Principles of Finance with Excel (Основы финансов c Excel)

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PFE, Chapter 13: The CAPM and SML page 30

Asset A Asset B

Mean return 30% 13%

Return sigma σ

40% 10%

Correlation

0.5

5.b. Stock C has a

of 1.3 and the portfolio of 75% C and 25% D has a

= 1.8. What

is the

of stock D?

5.c. Suppose the risk-free rate is r

= 5%, E(r

) = 15% and

= 25%.

•

You have $1,000 to invest. If you follow an optimal investment policy, and if you desire

to invest $500 in the risk-free asset, what is the mean and standard deviation of your

portfolio return?

•

Your sister also has $1,000 to invest, but wants to borrow another $1,000 in order to

make an investment of $2,000 in the market portfolio M. What will be the mean and

standard deviation of her portfolio return?

•

Which portfolio is better, yours or your sister’s?

Benninga, Portfolios with > 3 assets Page 31

Appendix: The CAPM with 3 or more assets

Introduction

This appendix generalizes the CAPM and SML discussion in this chapter. The first part

of the appendix discusses the portfolios of 3 assets. It will then be clear how to apply this to

portfolios of more than 3 assets.

In preparation for this chapter, we recall the messages of the Chapter 13. This appendix

is meant to confirm that all of these “messages” still hold, even if there are more than 2 risky

assets.

•

Calculation of the efficient frontier

•

Calculation of the Sharpe ratio.

•

Calculating the market portfolio—the portfolio of risky assets for which the Sharpe ratio

is maximized. This calculation also requires the risk-free rate r

•

Calculating the SML—this is a relation between the expected return of any asset, the

risk-free rate r

, and the expected return on the market portfolio E(r

()

(

)

the asset's

the expected

the market

is defined as

return of some

risk premium

asset . This can

be a single asset

or a portfolio.

if i Mf

beta

Cov r r

Var r

rr Err

↑





=+ −







  

This appendix can easily be skipped—its purpose is to demonstrate that all of the results of this chapter hold in a

more general setting. If you believe this already, go on to the next chapter.

Benninga, Portfolios with > 3 assets Page 32

Example

We start by considering a 3-asset problem. To describe 3 assets, we need to know the

expected return (or mean return—should standardize on one terminology), the variance, and all

the pairs of covariances. This data is described below.

ABCD

A 3-ASSET PORTFOLIO PROBLEM

Stock A Stock B Stock C

Mean 10% 12% 15%

Variance 15% 22% 30%

Cov(r

)

0.03

Cov(r

)

-0.01

Cov(r

)

0.02

Risk-free 6%

Suppose we form a portfolio of risky assets composed of proportion x

in asset A, x

in asset B,

and x

in asset C. Since the portfolio is fully invested in risky assets, it follows that x

= 1-x

-x

Portfolio return statistics: The expected return of the portfolio is given by:

()

(

)()

pAABBCC

rxErxErxEr=++

The calculation of the portfolio’s variance of return requires both the variances and the

covariances:

()

() ()

(

)

() () ()

222

2,2,2,

pA AB BC C

BABACACBCBC

Var r x Var r x Var r x Var r

x Cov r r x x Cov r r x x Cov r r

=+++

Here’s an example: The portfolio statistics are calculated in cells B17:B19:

Benninga, Portfolios with > 3 assets Page 33

ABCDEFG

Stock A Stock B Stock C

Mean 10% 12% 15%

Variance 15% 22% 30%

Risk-free 6%

Cov(r

)

0.03

Cov(r

)

-0.01

Cov(r

)

0.02

Portfolio proportions

0.6000

0.3000

0.1000 <-- =1-B13-B12

Portfolio statistics

Mean 0.1110 <-- =B12*B3+B13*C3+B14*D3

Variance 0.0894

Sigma 0.2990 <-- =SQRT(B18)

Sharpe ratio 0.1706 <-- =(B17-B5)/B19

<-- =B12^2*B4+B13^2*C4+B14^2*D4

+2*B12*B13*B7+2*B12*B14*B9+2*B13*B14*B8

A 3-ASSET PORTFOLIO PROBLEM

Cell B21 calculates the Sharpe ratio,

(

)

−

, for the particular portfolio. In Chapter 13, we

used Excel’s

Solver to find the portfolio with the maximum Sharpe ratio. We repeat this

procedure here:

Benninga, Portfolios with > 3 assets Page 34

Pressing

Solve gives the solution—the portfolio which maximizes the Sharpe ratio.

Given a risk-free rate r

= 6%, this portfolio is the market portfolio M.

Benninga, Portfolios with > 3 assets Page 35

ABCDEFG

Stock A Stock B Stock C

Mean 10% 12% 15%

Variance 15% 22% 30%

Risk-free 6%

Cov(r

)

0.03

Cov(r

)

-0.01

Cov(r

)

0.02

Portfolio proportions

0.2378

0.3575

0.4047 <-- =1-B13-B12

Portfolio statistics

Mean 0.1274 <-- =B12*B3+B13*C3+B14*D3

Variance 0.0918

Sigma 0.3030 <-- =SQRT(B18)

Sharpe ratio 0.2224 <-- =(B17-B5)/B19

<-- =B12^2*B4+B13^2*C4+B14^2*D4

+2*B12*B13*B7+2*B12*B14*B9+2*B13*B14*B8

A 3-ASSET PORTFOLIO PROBLEM

The security market line and

In Chapter 13 we showed that the asset’s

, defined as

()

Cov r r

Var r

= , relates the

asset’s expected return and the risk-free rate:

()

if Mf

Cov r r

rr Er r

Var r





=+ −





In the spreadsheet below, you can see that this is also true for our example. You need to know

how to compute the covariance for a combination of assets. In the equation below, we compute

the covariance between a the market portfolio, composed of proportions x

, x

, and x

of stocks

Benninga, Portfolios with > 3 assets Page 36

A, B, C and any generic portfolio (here composed of proportions y

, y

, and y

of these same

stocks):

()

()()()

222

pM AAA BBB CCC

BABBA BCBCCB ACACCA

Covrrxyxyxy

σσσ

=++

+++ ++ +

Now you can implement this, as shown in the spreadsheet:

ABCDEFG

Stock

Stock B Stock C

Mean 10% 12% 15%

Variance 15% 22% 30%

Risk-free 6%

Cov(r

)

0.03

Cov(r

)

-0.01

Cov(r

)

0.02

Portfolio proportions

0.2378

0.3575 <-- Now called the Market portfolio

0.4047

Market portfolio statistics

Mean 12.74% <-- =B12*$B$3+B13*$C$3+B14*$D$3

Variance 0.0918

Sigma 0.3030 <-- =SQRT(B18)

Market risk premium, E(r

)-r

0.0674 <-- =B17-B5

"Proof" of SML

Any portfolio, p

0.3

0.4

0.3

Mean, E(r

)

12.30% <-- =B25*$B$3+B26*$C$3+B27*$D$3

Covariance(p,M) 0.0858 <-- =B12*B25*B4+B13*B26*C4+B14*B27*D4+B7*(B12*

Portfolio beta, Cov(p,M)/Var(M) 0.9349 <-- =B30/B18

E(r

) by SML =r

+β

*[E(r

)-r

]

12.30% <-- =B5+B31*B21

<-- =B12^2*B4+B13^2*C4+B14^2*D4

+2*B12*B13*B7+2*B12*B14*B9+2*B13*B14*B8

THE SML WORKS FOR PORTFOLIOS OF 3 ASSETS!

When we say that the SML "works," we mean

that the expected portfolio return is

determined by the beta for

any

portfolio.

Benninga, Portfolios with > 3 assets Page 37

In cells B30:B31, we do the calculation for the

of any arbitrary portfolio. In cell B33

we show show that the r

[E(r

) – r

] calculates the expected return of the portfolio. Here are

some other examples, which show that the SML relation always holds:

Any portfolio, p

Mean, E(r

)

12.00%

Covariance(p,M) 0.0817

Portfolio beta, Cov(p,M)/Var(M) 0.8904

E(r

) by SML =r

*[E(r

)-r

]

12.00%

Any portfolio, p

-0.5

1.3

0.2

Mean, E(r

)

13.60%

Covariance(p,M) 0.1035

Portfolio beta, Cov(p,M)/Var(M) 1.1278

E(r

) by SML =r

*[E(r

)-r

]

13.60%

We conclude that:

Given the market portfolio M (defined as the Sharpe ratio maximizing portfolio), then for

any other asset or portfolio

p, the following relationship holds:

()

fMf

Cov r r

rr Err

Var r



=+ −



↑

Portfolios with more than 3 assets

We’ve repeated the calculations of this chapter for a portfolio of 3 assets. The primary

result which we’ve demonstrated is the SML:

If M maximizes the Sharpe ratio

(

)

−

, then for any asset or portfolio, the security

market line—which relates the asset’s expected return to its risk

—holds:

Benninga, Portfolios with > 3 assets Page 38

()

asset M

asset f M f

asset

Cov r r

rr Err

Var r



=+ −



↑

What if there are more than 3 risky assets? Everything we’ve said is still true—but

unfortunately the computations involved require techniques beyond the scope of this book.

My book Financial Modeling by Simon Benninga (MIT Press, 2000) contains details on how to do these

calculations for the general case with many assets using matrices.

PFE Chapter 15, Using SML and WACC page 1

CHAPTER 15: USING THE SML TO CALCULATE

A FIRM’S COST OF CAPITAL

this version: October 17, 2003

Chapter contents

Overview......................................................................................................................................... 2

15.1. The CAPM and the firm’s cost of equity—an initial example............................................. 3

15.2. Using the SML to calculate the cost of capital—calculating the parameter values.............. 9

15.3. A fully worked-out example—Hilton Hotels ..................................................................... 15

15.4. Computing the WACC using an asset

Asset

........................................................................ 22

15.5. Don’t read this section!....................................................................................................... 25

15.6. Some background information on equity betas .................................................................. 27

Summary....................................................................................................................................... 29

Exercises (unfinished)................................................................................................................... 30

Notice: This is a preliminary draft of a chapter of Principles of Finance with Excel by Simon Benninga

(http://finance.wharton.upenn.edu/~benninga/pfe.html). Check with the author (benninga@wharton.upenn.edu

)

before distributing this draft (though you will probably get permission), and check the website to make sure the

material is updated. All the material is copyright and the rights belong to the author.