Lets consider an individual making portfolio choice at time T , this choice involves two kind of assets: a
risky asset which consists of a broad portfolio of stocks (S&P500 index), whose gross return at time t per
f
unit invested at time t - 1 is 1+Rt , and a riskless asset whose gross return is 1+Rt . The decision maker
has access to the return histories over T periods, he knows in advance the future return of the riskless asset
that in accordance with standard practice is assumed to be constant.
Lets define rt = log(1 + Rt) and rtf = log(1 + Rtf),then we can describe the investor’s information set in
the following way:
It ≡ {rt}tT=0 , {rtf}tT=0 and {rf}tH=T +1 .
He invests one unit of saving, divided between an amount 1 - a in the safe asset and a in the risky asset,
and then he holds on to the portfolio until date H.
Let W (rt ,a) denote the value of the portfolio and suppose also that we are considering a self financing
portfolio. Thus, the value of the portfolio at time t = H is given by:
HH H
W(rt,a)=(1-a) Y (1+Rtf)+a Y (1+Rt)=(1-a)Hrtf +a X rt.
t=T +1 t=T +1 t=T +1
Lets also assume that utility depends only on the final value of the portfolio: U(W (rt, a)).Then, the
problem is to choose the best decision rule d that maps the observations contained in It into actions a, in
other words: the optimal share to invest in the risky asset. This decision rule is obtained by maximizing the
following expected utility:
Egr|s[U(W(rt,a))]
H
s.t Wt+H =(1-a)Hrtf+a rt and a ∈ [0, 1].
t=T +1
In order to simplify the analysis lets assume that H=1, such that the wealth form reduces to
W (ret ,a)=(1- a)rtf + art = rtf + a(rt - rtf )=C + aret ,
where C is a constant and ret is the excess return.
Example 1: (CARA investor) Assume that U(W (ret,a)) is the utility function of an investor with negative
exponential utility:
U(W (ret,a)) = - exp(-δW (ret, a))
Egr∣s [U(W(et,a))] = -KEgr∣s ∣exp(-δaet^,
where δ is the risk aversion parameter, K is the expected utility relative to the riskless asset and gr|s is the
distribution of the return rt given the regime s, which is unknown. Typically, the return distribution gr|s is
assumed to be Normal, consequently the expected utility results to be:
Eg l [U(W(et, a))] = - exp(-aδμ + 1 δ2a2σ2),
r|s 2
where μ = E(rt) and σ2 = Var(et) are the mean and the variance of the Normal distribution. In this case
the optimal share to invest in the risky asset is given by:
a(gr|s) = arg maax Egr|s [U(W(ret,a))]
18