STOCHASTIC

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be invested monthly for a year. With Buch a program it would be possible to avoid investing
the whole fund at the year's high. Dollar averaging as so applied produces a
diversification of timing which when accompanied by a diversification between issues
and between classes of investment makes for cautious risk spreading.
Similar statements about dollar averaging can be found elsewhere; for example, in
Cottle and Whitman [3, pp. 166-168, 180-182]. Examination of such statements indicates
the following general features. A given amount of a divisible asset is to be irreversibly
converted into another asset over a fixed number of periods. Usually, as in
the quotation, a given amount of dollars is to be sold for shares of stock. No firm
convictions are held that any available selling strategy will give a significantly higher
expected amount of the second asset at the end than any other. Instead, concern with
risk seems paramount. None of the possible outcomes is conceived of as a disaster. The
policy to be followed is nonsequential since it is not dependent on future price behavior.
The seller is too small to affect the market price of the asset. A fairly short period of
time is available to complete the conversion.
The fact that the sale of an asset whose price follows a stochastic process is involved
suggests that the literature on this topic may be relevant. Karlin [6], Samuelson [10],
and Taylor [11] have all studied this problem. A recent contribution is that of Hayes
[4]. Additional references may be found there, including references to the closely related
optimal stopping problem. In general, in this work the price of the asset has been
expected to rise with time and then fall to a given value at some horizon. The criterion
adopted has been to maximize the expected discounted value of the price obtained.
In the dollar averaging problem differences in the expected value of the alternatives
do not seem to be the significant factor. Instead, it is differences in the risk of the
alternatives.
The significance of risk suggests a relationship to the portfolio selection problem.
Under the nonsequential constraint there is a close similarity to the static portfolio
problem first considered by Markowitz [7]. If a sequential policy is allowed there is a
similarity to the sequential portfolio problems studied by Tobin [12] and by Mossin [8].
An essential difference, however, between the dollar averaging problem or the asset
sale problem and that of portfolio selection is the irreversible nature of the asset transfer.
In the asset sale problem the amount of the risky asset which may be held in later
periods depends on the amount which was held earlier. In the portfolio problem asset
transfers are fully reversible and constrained only by over-all budget requirements.
Another difference concerns the appropriate objective. Hedging against large regrets
appears to be a significant consideration in the dollar averaging problem and sometimes
in other asset sale problems. In previous work on portfolio selection this has not been
the case.
The possible significance of regret in the dollar averaging problem is foreshadowed
in the quotation. The author justifies the technique in part by stating that in this way
it is "possible to avoid investing the whole fund at the year's high". Consideration of
regret, however, has largely disappeared from current decision theory in favor of
expected utility maximization. Yet, if one suffers regret it seems no less rational to
consider it in decision making than it is to consider tomorrow's hangover when consuming
alcohol. Though one would like to purge regret from his psyche, it may be no
easier to eliminate than the hangover. In cases where the expected utility of the alternatives
are not too different, hedging against regret may attain overriding significance.
Situations where dollar averaging has been recommended may well fit such a case.
Even if one is not willing to consider regret in personal decision making another
578 PART V. DYNAMIC MODELS