Assessing Risk Utility Measurement

Assessing Risk Utility Measurement If you were presented with Options 1 and 2 in Figure, would you necessarily choose the option with the highest expected monetary value, Option 1? The second option has much to recommend it, offering the possibility of a higher return, and more importantly for most, offering a minimum return of £30. If the numbers represented £000, we suspect that few would choose Option 1. Suppose the same question were asked of a manager, only this time the figures represented profit estimates (again in £000). We suspect that most managers would also choose Option 2. The reasons would not be dissimilar to those of the personal choice. A justification might run something like this: I am not too concerned whether or not my profits are £100,000 or £120,000, but I am concerned about my profits showing £30,000 (Option 2) rather than £100,000 (Option 1). Obviously, there is quite a strong possibility of this happening if I choose Option 2. However, I judge this possibility to be less serious than reporting no profits, which is possible in Option 1. Despite the lesser chance of the latter event, I will choose Option 2 (principally to avoid zero returns). Note that when we originally looked at this problem (section 3.4.2), we were careful to say that the decision maker was equally happy with every £1 received. This would make the above statement impossible, for the decision maker is saying in this example that he is more concerned about the difference between £0 and £30, than £100 and £120. Empirical tests (e.g. Swalm (1966)) confirm that the reasoning we have presented here is the more common. Evidently, when faced with a choice between risky options, choosing the option with the highest expected value is not satisfactory. In particular, it would seem that not enough account is taken of the effect of the lower returns on the decision. This attitude can be represented by constructing a scale which would weight the difference between £100 and £120 less than the difference between f0 and £30. Then we might be able to assess the options in terms of expected value of the scores on the new scale. A suitable procedure was suggested by von Neumann and Morgenstern (1949), and the resultant measure was unfortunately given the name 'utility', a term which we shall prefix with the initials N-M to emphasize its special qualities (see section 3.4.4 for a review of utility). The scale is derived by asking a decision maker (D-M a number of hypothetical questions in the following manner: Analyst: If you were offered the following bet: 50 per cent chance of £150 and 50 per cent chance of £o How much would you want to receive for certain as minimum compensation for losing the chance to enter this bet? D-M : Well, say, £40 Analyst : If you were offered the following bet: 50 per cent chance of £40 and 50 per cent chance of £o How much would you want to receive etc? (as above) D-M: This is decidedly meaner than the last bet; in fact I'm not that interested in taking it, so I don't need much to dissuade me, or make me indifferent. About £5 would be my answer. Analyst: Finally, if you were offered the following bet: 50 per cent chance of £150 and 50 per cent chance of £40 How much would you want to receive etc? (as above) D-M: I would get at least £40 from the bet with a fair chance of an extra £110. I rather like this bet and would be a bit reluctant to lose it, so I reckon about £80. From these questions we are able to devise a scale in much the same way as a scale was devised for the value function. First, as with any scale, we need to set two points. Let us assign the number 0 to £0 and 12 to £150 (for reasons that will become clear later). From the first bet the decision maker expresses an equivalence between the bet and £40 for certain, therefore: The bet = certainty 0.5 (0) + 0.5 (12) = 1.0 (A) A =6 where A is, of course, the value that our scale places upon the £40. The second bet can now be represented as: 0.5 (6) + 0.5 (0) = 1.0 (B) B =3 B is £5, so our scale values B at 3. The third bet gives rise to the equation: 0.5 (12) + 0.5 (6) = 1.0 (C) C =9 The right hand side is the certainty equivalent, which is usually something less than the expected value, reflecting the fact that in general people are risk averse. Before we develop this point further, let us measure these results by constructing the scale. Typically, further questions, in the manner illustrated here, will yield answers whose points are then described by the line in Figure 3.12. In this particular case the line may be approximated by the equation: Rather than choose the option which maximizes expected Is, the choice will be better predicted by maximizing expected u's, for it is the reaction to the Is as measured by the u's that is significant. All we need do is convert the Is into us using the equation u= Lx; the original bet that we considered in Figure 3.11 can now be represented as in Figure 3.13. The result is as we anticipated. Although Option 1 has a higher expected value, the relatively greater concern of the decision maker for the lower return means that Option 2 would be chosen, as it has the higher N-M utility measure.