SK460 Finance Theory, Diderik Lund, 31 January 2001
CAPM: Implications, applications
For a portfolio: Variance the relevant risk measure.
For each security: Covariance with R~ the relevant r. m.
Reason: Covariance measures contribution to portfolio variM
ance.
Generally: Covariance with each agent's marginal utility.
But when mean-var preferences: Covariance with agent's wealth.
In equilibrium: All have same risky portfolio (composition).
Thus covariance with R~ M relevant for everyone.
Implication: Adding noise does not matter:
Suppose p~h1 = p~j1 + "~, where E (~") = 0 and "~ is stochastically
independent of (~pj 1 ; R~ M ). Can show ph0 = pj 0 :
1 ~
E (~pj 1 + "~) , cov(~
pj 1 + "~; RM )
ph0 =
1 + Rf
1 ~
~
=
E (~
pj 1 ) + E (~") , [cov(~pj 1 ; RM ) + cov(~"; RM )] = pj 0 :
1 + Rf
At this point: Do not get confused by jM .
Common confusion: Since jM = jM j M , some believe that
a higher j leads to a higher jM .
Have just shown a case where this is not true; instead:
hM =
hM
= jM < jM = jM :
hM hM j M
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SK460 Finance Theory, Diderik Lund, 31 January 2001
CAPM terminology: Systematic vs. unsystematic risk
Dene "~j through the equation
~
~
Rj = Rf + j RM , Rf + "~j :
Then the CAPM equation
~
~
E (Rj ) = Rf + j E (RM ) , Rf
implies that E (~"j ) = 0, and that
cov(~"j ; R~ M ) = cov(R~ j ; R~ M ) , j var(R~ M ) = jM , j M2 = 0:
This allows us to split j2 in two parts:
2 + 2 = + 2 :
j2 = j2M
j jM
"j
"j
First term called systematic risk. This is reected in the market
valuation. (More general term: Relevant risk or covariance risk.)
Second term called unsystematic risk. As we have seen, it is not
reected in market valuation. (More general term: Irrelevant risk.)
The sum of the two called total risk or variance risk. This is
relevant for portfolios, evaluated for being the total wealth of someone, but not for individual securities, to be combined with other
securities in portfolios.
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SK460 Finance Theory, Diderik Lund, 31 January 2001
Valuation of rms, value additivity
1
p0=
E (~
p 1 ) , cov(~
p 1 ; R~ ) V (~
p 1 );
1+R
i
f
i
i
M
i
denes valuation function V (), given some Rf ; R~ M .
p~i1 is value of share at t = 1, including dividends.
Consider rm nanced 100% by equity (no debt).
Firm's net cash ow at t = 1 goes to shareholders.
Plug in that net cash ow instead of p~i1.
Then formula gives value of all shares in rm.
What if p~i1 = ap~j1 + bp~k1?
Linearity of E () and cov() implies pi0 = apj0 + bpk0:
1 ~
E (~
pi1 ) , cov(~
pi1 ; RM )
pi0 =
1 + Rf
1 ~
=
E (ap~j 1 + bp~k1 ) , cov(ap~j 1 + bp~k1 ; RM )
1 + Rf
1 ~
~
=
aE (~pj 1 ) + bE (~
pk1 ) , [a cov(~
pj 1 ; RM ) + b cov(~
pk1 ; RM )]
1 + Rf
= apj 0 + bpk0 :
Diversication is no justication for mergers.
Diversication can be done by shareholders.
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SK460 Finance Theory, Diderik Lund, 31 January 2001
Investment project's rate of return
Consider (potential) real investment project:
{ Outlay I at t = 0.
{ Revenue p~ 1 at t = 1.
Project value? Should project be undertaken?
Assume 100% equity nanced. (Assume separate rm?)
Project's rate of return is (~p 1 , I )=I .
If use of restricted technology or resources: No reason for SML
I
I
equation to hold for this rate of return.
May earn above-normal expected rate of return.
However: Valuation of p~I 1 possible:
1 ~
E (~
pI 1) , cov(~
pI 1 ; RM )
pI 0 = V (~
pI 1 ) =
1 + Rf
denes pI 0 independently of I .
If pI 0 > I , undertake project. Net value pI 0 , I .
If pI 0 < I , drop project. Net value of opportunity is 0. Net
value of having to undertake project is negative.
Competition and free entry ) pI 0 = I .
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SK460 Finance Theory, Diderik Lund, 31 January 2001
Project valuation, contd.
If claim to p~ 1 costs p 0, this is equilibrium price.
Project is then on security market line (SML).
If project available at dierent cost I : Not at SML.
I
I
Project's k , must satisfy the equation
p~
E ( I 1 ) = 1 + Rf + [M , Rf ]k
pI 0
p~
= 1 + Rf + [M , Rf ] cov( I 1 ; R~ M )=M2 :
pI 0
Expressed in terms of exogenous variables (eliminating pI 0) this
becomes
(1 + Rf )
:
k = 2 E (~p 1 )
M cov(~p ;R~ ) , M + Rf
1
Only if I = pI 0 = V (~pI 1 ), will the project rate of return p~I 1=I , 1
satisfy the CAPM equation with k .
I
I
M
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SK460 Finance Theory, Diderik Lund, 31 January 2001
Stylized example of project valuation
Suppose project produces two commodities at t = 1.
One variable input is needed at t = 1.
Uncertain prices of input and of both commodities.
Uncertain quantities of input and of both commodities.
Net cash ow, t = 1: Y~ = P~ X~ + P~ X~ , P~ X~ :
For instance, i is milk, j is beef, k is labor.
(Warning: Many farms owned by non-diversied farmers. Then
i
i
j
j
k
k
CAPM does not apply.)
CAPM: V (Y~ ) = V (P~iX~i) + V (P~j X~ j ) , V (P~k X~k ).
Four points to be made about this:
{ Flexibility or not?
{ How to value a product of stochastic variables?
{ How to interpret valuation for negative term?
{ How to interpret valuation of, e.g., beef today?
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SK460 Finance Theory, Diderik Lund, 31 January 2001
Example, P~ X~ + P~ X~ , P~ X~ ; contd.
Flexibility
i
i
j
j
k
k
If any outlay at t = 0, those can not be cancelled later.
What if Y~ < 0?
Reasonable to assume: Each P~h and X~h always > 0.
Then: Y~ < 0 happens when P~k X~ k is large.
May be able to cancel project at t = 1 if Y~ < 0.
If such exibility, need option valuation methods. (Later.)
Then: Value at t = 1 will be 0, not Y~ , when Y~ < 0.
Assume now: No exibility. Committed to pay P~k X~ k .
For some projects, exibility is realistic. For others, not.
Partial exibility may also be realistic.
Valuation of product of stochastic variables
Quantity uncertainty often local, technical, meteorological. May
simplify valuation of P~ X~ expressions if assume: Each X~ h (h =
i; j; k) is stoch. indep. of (P~h ; R~ M ). Then: E (P~ X~ ) = E (P~ )E (X~ )
and
~ R~ M ) = E (P~ X~ R~ M ) , E (P~ X~ )E (R~ M )
cov(P~ X;
~
~
~
~
~
= E (X ) E (P RM ) , E (P )E (RM ) = E (X~ ) cov(P~ ; R~ M ) )
V (P~ X~ ) = E (X~ )V (P~ ), quantity uncertainty irrelevant.
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SK460 Finance Theory, Diderik Lund, 31 January 2001
Example, P~ X~ + P~ X~ , P~ X~ ; contd.
i
i
j
j
Valuation of negative term
k
k
1 ~
~
~
E (Pk ) , cov(Pk ; RM ) :
1 + Rf
If this covariance increases, then value increases.
High covariance between input price and R~M is good.
Reason: Project owners are committed to the expense.
Prefer expense is high when they are otherwise wealthier.
Prefer expense is low when they are otherwise poorer.
V (,P~k X~ k ) = ,E (X~ k ) Valuation of claim to commodity at t = 1
Might perhaps calculate V (P~j ) from time series estimates of
E (P~j ) and cov(P~j ; R~ M ).
\Value today of receiving one unit of beef next period."
In general not equal to price of beef today.
Would have equality if beef were investment object, like gold.
Instead V (P~j ) is present value of forward price of beef.
Usually lower than price of beef today.
More about this later in course.
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SK460 Finance Theory, Diderik Lund, 31 January 2001
CAPM: Some remarks on realism and testing
CAPM equation may perhaps be tested on time-series data.
More about this later in course: Roll's paper.
Need R , need R~ , need stability.
f
M
Existence of risk free rate
Interest rates on government bonds are nominally risk free.
With ination real interest rates are uncertain.
Real rates of return are what agents really care about.
Some countries: Indexed bonds, risk free real rates.
Alternative model: No risk free rate. Roll's paper.
Without Rf , still CAPM equation with testable implications.
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SK460 Finance Theory, Diderik Lund, 31 January 2001
CAPM: Some remarks on realism and testing, contd.
Observability of market portfolio
By denition, M portfolio contains all risky assets.
In real world, not all risky assets are traded.
Problems of asymmetric information prevent some trading.
E.g., many people own their homes.
In particular: Human capital. Slavery forbidden.
Implication: People not as well diversied as in model.
People's risky portfolios (in extended sense) dier.
Big problem, no good solution.
Stability of expectations, variances, covariances
CAPM says nothing testable about single outcome.
Need repeated outcomes, i.e., time series.
Outcomes must be from same probability distribution.
Requires stability over time.
A problem, perhaps not too bad.
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SK460 Finance Theory, Diderik Lund, 31 January 2001
CAPM: Some remarks on realism and testing, contd.
Empirical line often has too high intercept, too low slope.
Can nd other signicant variables:
{ Asset-specic variables in cross-section.
{ Economy-wide variables in time series.
If these determined at t = 0: Conditional CAPM.
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SK460 Finance Theory, Diderik Lund, 31 January 2001
Linear Factor Model
Developed by Stephen Ross 1976.
Also known as Arbitrage Pricing Theory (APT).
Theoretically an alternative to CAPM.
Main result has CAPM equation as special case.
Point of departure: Empirical observation:
R~ i = E (R~ i ) +
k
X
j
=1
bij F~j + "~i
where the bij 's are constants,
the F~j 's are called factors, e.g.,
the mkt. pf.s rate of r.,
the interest rate,
other macroeconomic variables.
NB: Not asset specic.
"~i is noise, indep. of other "~j 's.
Main assumption: All covariance between asset returns can be captured by a linear structure of a limited number of common stochastic factors, so that the remaining, asset-specic part is white noise,
uncorrelated across assets.
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SK460 Finance Theory, Diderik Lund, 31 January 2001
Linear Factor Model, contd.
If the number of assets, n, is large, there exist 0; : : : ; k so that
E (R~ i) = 0 +
k
X
j
=1
j bij :
Will not look at the proof.
Special cases:
If riskless asset exists, then 0 = Rf .
If the only factor is R~ M , Rf , then CAPM.
Remarks
Nice thing: Can accomodate empirical nding: Many factors matter. Also: Simpler assumptions on distributions and/or utility functions (than CAPM).
Problem: Not obvious what factors should be.
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