文档介绍：Ec2723, Asset Pricing I
Class Notes, Fall 2003
Static Portfolio Choice,
the CAPM, and the APT
John Y. Campbell1
First draft: July 30, 2003
This version: September 15, 2003
1 Department of Economics, Littauer Center, Harvard University, Cambridge MA 02138, USA.
Email john_******@.
The principle of participation
We begin by considering a choice between one safe and one risky asset. We make
only weak assumptions on preferences and the distribution of returns on the risky
asset. An investor with initial wealth w can invest in a safe asset with return r,or
a risky asset with return r + xe. Final wealth is
w(1 + r)+θxe = w0 + θx,e
where θ is the dollar amount (not the share of wealth) invested in the risky asset.
Theinvestor’sproblemis
Maxθ V (θ)=Eu(w0 + θxe).
The first-order condition is
V 0(θ∗)=Exue 0(w0 + θ∗xe)=0
and the second-order condition is
2
V 00(θ∗)=Exe u00(w0 + θ∗xe)=0,
which shows that the problem is well defined for a risk-averse investor.
We have
V 0(0) = Exue 0(w0),
which has the same sign as Exe. The investment in the risky asset should be positive
if it has a positive expected return. This is true for any level of risk aversion. Thus
we cannot explain non-participation in risky asset markets by risk aversion. We need
fixed costs of participation or a kink in the utility function that generates “first-order
risk aversion”.
Portfolio choice with a small risk
We consider a small risk
xe = kµ + ye
and assume k>-order condition is
E(kµ + ye)u0(w0 + θ∗(k)(kµ + ye)) = 0.
1
Differentiating . k,
2
µEu0(we)+θ∗(k)µE(kµ + ye)u00(we)+θ∗0(k)E(kµ + ye) u00(we)=0.
Evaluating at k =0,
µ 1
θ∗0(0) = 2 .
Eye A(w0)
Then a Taylor expansion for the investment in the risky asset gives
Exe 1
θ∗(k) θ∗(0) + kθ∗0(0) = 2 .
≈ E(xe Exe) A(w0)
−
We can divide θ by wealth to find the share of wealth invested in the risky asset.
Call this
θ∗(k) Exe 1
α∗(k)= 2 .
w0 ≈ E(xe Exe