STOCHASTIC

andom outcome, regret is defined to be the loss incurred by not following
the best policy for the specific outcome. Thus, if the decision maker employed
a reliable clairvoyant to inform him of the maximum price which would actually
be realized, the optimal policy would be to sell the entire asset at the highest
price. Failure to follow this policy would lead to regret, since profits would
be lower than necessary. Not having access to a clairvoyant, the decision maker
might choose instead to adopt a policy which leads to the smallest regret under
worst possible conditions. By following such a policy the decision maker can
be sure that even if events turn out badly, failure to adopt the policy would
have exposed him to potentially worse outcomes. This is a conservative policy,
but one which is clearly appropriate in cases where the decision maker is, for
example, accountable for his actions after the fact. It is, in general, a much
less conservative policy than always assuming the worst.
One apparently common policy recommendation for asset selling is that of
"dollar averaging." In such a policy, the decision maker sells equal quantities
of the asset in each period. (In the case of security buying, equal quantities of
dollars are "sold" in each period.) Pye shows that dollar averaging is an optimal
nonsequential policy when the maximum possible single-period price increases
and decreases are equal. In general, nonsequential minimax regret policies are
not unique, and dollar averaging is merely one of an infinite number of optimal
policies. In Exercise CR-13, another simple optimal nonsequential policy is
outlined. Pye also derives the optimal nonsequential selling strategy for a policy
of maximizing expected utility of total revenue. For any strictly concave utility
function, the optimal policy is to sell all of the asset on the last day. In general,
none of the infinitely many nonsequential minimax policies can have this form.
Thus, an expected utility of revenue principle cannot lead to dollar averaging.
Pye considers next the question of sequential minimax regret policies. Here
the decision at each stage is conditioned on actual, realized past outcomes.
Dynamic programming is used to derive the general form of an optimal policy.
In the nomenclature of the Ziemba paper of Part I, the state variable at stage t
consists of two dimensions: (1) the difference between the maximum price
occurring up through time / and the price at time t, and (2) the quantity of
asset still held at time t. The optimal return function at stage / is defined to
be the regret at time / given that an optimal minimax sequential policy is
followed at t and subsequently. A functional equation is derived for the optimal
return function using backward induction. Pye shows that the return function
is convex and decreasing in each state dimension, and for a fixed state, is
decreasing in time. These properties are used to derive the form of the optimal
policy, which is to sell at time t any portion of the asset in excess of a critical
quantity. This critical value is a decreasing function of the first state dimension
(i.e., the difference between maximum realized price up through t and the
price at t). A numerical example is given to show that the critical value is not,
442 PART V DYNAMIC MODELS