Asset allocation $($Answer all parts of this question.$)$

$($a$)(3$P$)$ The optimal allocation to risky assets in mean$-$variance analysis can co$-$

incide with the optimal asset allocation when maximizing the expected utility

of next period wealth, $E_0[u(W_1)].$ Briefly describe three sets of assumptions

that are consistent with mean$-$variance analysis.

$($b$)(3$P$)$ Explain and illustrate the motive for intertemporal hedging if real short

interest rates are stochastic.

$($c$)(4$P$)$ What are the implications of stochastic real interest rates for optimal long$-$

term portfolio choice if inflation risk is modest? Suppose that real $($inflation$-$

indexed$)$ bonds are not available.

$($d$)(10$P$)$ An investor maximizes expected utility of wealth in a three$-$period model,

i$.$e$.,$ he maximizes $E_0[u(W_2)].$ Assume that the investor has a logarithmic

utility function. There is a financial market with two assets, a riskless asset

with a constant return of $R^f$ and a risky asset whose return $R$ is log$-$normally

distributed:

$r^f=ln(R^f)$

$r_t=ln(R_t)$

$r_{1}N(\frac{?}{\text{b}\text{a}\text{r}}(r{)}_1,{}^2)$

$r_{2}N(\frac{?}{\text{b}\text{a}\text{r}}(r{)}_2,{}^2)$

Derive the optimal fractions ${}_0$ and ${}_1$ of wealth in period $0(W_0)$ and $1(W_1)$

invested in the risky asset.

Hint: In this two$-$asset market, one$-$period log portfolio returns can be approx$-$

imated by:

$r_p=r_f+(r_t-r_f)+\frac{1}{2}(1-){}^2$