FINM2401

C A P M A N D C O S T O F C A P I T A L

Last Week…

Lecture 10

2

�  Computing portfolio expected return and risk. ¡  Covariance and correlation provide diversification benefits.

� What we mean by an efficient portfolio. � What happens when we add a risk-free asset to a

portfolio. �  Tangent portfolio (highest Sharpe Ratio)

⎡ ⎤ −⎣ ⎦=σ

Sharpe Ratio p f

p

E R r

This Week

Lecture 10

3

�  Formalise CAPM ¡  How to compute beta (β) ¡  Security Market Line (SML) v. Capital Market Line (CML)

�  Applications of CAPM ¡  Equity cost of capital ¡  Beta estimation

�  Putting it all together to find cost of capital

Diversification with General Portfolios

�  Last Lecture, we saw that for a portfolio with arbitrary weights, the standard deviation is calculated as:

= × ×

↑ ↑ ↑

∑6 4 4 4 44 7 4 4 4 4 48

Security ’s contribution to thevolatility of the portfolio

Amount Total Fraction of ’sof held Risk of risk that is

common to

( ) ( ) ( , )

i

P i i i pi

ii i

P

SD R x SD R Corr R R

Lecture 9

4

( ) ( ) ( )( )

= =

= = σ

∑∑ ∑ ,

, ,

,

P P P i i Pi

i i P i i pi i

Var R Cov R R Cov x R R

x Cov R R x

Incremental Risk: Risk new security adds to portfolio

The contribution of investment i to the volatility of the portfolio depends on the risk that i has in common with the portfolio.

Efficient Portfolios

Lecture 10

5

�  Efficient portfolios provide highest return for a given level of risk or lowest risk for a given level of return. ¡  If you can increase a portfolio’s Sharpe Ratio by adding

another security, then the original portfolio is NOT efficient.

� What does adding a security do to the Sharpe Ratio of a portfolio?

Risk Premium of new security

Risk new security adds to portfolio

[ ],

Sharpe Ratio if p fi f

i i p p

E R rE R r −⎡ ⎤− ⎣ ⎦↑ >σ ρ σ

Efficient Portfolios

Lecture 9

6

Risk Premium: Suppose we give up the risk-free return and invest in asset. : In order to gain this excess return we are taking on more risk? Incremental Risk: How much additional risk are we taking by investing in the asset? If the “incremental” Sharpe Ratio is greater than the portfolio Sharpe Ratio, then the portfolio isn’t efficient because adding the new security can increase return.

Incremental Risk: Risk new security adds to portfolio

Risk Premium of new security

This incremental Sharpe ratio only makes sense if correlation is positive.

Efficient Portfolios

Lecture 10

7

�  Re-arranging:

�  If this inequality is true, then your original portfolio was NOT the tangent portfolio

[ ],

p fi f

i i p p

E R rE R r −⎡ ⎤− ⎣ ⎦>σ ρ σ

[ ] ( ),i i pi f p f

p

E R r E R rσ ρ

− > −⎡ ⎤⎣ ⎦σ

[ ] ( ),i i pi f p f

p

E R r E R rσ ρ

> + −⎡ ⎤⎣ ⎦σ

Efficient Portfolio and Required Returns

Lecture 10

8

�  Beta of Portfolio i with respect to Portfolio P

�  Increasing the amount invested in i will increase the Sharpe ratio of portfolio P if its expected return E[Ri] exceeds the required return ri , which is given by:

, ,2

i i P i PPi

P P

σ ρ σβ = =

σ σ

[ ]( )Pi f i P fr r E R r= + β × −

,,

i Pi P

i P

σρ =

σ σ

Try it out… Assume you own a portfolio of 25 different “large cap” stocks. You expect your portfolio will have a return of 12% and a standard deviation of 15%. A colleague suggests you add gold to your portfolio. Gold has an expected return of 8%, a standard deviation of 25%, and a correlation with your portfolio of -0.05. If the risk-free rate is 2%, will adding gold improve your portfolio’s Sharpe ratio?

Lecture 10 9

Solution P  The beta of gold with your portfolio is:

P  The required return that makes gold an attractive addition to your portfolio is:

P  Because the expected return of 8% exceeds the required return of 1.67%, adding gold to your portfolio will increase your Sharpe ratio.

Lecture 10 10

σ ρ × −β = = = −

σ, 25% 0.05 0.08333

15%P Gold Gold PGold

P

[ ]( )2% 0.08333 (12% 2% ) 1.67%

PGold f Gold P fr r E R r= + β −

= − × − =

Expected Returns and the Efficient Portfolio

Lecture 10

11

�  Expected Return of a Security ¡  A portfolio is efficient if and only if the expected return of

every available security equals its required return.

[ ] [ ]( )effi i f i eff fE R r r E R r= ≡ + β × −

Finding the Efficient Combination

Lecture 10

12

� What is the efficient combination of the large cap portfolio on slide 7 and gold? The risk-free rate is 2%.

Large Cap Portfolio Gold

E[R] 12% 8% SD[R] 15% 25% Correlation -0.05

Finding the Efficient Portfolio

Lecture 10

13

�  As you add gold to your portfolio, the covariance of gold to the resulting portfolio will change. ¡  If P is a portfolio of L(Large Caps) and G(gold), then

÷ Where xG and xL are the weights of L and G in P

¡  An excel spreadsheet can be used to compute the weights that maximize the Sharpe Ratio of P

¡  For these weights, adding more gold to your portfolio will not give sufficient return to compensate for the additional risk. ÷ The required return on gold will be its expected return of 8%

2, ,G P G G L G Lx xσ = σ + σ

Finding the Efficient Portfolio

Lecture 10

14

�  The combination that maximises the Sharpe Ratio is 24.17% gold, 100% large caps, -24.17% risk-free asset.

�  A copy of this spreadsheet will be posted on Blackboard �  You are not expected to solve for the weights, but should be

able to tell whether a given set of weights are optimal

xgold E[Rp] Var[Rp] SD[Rp] Sharpe  Ratio Cov(Rgold,Rp) Corr(Rgold,Rp) BetaReqd  Return  on  Gold

0.00% 12.00% 0.022500 15.000% 0.666667 -­‐0.001875 -­‐0.050000 -­‐0.083333 1.167%10.00% 12.60% 0.022750 15.083% 0.702773 0.004375 0.116024 0.192308 4.038%15.00% 12.90% 0.023344 15.279% 0.713413 0.007500 0.196352 0.321285 5.502%20.00% 13.20% 0.024250 15.572% 0.719221 0.010625 0.272919 0.438144 6.907%24.00% 13.44% 0.025200 15.875% 0.720652 0.013125 0.330719 0.520833 7.958%24.17% 13.45% 0.025243 15.888% 0.720654 0.013228 0.333031 0.524022 8.000%25.00% 13.50% 0.025469 15.959% 0.720600 0.013750 0.344635 0.539877 8.209%

The Capital Asset Pricing Model

Lecture 10

15

�  The Capital Asset Pricing Model (CAPM) allows us to identify the efficient portfolio of risky assets without having any knowledge of the expected return of each security.

�  Instead, the CAPM uses the optimal choices investors make to identify the efficient portfolio as the market portfolio, the portfolio of all stocks and securities in the market.

The CAPM Assumptions

Lecture 10

16

1.  Investors can buy and sell all securities at competitive market prices (without incurring taxes or transactions costs) and can borrow and lend at the risk-free interest rate.

2.  Investors hold only efficient portfolios of traded securities - portfolios that yield the maximum expected return for a given level of volatility.

3.  Investors have homogeneous expectations regarding the volatilities, correlations, and expected returns of securities.

Efficiency of the Market Portfolio

Lecture 10

17

�  Given homogeneous expectations, all investors will demand the same efficient portfolio of risky securities.

�  The combined portfolio of risky securities of all investors must equal the efficient portfolio.

�  Thus, if all investors demand the efficient portfolio, and the supply of securities is the market portfolio, the demand for market portfolio must equal the supply of the market portfolio.

Try it out.. P Suppose there are only three securities

available for investors: P  ABC – total market cap of $3million P  DEF – total market cap of $6million P  GHI – total market cap of $1million

P  If all the CAPM assumptions apply, what is the weight of DEF in the risky portion of your portfolio?

Lecture 10 18

Solution P  If all CAPM assumptions apply, you hold all

three securities in proportion to their market cap.

P Therefore DEF is 6/10 = 60% of the risky assets in your portfolio

P Note that you may also hold the risk free asset (with either positive or negative weight)

Lecture 10 19

The Capital Market Line

Lecture 10

20

0%

5%

10%

15%

20%

25%

30%

35%

0% 5% 10% 15% 20% 25% 30% 35%

rf

Market Portfolio Capital Market Line

[ ]( ): mkt f

p f pmkt

E R rCML E R r

−⎡ ⎤ = + σ⎣ ⎦ σ

The Capital Market Line

Lecture 10

21

�  The expected return and volatility of a portfolio on the capital market line are:

[ ] ( ) [ ] [ ]( )= − + = + −1xCML f Mkt f Mkt fE R x r xE R r x E R r

( ) ( )=xCML MktSD R xSD R

Determining the Risk Premium

Lecture 10

22

� Market Risk and Beta ¡  Given an efficient market portfolio, the expected return of an

investment is:

¡  The beta is defined as:

[ ] [ ]( )= = + β −1 4 44 2 4 4 43Risk premium for security

Mkti i f i Mkt f

i

E R r r E R r

( ) ( )( )

( )( )

Volatility of that is common with the market

i

, ,

i

Mkt i i Mkt i Mkti

Mkt Mkt

SD R Corr R R Cov R RSD R Var R×

β ≡ β = =

6 4 4 4 4 7 4 4 4 48

Try it out… P Assume the risk-free return is 5% and the

market portfolio has an expected return of 12% and a standard deviation of 44%.

P ATP Oil and Gas has a standard deviation of 68% and a correlation with the market of 0.91.

P What is ATP’s beta with the market? P Under the CAPM assumptions, what is its

expected return?

Lecture 10 23

Solution

Lecture 10 24

σ × ρ ×β = = =

σ,

i0.68 0.91

1.410.44

i i mkt

mkt

[ ] [ ]( )( )

= + β −

= + − =5% 1.41 12% 5% 14.87%

Mkti f i Mkt fE R r E R r

The Security Market Line

Lecture 10

25

�  There is a linear relationship between a stock’s beta and its expected return (See figure on next slide). The security market line (SML) is graphed as the line through the risk-free investment and the market.

¡  According to the CAPM, if the expected return and beta for individual securities are plotted, they should all fall along the SML.

Security Market Line

0%

5%

10%

15%

20%

25%

0% 10% 20% 30%

Ex

pec

ted

Ret

urn

Volatility

Lecture 10

26

rf

Market Portfolio

0%

5%

10%

15%

20%

25%

0 0.5 1 1.5

Ex

pec

ted

Ret

urn

Beta

Security Market Line

Market Portfolio

rf

CML

The Security Market Line

Lecture 10

27

�  Beta of a Portfolio ¡  The beta of a portfolio is the weighted average beta of the

securities in the portfolio.

( )( )

( )( )

( )( )

,,

,

i i MktP Mkt iP

Mkt Mkt

i Mkti i ii i

Mkt

Cov x R RCov R RVar R Var R

Cov R Rx xVar R

β = =

= = β

∑

∑ ∑

Try it out… P Suppose the stock of the 3M Company (MMM)

has a beta of 0.69 and the beta of Hewlett-Packard Co. (HPQ) stock is 1.77.

P Assume the risk-free interest rate is 5% and the expected return of the market portfolio is 12%.

P What is the expected return of a portfolio of 40% of 3M stock and 60% Hewlett-Packard stock, according to the CAPM?

Lecture 10 28

Solution

Lecture 10 29

= + β −[ ] ( [ ] )  Mkti f i Mkt fE R r E R r

β = β = + =∑ (.40 )(0.69) (.60 )(1.77) 1.338P i iix

= + − =[ ] 5% 1.338(12% 5% ) 14.37%iE R

CAPM Summary

Lecture 10

30

�  The market portfolio is the efficient portfolio. ¡  This means the market portfolio is the tangent portfolio and all

investors hold the market portfolio plus the risk free asset.

�  The risk premium for any security is proportional to its beta with the market. ¡  This means that beta is the relevant measure of risk and all

securities will lie on the Security Market Line.

Cost of Capital

Lecture 10

31

�  Cost of capital will depend on systematic risk. �  Since a firm is financed by both debt and equity, its

cost of capital will be an average of its cost of debt and its cost of equity.

The Equity Cost of Capital

Lecture 10

32

�  The Capital Asset Pricing Model (CAPM) is a practical way to estimate.

�  The cost of capital of any investment opportunity equals the expected return of available investments with the same beta.

�  The estimate is provided by the Security Market Line equation:

( )Risk Premium for security

i f i Mkt f

i

r r E R r= + β × −⎡ ⎤⎣ ⎦1 4 4 4 2 4 4 43

Try it out… P  Suppose you estimate that Wal-Mart’s stock has a

volatility of 16.1% and a beta of 0.20. A similar process for Johnson & Johnson yields a volatility of 13.7% and a beta of 0.54. P  Which stock carries more total risk? P  Which has more market risk?

P  If the risk-free interest rate is 4% and you estimate the market’s expected return to be 12%, calculate the equity cost of capital for Wal-Mart and Johnson & Johnson. P  Which company has a higher cost of equity capital?

Lecture 10 33

Solution P Total risk is measured by volatility. Wal-Mart

stock has more total risk than J&J. P Systematic risk is measured by beta. Johnson

& Johnson has a higher beta, so it has more market risk than Wal-Mart.

P Johnson & Johnson’s equity cost of capital is

P The equity cost of capital for Wal-Mart is

Lecture 10 34

( )= + − =& 4% 0.54 12% 4% 8.32%J Jr

( )= + − =4% 0.20 12% 4% 5.6%WMTr

The Market Portfolio

Lecture 10

35

�  The market portfolio is a value-weighted portfolio of all securities in the market.

� Weights are based on Market Capitalisation (share price times number of shares outstanding).

� Owning the market portfolio means owning the same percentage of every company in the market.

�  This is a passive portfolio – you don’t need to re-balance it in response to price movements.

Market Indexes

Lecture 10

36

�  S&P 500 ¡  A value-weighted portfolio of the 500 largest U.S. stocks

� Wilshire 5000 ¡  A value-weighted index of all U.S. stocks listed on the major

stock exchanges

�  Dow Jones Industrial Average (DJIA) ¡  A price weighted portfolio of 30 large industrial stocks

�  S&P/ASX 200 ¡  A value-weighted portfolio of the 200 largest Australian stocks.

Approximately 78% of total Market Cap on the ASX

The Market Risk Premium

Lecture 10

37

�  Determining the Risk-Free Rate ¡  The yield on Commonwealth Government Bonds ¡  Match term of bond to term of investment you are valuing

�  The Historical Risk Premium ¡  Estimate the risk premium (E[RMkt]-rf) using the historical

average excess return of the market over the risk-free interest rate

The Market Risk Premium

�  Using historical data has two drawbacks: ¡  Standard errors of the estimates are large ¡  Backward looking, so may not represent current expectations.

� One alternative is to solve for the discount rate that is consistent with the current level of the index.

1

0

DividendYield + ExpectedDividendGrowthRateMktDivr gP

= + =

Lecture 10

38

Estimating Beta from Historical Returns

Lecture 10

39

�  Recall, beta is the expected percent change in the excess return of the security for a 1% change in the excess return of the market portfolio.

�  Consider Commonwealth Bank and how it changes with the market portfolio (using the S&P/ASX200 index as a proxy).

Monthly Excess Returns: CBA and ASX200

Lecture 10

40

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

May-07

Sep-07

Jan-08

May-08

Sep-08

Jan-09

May-09

Sep-09

Jan-10

May-10

Sep-10

Jan-11

May-11

Sep-11

Jan-12

ASX200 CBA

Scatterplot of Monthly Excess Returns

Lecture 10

41

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2

CB

A R

etu

rn

Market Return

Using Linear Regression

Lecture 10

42

�  Linear Regression ¡  The statistical technique that identifies the best-fitting line

through a set of points.

÷ αi is the intercept term of the regression. ÷ βi(RMkt – rf) represents the sensitivity of the stock to

market risk. When the market’s return increases by 1%, the security’s return increases by βi%.

÷ εi is the error term and represents the deviation from the best-fitting line and is zero on average.

( ) ( )i f i i Mkt f iR r R r− = α + β − + ε

Using Linear Regression

�  Alpha ¡  Since E[εi] = 0:

÷ αi represents a risk-adjusted performance measure for the historical returns.

¢  If αi is positive, the stock has performed better than predicted by the CAPM.

¢  If αi is negative, the stock’s historical return is below the SML.

( ) {Distance above / below the SMLExpected return for from the SML

i f i Mkt f i

i

E R r E R r= + β − + α⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦1 4 4 44 2 4 4 4 43

Lecture 10

43

The Debt Cost of Capital

�  Debt Yields ¡  Yield to maturity is the IRR an investor will earn from holding

the bond to maturity and receiving its promised payments. ¡  If there is little risk the firm will default, yield to maturity is a

reasonable estimate of investors’ expected rate of return. ¡  If there is significant risk of default, yield to maturity will

overstate investors’ expected return.

Lecture 10

44

The Debt Cost of Capital

�  If there is a chance of default, the expected return on the debt will be:

�  The importance of the adjustment depends on the riskiness of the bond.

Lecture 10

45

( ) ( )( )

1

Yield to Maturity Prob default Expected Loss Ratedr p y p y L y pL= − + − = −

= − ×

Debt Betas

Lecture 10

46

�  Alternatively, we can estimate the debt cost of capital using the CAPM.

�  Debt betas are difficult to estimate because corporate bonds are traded infrequently.

� One approximation is to use estimates of betas of bond indices by rating category.

Try it out… P  DEF Corp. has outstanding 10 year bonds with a B

rating and a yield to maturity of 7.5%. P  The market risk premium currently stands at 4%,

and the risk free rate is 3%. P  Your research shows that the average beta for B

rated debt with 10-15 years to maturity is 0.40. P  You also find that the probability of default for B

rated debt given current economic conditions is 6%, with an expected loss rate of 40%.

P  Estimate the expected return on DEF Corp. bonds.

Lecture 10 47

Solution P Expected Return:

P  rd = y – pL P  7.5% – 0.06 × 40% = 5.1%

P CAPM P  rd = rf + βd (mkt risk premium) P  3% + 0.4 × 4% = 4.6%

Lecture 10 48

A Project’s Cost of Capital

Lecture 10

49

�  All-equity comparables ¡  Find an all-equity financed firm in a single line of business that

is comparable to the project. ¡  Use the comparable firm’s equity beta and cost of capital as

estimates

�  Levered firms as comparables

Asset (unlevered) cost of capital

Lecture 10

50

�  Expected return required by investors to hold the firm’s underlying assets.

� Weighted average of the firm’s equity and debt costs of capital

�  Can also estimate rU using an asset beta:

U E DE Dr r rE D E D

= ++ +

U E DE DE D E D

β = β + β+ +

Cash and Net Debt

Lecture 10

51

�  Some firms maintain high cash balances �  Cash is a risk-free asset that reduces the average

risk of the firm’s assets �  Since the risk of the firm’s enterprise value is what

we’re concerned with, leverage should be measured in terms of net debt. ¡  Net Debt = Debt – Excess Cash and short-term investments

Project Risk and Cost of Capital

Lecture 10

52

�  Cost of capital should reflect the systematic risk of the project ¡  Firm asset betas or asset cost of capital will be an average of

the systematic risk of all projects in the firm. This may not reflect the systematic risk in an individual project.

Project Risk and Cost of Capital

�  Another factor that can affect market risk of a project is its degree of operating leverage

�  Operating leverage is the relative proportion of fixed versus variable costs

�  A higher proportion of fixed costs increases the sensitivity of the project’s cash flows to market risk ¡  The project’s beta will be higher ¡  A higher cost of capital should be assigned

Lecture 10

53

Weighted Average Cost of Capital

Lecture 10

54

�  Taxes – A Big Imperfection ¡  When interest payments on debt are tax deductible, the net

cost to the firm is given by: Effective after-tax interest rate =

¡  This is reflected in the after-tax weighted average cost of capital (WACC):

( )1wacc E D CE Dr r rE D E D

= + − τ+ +

( )1 Cr − τ

The Weighted Average Cost of Capital

�  How does rwacc compare with rU? ¡  Unlevered cost of capital (or pretax WACC) is:

÷ Expected return investors will earn by holding the firm’s assets ÷  In a world with taxes, it can be used to evaluate an all-equity

project with the same risk as the firm.

¡  In a world with taxes, WACC is less than the expected return of the firm’s assets. ÷ With taxes, WACC can be used to evaluate a project with the

same risk and the same financing as the firm.

Lecture 10

55

Final Thoughts on the CAPM

Lecture 10

56

�  There are a large number of assumptions made in the estimation of cost of capital using the CAPM.

�  How reliable are the results? ¡  CAPM is practical, easy to implement, and robust. ¡  CAPM requires managers to think about risk in the correct

way.