Density Estimation and Combination under Model Ambiguity

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+R_t , and a riskless asset whose gross return is 1+R_t . 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 r_t^f = log(1 + R_t^f),then we can describe the investor’s information set in
the following way:

It ≡ {rt}_t^T₌₀ , {r_t^f}_t^T₌₀ and {r^f}_t^H_{=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+Rt^f)+a ^Y (1+Rt)=(1-a)Hrt^f +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 (r_t, 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)Hrt^f+a      r_t 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)r_t^f + art = r_t^f + a(rt - r_t^f )=C + aret ,

where C is a constant and re_t 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 (re_t,a)) = - exp(-δW (re_t, 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 g_r|_s is the
distribution of the return rt given the regime s, which is unknown. Typically, the return distribution g_r|_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(r_t) 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 m_aax Eg_r|s [U(W(ret,a))]

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