Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 35 / 2

Indifference means equality of the risky expected utility and the utility of the sure reduced wealth. A second-order Taylor expansion, using $\mathbb{E}\varphi =0$, givesFor the sure alternative,Equating these and retaining the leading terms produces the small multiplicative risk premiumFor increasing concave utility functions this is positive. The approximation assumes genuinely small shocks with controlled Taylor remainders and positive final wealth; small variance alone would not justify it for rare large shocks.

Constant relative risk aversion coefficient means that $\alpha =-W{U}^{\prime \prime }(W)/U^{\prime }(W)$ does not depend on wealth, for $W>0$ and $U^{\prime }(W)>0$. The defining differential equation integrates asA second integration therefore givesThe additive constant and positive scale do not affect expected utility preferences. Since $U^{\prime \prime }(W)=-\alpha c{W}^{-\alpha -1}$, positive $\alpha$ gives risk aversion, zero gives risk neutrality, and negative $\alpha$ gives risk seeking. These are respectively concave, affine and convex utility functions. The usual CRRA utility class is the positive-$\alpha$ case.

Use $w$ for current wealth and $\pi =w-B$ for the dollar position in the risky asset. Its controlled dynamics have drift $rw+(\mu -r)\pi$ and diffusion coefficient $\sigma \pi$. Over a short time step $h$, the dynamic programming principle givesApply Itô formula to the second term, divide by $h$, and let $h\downarrow 0$. The Hamilton-Jacobi-Bellman equation isFor the increasing concave value function, $J_w>0$ and $J_{ww}<0$, so the expression is a strictly concave quadratic in the risky position. Its derivative vanishes atSubstitution of this maximizer yieldsThere is a running utility reward but no consumption withdrawal in the wealth dynamics. The required concavity is also verified explicitly by the positive coefficient in part (d), so this stationary point really is the maximum.

The power-wealth investment with running utility objective is homogeneous of degree $\gamma$ in wealth: scaling both wealth and positions by $a>0$ scales the reward by $a^{\gamma }$. Accordingly set $J(w,t)=A(t)w^{\gamma }$. ThenFor completeness the remaining equation is $A^{\prime }+\kappa A+e^{-\rho t}=0$, $A(T)=1$, whereThis confirms $J_{ww}<0$. The constant optimal risky fraction also makes wealth a positive geometric Brownian motion when started positive.

Financially the relative risk aversion coefficient is $1-\gamma$. Increasing $\gamma$ reduces risk aversion, raises the risky fraction, and decreases the bond fraction: its derivative is $-(\mu -r)/[{\sigma }^2(1-\gamma {)}^2]<0$. If the risky fraction is below one, some wealth is lent; at one, none is invested in bonds; above one, $B^{\ast }<0$ represents borrowing to leverage the risky position. As $\gamma \uparrow 1$ the risky fraction becomes unbounded, consistent with the absence of a finite unconstrained optimum for linear utility and positive excess drift. As $\gamma \downarrow 0$ the limiting fraction is $(\mu -r)/{\sigma }^2$, the logarithmic-utility allocation obtained by the appropriately rescaled power-utility limit; the literal objective at $\gamma =0$ would be constant and would not select a unique strategy.