have the same optimal riskyportfolio given the risk-free rate.

100%

bonds

100%

stocks

rf

return

Optimal

Risky

Porfolio

CML


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The separation property implies that portfolio choice

can be separated into two tasks: (1) determine theoptimal risky portfolio, and (2) selecting a point on the

CML.

100%

bonds

100%

stocks

rf

return

Optimal

Risky

Porfolio

CML


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Optimal Risky Portfolio with a Risk-Free

Asset

By the way, the optimal risky portfoliodepends on the risk-free rate as well asthe risky assets.

100%

bonds

100%

stocksreturn

First

Optimal

Risky

Portfolio

Second Optimal

Risky Portfolio

CML0 CML

1

0

fr

1

fr


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50

Market risk, which is relevant for stocksheld in well-diversified portfolios, is defined

as the contribution of a security to theoverall riskiness of the portfolio.

It is measured by a stocks beta coefficient,which measures the stocks volatility

relative to the market. What is the relevant risk for a stock held in

isolation?

How is market risk measured for

individual securities?


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Definition of Risk When Investors

Hold the Market Portfolio Researchers have shown that the best

measure of the risk of a security in a large

portfolio is the beta ()of the security. Beta measures the responsiveness of a

security to movements in the market

portfolio.)(

)(2

,

M

Mi

iRRRCov

=


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How are betas calculated? Run a regression with returns on the stock

in question plotted on the Y axis andreturns on the market portfolio plotted onthe X axis.

The slope of the regression line, whichmeasures relative volatility, is defined asthe stocks beta coefficient, or.

Analysts typically use four or five years of

monthly returns to establish the regressionline. Some use 52 weeks of weeklyreturns.


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Estimating with regression

S

ecurityReturn

s

S

ecurityReturn

s

Return onReturn on

market %market %

RRii == ii ++ iiRRmm ++ eeii

Slope =Slope = iiChara

cteris

ticLine

Character

istic

Line


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54

Use the historical stock returns tocalculate the beta for XYZ.

Year Market XYZ

1 25.7% 40.0%

2 8.0% -15.0%

3 -11.0% -15.0%4 15.0% 35.0%

5 32.5% 10.0%

6 13.7% 30.0%

7 40.0% 42.0%

8 10.0% -10.0%

9 -10.8% -25.0%

10 -13.1% 25.0%


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Calculating Beta for KWE

RXYZ = 0.83R M + 0.03

R2

= 0.36-40%

-20%

0%

20%

40%

-40% -20% 0% 20% 40%

R M

RXYZ


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Estimates of for Selected

StocksStock Beta

Bank of America 1.55

Borland International 2.35

Travelers, Inc. 1.65

Du Pont 1.00

Kimberly-Clark Corp. 0.90

Microsoft 1.05

Green MountainPower

0.55

Homestake Mining 0.20

Oracle, Inc. 0.49


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If b = 1.0, stock has average risk. If b > 1.0, stock is riskier than average. If b < 1.0, stock is less risky than average. Most stocks have betas in the range of

0.5 to 1.5. Can a stock have a negative beta?

How is beta interpreted?


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Relationship between Risk and ExpectedReturn (CAPM)

Expected Return on the Market:

Expected return on an individual security:

PremiumRiskMarket+= FM RR

)(F

MiF

i RRRR +=

Market Risk Premium

This applies to individual securities held within well-diversified portfolios.


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Expected Return on an IndividualSecurity

This formula is called the Capital Asset PricingModel (CAPM)

)(F

MiF

i RRRR +=

Assume i = 0, then the expected return isRF.

Assumei

= 1, then Mi RR =

Expectedreturn on

a security

=Risk-free

rate+

Beta of the

security

Market risk

premium


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Relationship Between Risk & ExpectedReturn

Expecte

dreturn

)(

FM

iFi RRRR +=

FR

1.0

MR

)(F

MiF

i RRRR +=


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Relationship Between Risk & ExpectedReturn

Expected

re

turn

%3=FR

%3

1.5

%5.13

5.1 =i %10=MR

%5.13%)3%10(5.1%3 =+=iR


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Summary and Conclusions

The contribution of a security to the risk of a well-diversified portfolio is proportional to the covariance of thesecurity's return with the markets return. This contribution

is called the beta.

The CAPM states that the expected return on a security is

positively related to the securitys beta:

)(

)(2

,

M

Mi

iR

RRCov

=

)(F

MiF

i RRRR +=

Risk-Return Corporate Finance

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