STOCHASTIC

(c) Can the result in (b) be true if E log XJX„ _ i < 0 ?
(d) Provide a counterexample to the "false corollary": "If maximizing the geometric
mean almost certainly leads to a better outcome, then the expected value utility of its
outcomes exceeds that of any other rule provided it is sufficiently large."
(e) Attempt to determine a method for constructing utility functions over final wealth
that satisfy the false corollary. [Hint: Suppose U is bounded.]
(f) Provide an example to verify the theorem in Section II.
(g) Give an example of increasing inequivalent utility functions U and V that have
distinct optimal strategies for specific Bernoulli investments.
(h) Can the strictly concave assumption be dropped from the statement of the theorem
in Section II?
(i) Show that the almost certainly transitivity result in Section II does not hold if the
time horizon is finite.
(j) Refer to Section IV. What are the higher order terms in the Taylor series expansion ?
When will they be "small" ?
Suppose there are three investments having mean returns of (1.1,1.05,1.20) per dollar
invested and variance-covariance matrix of returns equal to
0.15 0 -0.2 \
0.0 0 0.0 I
-0.2 0 0.5 /
(k) Compute Markowitz's efficient surface.
(1) Compute Thorp's efficient surface.
(m) Compare the surfaces.
(n) Prove the lemma: An E log X optimal portfolio is never stochastically dominated.
(o) Comment on the statement: "A characteristic of the Kelly criterion is that as risk
decreases and expectation rises, the optimal fraction of assets to be invested in a single
situation may become 'large'" in light of the fact that the optimal asset proportions,
say A,, satisfy X, — .E(Ajr,/£Air,), where r, is the gross return from investment /'.
22. This exercise presents an extension of Breiman's model to allow for more general
constraint sets and value functions. The development extends Theorem 1 and indicates
that no strategy has higher expected return than the expected log strategy in any period.
The model applies to common investment circumstances that include borrowing, transactions
and brokerage costs, taxes, possibilities of short sales, and so on. Suppose initially that the
value function has stochastic constant returns to scale: VN = £f X^r*, as in Breiman's
formulation. It will be convenient to utilize the superscripts N to denote the investment period
under consideration. Let AN