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

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

i$.$e$.,$ he maximizes $E_t[u(W_{t+1})].$ Assume that the investor has a logarithmic

utility function, $ln(x).$ 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:

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

$r=ln(R)$

$rN((\frac{?}{\text{b}\text{a}\text{r}}(r)),{}^2)$

Derive the optimal fraction $$ of current wealth $W_t$ invested in the risky asset.

Hint: In this two$-$asset market, log portfolio returns can be approximated by:

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

$($b$)(10$P$.)$ A general advice for long$-$term investors is that they should hold a higher

fraction of equities compared to short$-$term investors. Explain the two main

reasons for this advice.