theory versus practice in risk analysis: a reply

Theory versus Practice in Risk Analysis: A ReplyAuthor(s): Willis R. Greer, Jr. and Ted D. SkekelSource: The Accounting Review, Vol. 50, No. 4 (Oct., 1975), pp. 839-843Published by: American Accounting AssociationStable URL: http://www.jstor.org/stable/245251 .

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Theory Versus Practice in Risk Analysis: A Reply

Willis R. Greer, Jr. and Ted D. Skekel

T HE comment by Hoskins (Hoskins, 1975) represents a significant ad- vance in modeling actual decision

processes. However, we must add that Hoskins' findings support rather than dis- pute the conclusion of the original study that "there appears to be substantial con- flict between the decision processes used by actual decision makers and existing utility theory." Our intentions in writing this response are (1) to demonstrate why this is so and (2) to expand further on the Hoskins model.

First let it be restated that the model in the original study

U(9, u,) = a- ba6

was not designed to produce decisions which closely paralleled actual decisions. (In fact, a much better "match" could have been achieved by manipulating the parameters.) Rather, the model was built in the shape of classical utility theory with parameters which were computed from an explicit statement by the firms as to their attitude toward risk-taking. The principal conclusion of the original research was that actual decisions are inconsistent with such a model.

To demonstrate more transparently the differences between the decisions made using the original model and the actual decisions, consider the "difference matrix" shown in Table 1. Each "D" in the

table represents a firm/investment inter- section where a difference occurred in the original study. These "D's," then, provide a detailed listing of the actual decisions which were inconsistent (for whatever reason) with the decisions generated using the original model.

It should be clear to the reader that the model of the original study is consistent with a utility function of a shape shown in Figure 1 and that this is defined as "existing" or "classical" utility theory. In the original study the actual decisions seemed to be resulting from use of a utility function of nonclassical shape-one which had a "kink or discontinuity." Moreover, in the original study a comment was made that "the exact derivation of such func- tions is left to others."

Such a derivation is exactly what Hos- kins has provided: he has derived a kinked utility function. Hoskins' model,

U = - a(Sh)

is a utility function with a shape as shown in Figure 2. Notice the function is curvilinear below h and linear above h.

Now consider the assignment of a value to the parameter a in Hoskins' model. Since his objective is not to build a theo-

Willis R. Greer, Jr. is A ssociate Professor of Accounting and Ted D. Skekel is a Ph.D. candidate, University of Oregon.

839



840 The Accounting Review, October 1975

TABLE 1

U(x, a-) =ax-bao DECISIONS VERSUS AcTUAL DECISIONS

Investment Total Differences

Firm A B C D E F G H I J K L M N O P Q R S T by Firm

1 D 1 2 D D D D D D 6 3 D D D D D D D D 8 4 D D D D D D 6 5 D 1 6 D D D D D D D D D 9 7 D D 2 8 D D D 3 9 D D D D 4

10 D D D 3 11 D D D D 4 12 D D D 3 13 D D D D D D 6 14 D 1 15 D D D D D D D 7 16 D D D D 4 17 D D D D D D D D D D D D 12 18 D D 2 19 0 20 D D D D 4 21 D D 2 22 D D 2 23 D D 2 24 D D D D D 5 25 D D 2 26 D D 2 27 D D D D D 5

Total Differ- ences by

Investment 4 1 4 4 10 4 7 0 7 2 4 12 3 0 5 0 12 4 9 14 106

FIGURE 1

CLASSICAL UTILITY FUNCTION

U

0-01~~

FIGURE 2 HOSKINS' MODEL

U



Greer and Skekel: Risk Analysis: A Reply 841

TABLE 2

U = X-a(SA) DECISIONS VERSUS ACTUAL DECISIONS



1 0 2 D D D D 4 3 D D D D D 5 4 D D D D D D 6 5 D D D 3 6 D D D D D D D D D 9 7 D D D 3 8 D D 2 9 0

10 D D D D D 5 11 D D 2 12 0 13 D D D 3 14 0 15 D D D D D D 6 16 0 17 D D D D D D D 7 18 0 19 D D 2 20 0 21 0 22 D D 2 23 D D D 3 24 D D D D 4 25 0 26 0 27 D 1


Investment 4 0 0 0 0 0 7 0 7 2 4 6 0 1 0 1 14 0 9 12 67

retical model but rather to parallel as closely as possible the actual decisions, the most logical way to assign a value to a would be to use that value which would minimize the number of "differences by firm" for each firm. (Of course, these differences might further be reduced in number by altering h, but we chose not to explore this possibility.)

If the values of a are so determined, and if Hoskins' model is then used to remake the decisions over the entire data set, the difference matrix shown in Table 2 results. Notice this difference table represents a considerable improvement over Table 1. Hoskins' model is a closer parallel to real-world decision making than is a model constructed from basic risk attitudes in accordance with classical utility theory.

In examining Figure 2, though, we were bothered by the fact that Hoskins' model is linear for dollar amounts above h. While decision makers clearly are not as averse to risk when all contingent outcomes lie above this value, we would hypothesize that they do not necessarily become ex- pected value adherents as soon as h is exceeded.

Consider, for instance, Investment 0. This is a special case in that jc= xo and Sh= zero. (Notice this is why no differences are recorded for Investment 0 in Table 2. It is "impossible" to make a "wrong" decision for Investment 0 if the utility function is linear.) We would therefore expect a risk-neutral decision maker to flip a coin. The actual results, though, were that five decision makers selected the



842 The Accounting Review, October 1975

TABLE 3

U=9-a(SA) -jb[l/(9-xO)2] DECISIONS VERSUS ACTUAL DECISIONS



1 D 1 2 D D D D 4 3 0 4 0 5 D D D 3 6 D 1 7 D D 2 8 D 1 9 0

10 D D 2 11 D D 2 12 0 13 D D D D 4 14 D 1 15 0 16 0 17 0 18 0 19 0 20 0 21 0 22 D 1 23 D 1 24 0 25 D 1 26 0 27 D D 2


Investment 1 0 0 0 0 0 0 0 0 1 1 6 0 1 5 1 2 1 2 5 26

risk alternative and fifteen chose "no risk." From the binomial distribution we find,

P(R?< 5 n = 20, P =)=0.021

Therefore one might safely conclude that typical decision makers in this sample were not risk-neutral at dollar values above h.

We next tried to determine what might be a best-fit curvilinear shape to assign to the model for dollar amounts above h. After considerable trial-and-error-type conjecture (which, of course, does not guarantee optimality) we added to Hos- kins' model a negatively signed risk aversion term,

-XO)2

which was used only in assigning utility to

the risky alternatives. As the spread between xc and xo narrows, the relative utility of the risky alternative diminishes. A dummy variable, j, causes the risk aversion term to have an effect on the investment's utility only when SA= zero. In our new model,

U = - a(Sh) -jb[1/(2 - xo)2J

wherej= 1 if Sh=O

j=O if Sh>O

This new model was used to construct the difference matrix shown in Table 3. The result is a significant improvement over the Hoskins' model and is consistent with a utility function of the general shape (page 503) discussed in the original study. A comparison of Table 2 to Table 3 shows



Greer and Skekel: Risk Analysis: A Reply 843

FIGURE 3

THE REVISED MODEL

U

that the linear portion of Hoskins' model (Table 2) gave rise to 59 differences while the new model (Table 3) resulted in only 18 differences for the same investment decisions (i.e., those decisions involving a zero Sh value for the risky alternative).

Our final conclusion is that the decision makers seemed to be using a curvilinear, kinked utility function such as the one shown in Figure 3. This function matches the concept discussed in the original study. For dollar values below h our new model is a better parallel to actual decision processes than is classical theory. For dollar values above h, our new model is a better parallel to actual decision processes than is either classical theory or Hoskins' model.

REFERENCE

Hoskins, C. G., "Theory Versus Practice in Risk Analysis: An Empirical Study: A Comment," THE

ACCOUNTING REVIEW (October 1975), pp. 835-8.



theory versus practice in risk analysis: a reply

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