Past exam of the mathematics course of the University of Cambridge / 2005 / ii / Paper 2 / 28J (Stochastic Financial Models)

Let ${\mathcal{F}}_n$ represent the information at decision time $n$ and $E_n$ conditional expectation given it. $V_n(w)$ is the largest conditional expected utility from the remaining consumption opportunities with current wealth $w$; it may depend on the information state as well as $w$. The controls $c$ and $\theta$ are chosen using current information. In the usual allocation interpretation $0\le c\le w$ and $0\le \theta \le 1$, and next wealth is $(w-c)[r+\theta (X_{n+1}-r)]$. Other borrowing/short-selling rules require explicitly replacing this admissible set while preserving solvency.

The Bellman equation follows by separating current utility from future utility: after any current decision the best continuation has conditional value $V_{n+1}$, and maximizing over the current admissible decisions gives the stated recursion. At the last opportunity all wealth is consumed, because utility is increasing, giving $V_N(w)=U(w)$.

There are two printed qualifications. Strict convexity is not needed for this recursion, but it conflicts with the concave power utility in part (ii) and does not support the interior maximizing first-order conditions in part (iii). Also the displayed recursion starting at $n=0$ includes consumption at times $0,\dots ,N$, whereas the preceding sum starts at one. The same recursion represents that sum when begun at time one; alternatively relabel the consumption dates. Below the policy is given for the recursion's remaining horizon, so this indexing discrepancy does not leave the optimization unspecified.

Here $U^{\prime }(x)=x^{-R}>0$ and $U^{\prime \prime }(x)=-R{x}^{-R-1}<0$: the specified utility is strictly concave, contrary to the word “convex” in part (i). Assume its expected values are finite on the admissible set. Define $G_{\theta }=r+\theta (X-r)$ and choose a constant portfolio fraction ${\theta }_{\ast }$ maximizing $E[U(G_{\theta })]$. Equivalently maximize $E[G_{\theta }^{1-R}]$ for $0<R<1$ and minimize it for $R>1$. An interior optimum satisfiesFor a constrained endpoint optimum the corresponding derivative has the appropriate one-sided sign. The objective is concave in $\theta$ and strictly concave if $P(X\ne r)>0$, so this condition or its endpoint version determines the optimum. Put $\kappa =E[G_{{\theta }_{\ast }}^{1-R}]$ and $\gamma ={\kappa }^{1/R}>0$.

Backward induction with independence of the returns gives $V_n(w)=A_n{w}^{1-R}/(1-R)$ and $A_N=1$. Maximizing the consumption expressiongives $c^{-R}=A_{n+1}\kappa (w-c{)}^{-R}$, henceWriting $b_n=A_n^{1/R}$ yields $b_N=1$ and $b_n=1+\gamma b_{n+1}$. Therefore the fully explicit horizon dependence isUse $b_n=(1-{\gamma }^{N-n+1})/(1-\gamma )$ when $\gamma \ne 1$, and $b_n=N-n+1$ when $\gamma =1$. The investor consumes this fraction, invests the remaining wealth with the same optimal portfolio mix at each date, and consumes all remaining wealth at $N$. If the stock is identical to the bank almost surely, every portfolio fraction is equivalent.

The equalities are valid at a differentiable interior optimum under the intended concave formulation, but not under the printed assumptions alone. To derive the intended result, differentiate the finite conditional sum in the Bellman objective. For an interior portfolio fraction with positive invested wealth, its derivative givesThe consumption derivative gives $U^{\prime }(c_n^{\ast })=E_n[V_{n+1}^{\prime }(w_{n+1}^{\ast })G_{{\theta }_n^{\ast }}]$. Since $G_{\theta }=r+\theta (X-r)$, the first equality cancels its risky part and yields the second requested equality. The envelope derivative with respect to wealth gives $V_n^{\prime }(w_n^{\ast })=E_n[V_{n+1}^{\prime }(w_{n+1}^{\ast })G_{{\theta }_n^{\ast }}]$, giving the third. Thus, under these added hypotheses,The first prices the excess stock payoff using future marginal utility; the second equates marginal utility from consuming now to that from saving in the bank; the third identifies the marginal value of wealth with that common quantity. At portfolio or consumption boundaries these are replaced by one-sided optimality inequalities. Finiteness of the sample space justifies differentiating a finite sum; it does not ensure differentiability of a supremum or an interior optimum.

A concrete counterexample satisfies the printed convex assumptions and attains every supremum. Take one remaining decision, a one-point sample space, a deterministic stock return $X=x>r>1$, $U(c)=c^2$, and controls in $[0,w]\times [0,1]$. The optimum is $c^{\ast }=0,{\theta }^{\ast }=1$, since the objective $c^2+x^2(w-c{)}^2$ is convex in $c$ and its larger endpoint value is $x^2{w}^2$. Then $w_{n+1}^{\ast }=xw$ and $V_{n+1}^{\prime }=2xw$. The first claimed expectation is $2xw(x-r)>0$; the second compares $2rxw>0$ to $U^{\prime }(0)=0$; the third compares $2rxw$ to $V_n^{\prime }(w)=2x^{2}w$. All three fail. Consequently the requested equalities cannot be proved as printed; the derivation above supplies their precise intended interior version.