Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 23 / 1

The no-borrowing portfolio constraint is $0\le x\le T$. Net asset returns must be added to the original principal. Thus the two final portfolio wealths areThe expected utility isThe probabilities are those of the investor's beliefs; no risk-neutral measure is being used in this preference calculation.

Risk neutrality means indifference between a random portfolio wealth and its certain expected value. For increasing twice differentiable expected utility preferences, this corresponds to an affine function $u(w)=Aw+C$ with $A>0$ on the relevant wealth interval. Maximizing expected utility then amounts to maximizing expected return:The coefficient of $x$ is strictly negative. Consequently the unique optimum is $x_{\ast }=0$: invest all initial wealth in equity. Risk neutrality here is a preference property, distinct from the pricing use of a risk-neutral measure.

Risk aversion means that a sure mean is preferred to the corresponding risky portfolio wealth. An increasing concave utility function, with $u^{\prime }>0$ and $u^{\prime \prime }\le 0$, has this property by the Jensen inequality; strictly concave utility gives strict preference for nondegenerate risks. Differentiating the expected utility from part (a) givesAn interior optimum therefore satisfiesThis balances the expected loss of marginal utility in the good state against its expected gain in the bad state. The borrowing constraint also requires checking the endpoints: $x_{\ast }=0$ if $U^{\prime }(0)\le 0$, while $x_{\ast }=T$ would require $U^{\prime }(T)\ge 0$. At $x=T$ both state wealths equal $T(1+r)$, soHence an optimum never puts all wealth into deposits. If $U^{\prime }(0)>0$, continuity and the displayed endpoint sign give an interior root, and concavity makes every such root globally optimal. Under strict concavity it is unique.

Assume first that $T>0$ and $-1<b<r<g$, so the logarithmic utility is defined throughout the allowed portfolio interval. Put $A=g-r>0$ and $B=r-b>0$. ThenEquating this to zero and clearing the positive denominators givesThe second derivative is strictly negative, and $U^{\prime }(T)<0$ by part (c), so $x_0<T$. The two-state logarithmic portfolio with a borrowing constraint is thereforeIn particular, the threshold for all equity isFor $p<p_c$ the optimum is interior. The probability bound printed in the PDF has $1+r$ where the all-equity threshold has $1+b$. Since $1+r>1+b$, that printed lower bound is strictly below $p_c$ and does not determine which of these two cases occurs. For example, with $(b,r,g)=(0,0.1,0.3)$ its lower bound is $13/35$, while $p_c=13/33$. At $p=0.38$ all the given inequalities hold and $x_{\ast }=0.23T$; at $p=0.4$ they also hold and $x_{\ast }=0$. Thus the boxed constrained formula applies to the printed data without replacing its probability assumption.

If bad-state gross wealth can be nonpositive, the logarithmic utility domain must instead be imposed explicitly. With $1+r>0$, $b\le -1$, and $0<p<1$, admissibility requires $x>-T(1+b)/(r-b)$; the same stationary root lies inside this positive-wealth interval and is optimal. If even the fully safe gross return is nonpositive, there is no admissible positive-wealth portfolio. These domain issues are implicit in using $\log w$.