Financial Econometrics - sogang.ac.krinchoi.sogang.ac.kr/NFUpload/nfupload_down.php?tmp_name=... ·...

Financial EconometricsChapter 7: Capital Asset Pricing Model

In Choi

Sogang University

In Choi (Sogang University) Chapter 7: Capital Asset Pricing Model 1 / 34

References

Chapter 5 of Campbell, Lo and MacKinlay (1997)

Chapters 5 and 20 of Cochrane (2001)

Chapters 5 and 8 of Cuthbertson and Nitzsche (2004)

Fama E.F. and K.R. French (2004): The Capital Asset Pricing Model:Theory and Evidence, Journal of Economic Perspectives, 18, pp.25—46.


What does CAPM imply?

The expected return of an asset must be linearly related to thecovariance of its return with the return of the market portfolio.

CAPM quantifies risk and the reward for bearing it.


What is CAPM?

CAPM was developed by Markowitz (1959), Sharpe (1964; Journal ofFinance) and Lintner (1965; Review of Economics and Statistics).

CAPM is a single-period model.

The Sharpe-Linter CAPM

E (Ri ) = Rf + βim(E (Rm)− Rf ) (1)

βim =Cov(Ri ,Rm)Var(Rm)

Ri : return on asset iRm : return on the market portfolioRf : return on risk-free asset


What is CAPM?

Let Zi = Ri − Rf and Zm = Rm − Rf . Then, if the risk-free rate isnonstochastic, (1) becomes

E (Zi ) = βimE (Zm)

βim =Cov(Zi ,Zm)Var(Zm)

.

In order to test for CAPM, regress Zi on Zm and see if(i) the intercept is zero(ii) β completely captures the cross-sectional variation of expectedexcess returns(iii) the market risk premium is positive.


What is CAPM?

Betas for individual stocks are calculated by using the time seriesregression model

Zit = αim + βimZmt + εit .

Zit : realized excess return for stock i at time t.Zmt : realized return for the market portfolio at time t (e.g., S&P 500index, KOSPI index, Hangseng Index, etc.)


What is CAPM?

Betas for individual stocks are typically reported in Web pages forstocks. Higher betas implies higher level of sensitivivity of stocks tomarket movements.

The CAPM can be used to compute expected stock returns. Thisworks only when the CAPM provides a good description of the data.


Effi cient-Set Mathematics

There are N risky asset returns denoted by an N × 1 vector R.E (R) = µR and Var(R) = E (R − µ)(R − µ)′ = ΩR (ΩR is calledthe variance-covariance of the random vector R).


Effi cient-Set Mathematics

Notation

ωa : N × 1 vector of portfolio weights for an arbitrary portfolio a withweights summing to oneMean of the portfolio with weights ωa : µa = ω′aE (R) = ω′aµRVariance of the portfolio with weights ωa : σ2a = ω′aΩRωaCovariance between any two portfolios a and b : ω′aΩRωb


Effi cient-Set MathematicsNo risk-free asset

Portfolio p is the minimum-variance portfolio of all portfolios withmean return µp if its portfolio weight vector (ωp) is the solution tothe following constrained optimization:

minω

ω′ΩRω

subject to

ω′µR = µp ;

ω′i = 1 (i = [1, ..., 1]′).



The solution of this problem1 is

ωp = g + hµp

where g and h are N × 1 vectors

g =1D

[B(Ω−1R i)− A(Ω−1R µR )

]h =

1D

[C (Ω−1R i)− A(Ω−1R µR )

]and A = i′Ω−1R µR , B = µ′RΩ−1R i, C = i

′Ω−1R i and D = BC − A2.

1This can be solved by the Lagrangian approach.In Choi (Sogang University) Chapter 7: Capital Asset Pricing Model 11 / 34


For a given µp , letting Rp = ω′pR and σ2p = Var(Rp),

σ2p = E (ω′pR −ω′pµR )(ω′pR −ω′pµR )

′ (2)

= ω′pΩRωp

=Cµ2p − 2Aµp + B

BC − A2 .

Plotting this in the space of (σ, µ), we obtain mean-variance frontier.The MV frontier is the locus of the minimum variance for a givenexpected return.



Minimizing (2) with respect to µp gives µp = A/C . This means thatthe global minimum variance is attained when

µ = A/C =i′Ω−1R µRC

=

(1C

Ω−1R i)′

µR .

This implies the weights of the global minimum variance portfolio is

ωg =1C

Ω−1R i

and its variance is

σ2g = ω′gΩRωg =1C 2i′Ω−1R i =

1C,

because i′Ω−1R i′ = C that is derivable from i′ωg = 1.



For each minimum-variance portfolio p, except the globalminimum-variance portfolio g , there exists a uniqueminimum-variance portfolio that has zero covariance with p (calledzero-beta portfolio with respect to p). This can be proven using thefact p and op correspond to the two solutions of equation (2).


Effi cient-Set MathematicsWith a risk-free asset

Now introduce a risk-free asset.

Given a risk-free asset with return Rf , consider the problem ofchoosing the minimum-variance portfolio with expected return µp .That is, we consider the contrained optimization problem

minω

ω′ΩRω

subject to

ω′µR + (1−ω′i)Rf = µp ; (i = [1, ..., 1]′).

The resulting solution is the optimal portfolio for risky assets. Theweight for the risk-free asset is obtained from this.



The solution is

ωp =µp − Rf

(µR − Rf i)′Ω−1R (µR − Rf i)×Ω−1R (µR − Rf i)

= cp × ω.

This shows that weights for the risky assets are proportional to thevector ω. (Note: cp is a constant.)



If only risky assets are owned, we should have i′ωq = 1. Thus, therisky-asset-only portfolio should be

ωq =Ω−1R (µR − Rf i)i′Ω−1R (µR − Rf i)

.

This is called the tangency portfolio. We have ωp = ωq iff µp = µRifor i = 1, ...,N (i.e., µR = µp × i).If µp = Rf , no risky assets will be owned.



Suppsoe that an investor allocates his fund between risk-free and riskyportfolio ωq . Given the optimal portfolio ωq for risky assets and therisk-free asset with return Rf , the asset return of the combinedportfolio is

Rp = (1− x)Rf + xRqwhich gives

µp = (1− x)Rf + xµq (3)

andVar(Rp) = x2Var(Rq). (4)



Equation (4) gives

x =σpσq.

Plugging this into equation (3), we obtain

µp = Rf +(

µq − Rfσq

)σp .



All effi cient portfolios lie along the line from the risk-free assetthrough portfolio q.

Portfolio q is a fixed bundle of risky assets held by all investorsregardless of their preferences. Hence, it is called the market portfolio.

Straight line connecting Rf and q and going beyond is called thecapital market line (CML).



Investor preferences determine where they will reside along the CML.Risk-averse investors will locate themselves in between Rf andq. Risk-loving investors will go beyond q.

An extreme investor who does not want any risk will reside at pointRf .

For any portfolio a, µa−Rfσa

is called the Sharpe ratio. It measures theexpected excess return per unit risk or it is the price of risk.



µq−Rfσq

is called the market price of risk.

Portfolio q has the maximum Sharpe ratio of all portfolios of riskyassets. This mean that no one will stay along the line Rf → a.


Derivation of CAPM

CAPM can be derived using the effi cient set mathematics.

Suppose that an investor decides to allocate xi to a security i withmean return µi and variance σ2i and 1− xi to the market portfolioq. This portfolio’s expected return is

µp = xiµi + (1− xi )µq ;

σp =√x2i σ2i + (1− xi )2σ2q + 2xi (1− xi )σiq .


Derivation of CAPM

But moving off the market portfolio is not an optimal investmentstrategy and thus it should be xi = 0. At xi = 0,

∂µp∂σp|xi=0=

(µi − µq

)σq

σiq − σ2q.2

But this should be equal to the slope of the CML

µq − Rfσq

.

Thus, we obtainµi = Rf +

σiqσ2q(µq − Rf ).

2Consider∂µp∂xi

∂xi∂σp

evaluated at xi = 0. Note also that σp = σq at xi = 0.In Choi (Sogang University) Chapter 7: Capital Asset Pricing Model 24 / 34

Testing for CAPMTesting for a zero intercept term

Let Zt be an N × 1 vector of excess returns for N assets at time t.According to the Sharpe-Lintner CAPM,

Zt = α+ βZmt + εt

where α = 0, Zmt is the time period t market portfolio excess returnand εt is an idiosyncratic error (∼ iid(0,Σ)).



Testing for CAPM amounts to testing the null hypothesis

H0 : α = 0.

Of course, this is only an aspect of CAPM.

In addition, we need to know whether β > 0 and whether there areother factors that affect the expected return.



Let

α = Z − βZm ;

β =∑Tt=1(Zt − Z )(Zmt − Zm)

∑Tt=1(Zmt − Zm)2

;

Z =1T

T

∑t=1Zt ; Zm =

1T

T

∑t=1Zmt ;

Σ =1T

T

∑t=1(Zt − α− βZmt )(Zt − α− βZmt )′.



Wald statistic for H0 : α = 0 is defined by

J = α′[Var(α)]−1 α

= T

[1+

Z 2m1T ∑T

t=1(Zmt − Zm)2

]−1α′Σ−1 α.

When T is large, J ∼ χ2(N).



There are other types of tests available (cf. Campbell, Lo andMacKinlay).

Time series empirical evidence is not favorable to CAPM as shown inTable 5.3 of Campbell, Lo and MacKinlay.


Testing for CAPMEvidence based on the cross-sectional regression

Figure 20.8 of Cochrane (2001, p.436) plots returns and betas of 10portfolios of NYSE stocks sorted by size, and those of portfolios madeup of corporate bonds and long-term government bonds.


Testing for CAPMEvidence based on the cross-sectional regression

We find:

(i) Portfolios with higher average returns have higher betas as CAPMpredicts.(ii) The smallest firms seem to earn an average return a few percentagepoints too high given their betas (small-firm effect).(iii) The long-term and corporate bonds have mean returns in line withtheir low betas, despite their standard deviations nearly as high asthose of stocks.


Capital budgeting and CAPMCapital budgeting

Suppose that a company is considering new projects that will providecash streams. The projects will be financed by equities only.

The question is which projects should be pursued and which shouldnot be.

If a projcect provides a cash stream which has positive present value,it should be pursued.

When calculating the present value, many companies in the real worlduse the discount rate that is implied by CAPM.

Why? The expected return implied by CAPM is the opportunity costof equity.



ExampleA company’s expected return calculated by CAPM is 16.495%. A newproject will provide cash $14 million next year, while its cost is $10 million.The present value of the project is

141+ 0.16495

− 10 = 2.0177.

The calculation shows that this project is worth pursuing.



If a company’s projects are financed by debts and preferred stocks,the cost of capital is easier to calculate: it is the cost of borrowing.

Example(Weighted average cost of capital) A company’s expected returncalculated by CAPM is 16.495%,and the cost of borrowing for thecompany is 8% per annum. A new project will provide cash $14 millionnext year, while its cost is $10 million out of which $4 million will befinanced by equities and $6 million by debt. The average cost of capital is

46+ 4

× 0.16495+ 66+ 4

× 0.08 = 0.11398.

The present value of the project is

141+ 0.11398

− 10 = 2.5676.

Again, this project is worth pursuing.In Choi (Sogang University) Chapter 7: Capital Asset Pricing Model 34 / 34

Financial Econometrics - sogang.ac.krinchoi.sogang.ac.kr/NFUpload/nfupload_down.php?tmp_name=... ·...

Documents

Transcript of Financial Econometrics - sogang.ac.krinchoi.sogang.ac.kr/NFUpload/nfupload_down.php?tmp_name=... ·...