SØK460 Finance Theory, Diderik Lund, 6 February 2001
A closer look at the CAPM
This and the next lecture:
• CAPM as equilibrium model? Mossin (1966).
• CAPM without risk free rate. Roll (1977) app., pp. 158—165.
• Testing the CAPM? Roll (1977) main text.
The need for an equilibrium model
• What will be effect of merging two firms?
• What will be effect of a higher interest rate?
• Could interest rate exceed E(R̃mvpf ) (min-variance-pf)?
• What will be effect of taxation?
Need equilibrium model to answer this. Partial equilibrium: Consider stock market only.
Typical competitive partial equilibrium model:
• Specify demand side: Who? Preferences?
• Leads to demand function.
• Specify supply side: Who? Preferences?
• Leads to supply function.
• Each agent views prices as exogenous.
• Supply = demand gives equilibrium, determines prices.
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SØK460 Finance Theory, Diderik Lund, 6 February 2001
Repeating assumptions so far:
• Two points in time, beginning and end of period, t = 0, 1.
• Competitive markets. No taxes or transaction costs.
• All assets perfectly divisible.
• Agent i has exogenously given wealth W0i at t = 0.
• Wealth at t = 1, W̃ i, is value of portfolio composed at t = 0.
• Agent i risk averse, cares only about mean and var. of W̃ i.
• Portfolio composed of one risk free and many risky assets.
• Short sales are allowed.
• Agents view Rf as exogenous.
• Agents view probability distn. of risky R̃j as exogenous.
• Everyone believe in same probability distributions.
Main results:
• CAPM equation, R̃j = Rf + βj [E(R̃m ) − Rf ].
• Everyone compose risky part of portfolio in same way.
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SØK460 Finance Theory, Diderik Lund, 6 February 2001
Partial equilibrium model of stock market
• Main contribution of Mossin’s paper.
• But have seen: Some important results without this.
Maintain all previous assumptions. Add these:
• The number of agents is m, i = 1, . . . , m.
• The number of different assets is n, j = 1, . . . , n.
• Before trading at t = 0, all assets owned by the agents: X̄ji .
• After trading at t = 0, all assets owned by the agents: Xji .
• Agents own nothing else.
• Asset values at t = 1, p̃j1, exogenous prob. distribution.
• One of these is risk free, in Mossin this happens for j = n.
• Choose units (for risk free bonds) so that p̃n = pn = 1.
• Asset values at t = 0, pj0, endogenous.
• But each agent views the pj0’s as exogenous.
• Thus each agent views R̃j = p̃j1/pj0 − 1, as exogenous.
• W0i consists of asset holdings, W0i =
Pn
i
j=1 pj0 X̄j .
• Thus each agent views own wealth, W0i , as exogenous.
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SØK460 Finance Theory, Diderik Lund, 6 February 2001
Interpretation of model setup
• Pure exchange model. No production. No money.
• Utility attached to asset holdings.
• Market at t = 0 allows for reallocation of these.
• Pareto improvement: Agents trade only what they want.
• At t = 1 no trade, only payout of firms’ realized values.
Mossin’s results
• No attention to existence and uniqueness of equilibrium.
• Walras’s law: Only relative prices determined.
• Choose pn = q, 1 + Rf = 1/q. Then other prices determined.
• Mossin’s eq. (14): Portfolio of risky assets same for all.
• Existence of linear (σ, µ) opportunity set, CML.
• Implies: MRS between σ and µ same for all.
• No individual choice whether to locate at CML or off CML: If
choose off CML, model does not hold, and CML does not exist!
Detail: Mossin does not derive σ, µ preferences from expected utility. Thus he cannot rule out that indifference curves could meet σ
axis non-horizontally. Thus he says (p. 778, last para) that extreme
risk aversion would lead to choice of risk free asset only.
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SØK460 Finance Theory, Diderik Lund, 6 February 2001
Notation
Mossin, Roll and C&W use very different notation:
Mos- C&W’s and
Roll’s
sin’s
my notation
not.
xij
Xji
i’s holding of asset j after trade
x̄ij
X̄ji
i’s holding of asset j before trade
µj
E(p̃j1 )
Expected price of asset j at t = 1
σj2
var(p̃j1 )
Variance of same
1/q
1 + Rf
pj
pj0
Price of asset j at t = 0
wi
W0i
i’s wealth at t = 0
y1i
E(W̃ i )
i’s expected wealth at t = 1
y2i
mj
Vj
Rj
y1i
W0i
y2i
W0i
q
1 + rF
Explanation
Risk free interest rate plus one
var(W̃ i )
Std. deviation of same
E(R̃j ) − Rf
rj − rF Asset j’s expected excess return
P
i
i Xj p̃j1 )
var(
P
i
i Xj pj0
Variance of asset j (all shares)
Total value of asset j at t = 0
µp
rp
Expected rate of return on portfolio
σp
σp
Std. dev. of rate of return on pf.
wj
xjp
Value weight of asset j in pf. p
E(Rz )
rz
Exp. rate of return on zero-β pf.
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SØK460 Finance Theory, Diderik Lund, 6 February 2001
Equilibrium response to increased risk free rate?
• Previous results:
¸
1 ·
E(R̃M ) − Rf
E(p̃j1) − λ cov(p̃j1, R̃M ) , with λ =
pj0 =
,
1 + Rf
var(R̃M )
¸
cov(R̃j , R̃M ) ·
E(R̃M ) − Rf .
E(R̃j ) = Rf +
var(R̃M )
• None of these have only exogenous variables on right-hand side.
• In both, R̃M on right-hand side is endogenous.
• Consider hyperbola and tangency in σ, µ diagram:
— If Rf is doubled, tangency point seems to move up and
right.
— Upward movement seems to be less than a doubling of
E(R̃M ).
— But this relies on keeping hyperbola fixed.
— CAPM equation shows that E(R̃j ) is likely to change.
— True for all risky assets, thus entire hyperbola changes.
• To detect effect of ∆Rf , need only exog. variables on RHS.
• Not part of this course.
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SØK460 Finance Theory, Diderik Lund, 6 February 2001
What happens if two firms merge?
• From our previous result,
¸
1 ·
E(R̃M ) − Rf
pj0 =
E(p̃j1) − λ cov(p̃j1, R̃M ) , with λ =
,
1 + Rf
var(R̃M )
have shown value of merged firm equals sum of previous two
values.
• Previously this was an approximate result.
• Relied on “smallness”: Assumed R̃M unchanged.
• Mossin asks same question on pp. 779—781.
• More satisfactory analysis: Does not rely on R̃M unchanged.
• Get exact result: The same.
• Shows explicitly that R̃M unchanged.
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SØK460 Finance Theory, Diderik Lund, 6 February 2001
Roll’s appendix
Main points (in relation to our course):
• Consider n risky assets, n > 2, no risk free asset. Then the
frontier portfolio set is an hyperbola. (Mentioned without proof
on p. 12, 24 January.)
• Can derive version of CAPM without risk free asset. Important
if, e.g., there is uncertain inflation. (C&W, sect. 7G1, p. 205.)
The version mentioned in the second point is important to understand Roll’s main text, and much of the CAPM literature.
Market portfolio plays important role also in that version of the
model, even though it is not equal to the risky part of everyone’s
portfolio.
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SØK460 Finance Theory, Diderik Lund, 6 February 2001
Differentiation of vectors and matrices
(See chapter 23 of the math manual by Sydsæter, Strøm and Berck.)
• Derivative of a (scalar) function with respect to an n × 1 vector
is the n × 1 vector of derivatives with resp. to each element.
• Derivative of a (scalar) function with respect to an 1 × n vector
is the 1 × n vector of derivatives with resp. to each element.


∂f  ∂f
∂f 
=
,...,
.
x = (x1, . . . , xn ) =⇒
∂x
∂x1
∂xn
• Derivative of scalar product: a and x both are n × 1:
∂ 0
(a · x) = a0. (0 denotes transpose.)
∂x
(Generalizes scalar d(by)/dy = b.)
• Derivative of m × 1 vector w.r.t. n × 1 vector is m × n matrix
of derivatives.
• Derivative of product of matrix and vector: A is m × n, x is
n × 1:
∂
(Ax) = A.
∂x
• Derivative of quadratic form: A is n × n, x is n × 1:
∂ 0
(x Ax) = x0(A + A0).
∂x
(Generalizes scalar d(by 2)/dy = 2by.)
• Symmetric version of the same: A = A0 is n × n symmetric:
∂ 0
(x Ax) = 2x0A.
∂x
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SØK460 Finance Theory, Diderik Lund, 6 February 2001
Roll: Derivation of frontier portfolio set
(Will now use Roll’s notation, but not his terminology. Roll calls
the set “the efficient portfolio frontier” or just the “efficient set.”)
• Exist n risky assets (securities).
• Xp = kxipk, the n × 1 vector of portfolio weights.
• Must have Xp0 ι =
Pn
i=1 xip
= 1 (where ι = (1, 1, . . . , 1)0).
• Fundamental data, exogenous in minimization problem, are
• R = krik n × 1 vector of mean rates of return
• V = kσij k n × n cov. matrix of rates of return
These can be either population values (from a probability
distribution) or they can be sample product moments (from a
frequency distribution) (Roll, p. 159). Important for argument
in main text.
• Mean of (r.o.r. of) portfolio p is rp = X 0 R =
Pn
i=1 xi ri .
• Variance of (r.o.r. of) portfolio p is σp2 = X 0V X =
• Covar. of (r.o.r. of) 2 pf.s is σp1p2 = Xp0 1 V Xp2 =
Pn
Pn
i=1
i=1
Pn
j=1 xi xj σij .
Pn
j=1 xip1 xjp2 σij .
• Use matrix notation in solution for frontier portfolio set:
• For any value of rp: Choose X to obtain minimum σp2.
• Lagrangian L = X 0 V X − λ1(X 0R − rp) − λ2(X 0ι − 1).
• F.o.c.: ∂L/∂X = 2V X − λ1R − λ2ι = 0.
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SØK460 Finance Theory, Diderik Lund, 6 February 2001
• F.o.c. imply V X = 12 (λ1R + λ2ι), which implies


1
λ1 

.
X = V −1(R ι) 
λ2
2
• Here: X expressed as function of R, V, λ1, λ2.
• Want instead: X expressed as function of R, V, rp.
• Roll shows (p. 160) this can be done:
• Define A ≡ (R ι)0 V −1(R ι), a 2 × 2 matrix.
• Then


rp 

.
X = V −1(R ι)A−1 
1
• Matrix A summarizes useful information about R and V :


A=

r1 · · · rn 
V
1 ··· 1



−1 



r1
...
rn



1

... 
 a b 
 ≡ 



b c
1
where a = R0 V −1R, b = R0 V −1ι, c = ι0 V −1ι.
• Will also use
A−1

• Proof of this is straightforward:
A · A−1

1  c −b 

.
=
ac − b2 −b a


1  ca − b2
0


 = I2 .
=
2
0
−b + ac
ac − b2
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SØK460 Finance Theory, Diderik Lund, 6 February 2001
Roll’s Corollaries 1 and 2 (partly)
• Can now show that frontier portfolio set is parabola in (σ 2, r)
diagram, thus hyperbola in (σ, r) diagram.
• Plug formula for frontier portfolio vector (as function of rp) into
general formula for portfolio variance:
σp2 = X 0 V X = (rp 1)A−1(R ι)0V −1V V −1(R ι)A−1(rp 1)0
= (rp 1)A−1AA−1(rp 1)0 = (rp 1)A−1(rp 1)0


a − 2brp + crp2
1  c −b 
0

 (rp 1) =
= (rp 1)
.
ac − b2 −b a
ac − b2
• Minimization w.r.t. rp gives the minimum variance portfolio:
b
−b + crp = 0 =⇒ rp = .
c
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SØK460 Finance Theory, Diderik Lund, 6 February 2001
Roll’s Corollaries 3 and 3A
• For every frontier portfolio, except the min-variance portfolio,
exists another frontier portfolio with an uncorrelated rate of
return.
• (The importance will become clear later.)
• Proof: Use general covariance formula, set equal to zero, solve.
• Call the uncorrelated portfolio Xz , z for “zero-beta.”
• Covariance is (use similar transformations as for variance above):


1  c −b   rp


Xz0 V Xp = (rz 1)
1
ac − b2 −b a



which is equal to zero if and only if rz = (a − brp)/(b − crp).
• Roll also proves (p. 163) that one of ((σp, rp), (σz , rz )) is on
upper half of hyperbola, the other is on the lower half, i.e.:
rz < (b/c) < rp or rp < (b/c) < rz .
• Then he shows (second part of Corollary 3A): In (σ, r) diagram,
a tangent to the frontier portfolio set at (σp, rp) intersects the
r axis at rz .
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SØK460 Finance Theory, Diderik Lund, 6 February 2001
Roll’s proof of Corollary 3.A
Proof of second part of Corollary 3.A (pp. 163—164) is misleading.
The slope of a tangent to the efficient set in the (σ, r)-diagram is
the derivative of the square root of equation (A.11), i.e., of σp, not
of σp2.
We need to take the inverse of this derivative, since the derivative
itself would be the slope to the tangent when rp is on the horizontal
axis, while we want σp on the horizontal axis.
We find
dσp
1
−2b + 2crp
r
=√
.
·
drp
ac − b2 2 a − brp + crp2
The second part of Corollary 3.A states that a tangent (with a slope
equal to the inverse of this expression) through the point (σp, rp),
which is the location of some (any) efficient portfolio, should intersect the vertical axis at rz , which is known from (A.15),
rz = (a − brp)/(b − crp).
Before we know the value of r at the intersection, let us just call it
ri , which must satisfy (rp − ri)/σp = (dσp/drp)−1.
We can solve this equation for ri and plug in the square root of
(A.11) for σp and the expression we just found for the derivative.
Some terms cancel out and we find that ri = (a − brp )/(b − crp),
which is the expression for rz from (A.15).
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SØK460 Finance Theory, Diderik Lund, 6 February 2001
Roll’s Corollary 5
• All frontier portfolios can be written as linear combination of
any pair of (different) frontier portfolios.
• Follows from the linearity of the frontier portfolio equation


X = V −1(R ι)A−1 

rp 
,
1
where all terms on the RHS are constants, except the rp.
• If Xp1 and Xp2 are frontier portfolios, and we want the frontier
portfolio with expected rate of return equal to r3 = αr1 + (1 −
α)r2, then we compose the portfolio

αr1 + (1 − α)r2

Xp3 = V −1(R ι)A−1 
1


αV −1(R ι)A−1 


r1 
r2
−1
+(1−α)V

(R ι)A−1 
1
1






=
= αXp1+(1−α)Xp2.
• Also for linear combination of many frontier portfolios.
• If all weights are between zero and one, then weighted sum will
have expected rate of return between the extremes of which the
sum consists.
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SØK460 Finance Theory, Diderik Lund, 6 February 2001
Derivation of the zero-beta CAPM
Copeland and Weston (pp. 205—208) describe a version of the CAPM
in which there is no riskless asset. They give no proof, but from
Roll’s appendix, the derivation of the main results is quite straight
forward:
• No riskless asset ⇒ everyone holds portfolio on upper half of
the hyperbola known as the frontier portfolio set.
• The market portfolio is a convex combination of the portfolios
of all agents.
• Corollary 5: Any convex combination of portfolios on the upper
half of the hyperbola, is itself efficient, i.e., located on the upper
half of the hyperbola. ⇒ The market portfolio is efficient.
• Can then do exactly the same reasoning as in Copeland and
Weston (pp. 196—198) with the little and the big hyperbola,
this time with the slope (rm − rz )/σm instead of (rm − rf )/σm
(which Copeland and Weston call (E(R̃m )−Rf )/σm on p. 197).
• Conclusion: See p. 207 of Copeland and Weston, in particular
equation (7.25),
E(R̃i) = E(R̃z ) + βi [E(R̃m) − E(R̃z )].
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