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Appendix A: a Deeper Dive Into SP/A Theory

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Appendix A: a Deeper Dive Into SP/A Theory Appendix A: A Deeper Dive into SP/A Theory For the benefit of readers interested in a deeper discussion of Lola Lopes’s SP/A framework, this Appendix discusses several topics. The most important of these is a discussion about the mathematical structure of Lopes’s formal model. The model is contrasted with a heuristic approach known as “the priority heuristic,” which is simpler in its informational requirements and implementation. The priority heuristic conforms to what is known as a “fast and frugal” heuristic. The Appendix also extends the discussion to include two different types of risky alternatives. One risk involves a uniform probability distribution, and the other involves a bimodal probability distribution. Uniform probabil- ities are typically associated with positions in securities that are very poorly understood, but have covered call features. The second risk involves bimo- dality, meaning that the probability density function has a “barbell” prop- erty, with the most likely outcomes being extreme, either very unfavorable or very favorable. Positions that are bets on discrete events tend to feature this payoff pattern: either the bet pays off well because an idea “works” or the bet pays little because the idea “does not work.” The next section describes the full range of alternatives that Lopes used in her experiments. Six Risky Alternatives Lopes used six different probability density functions in her experiments, the shapes of which are illustrated in Figure A.1 . In order they are the Risk Floor (RF), Peaked (PK), Short Shot (SS), Uniform (U), Bimodal (BM), and Long S h o t ( L S ) . The shapes depicted in Figure A.1 are probability density functions (pdfs). The horizontal axis displays possible payoffs, and the height of a bar indicates the probability with which the particular payoff will occur. For example, Figure A.1 indicates that for RF, the probability associated with a payoff of $770 is 31%. As you can easily see, RF and PK in Figure A.1 are simply scaled-down versions of RF and PK that are displayed in Figure 2.1 . Risk Floor 35 30 25 20 15 10 Probability (%) Probability 5 0 $770 $935 $1,089 $1,254 $1,419 $1,573 $1,727 $1,892 $2,057 $2,200 Payoff 35 Peaked 30 25 20 15 Probability (%) Probability 10 5 0 $0 $143 $286 $440 $583 $726 $880 $1,023 $1,166 $1,309 $1,452 $1,606 $1,749 $1,892 $2,046 $2,200 Payoff 35 Short Shot 30 25 20 15 Probability (%) Probability 10 5 0 $0 $143 $308 $473 $627 $781 $946 $1,111 $1,265 $1,430 Payoff Probability (%) Probability (%) 10 15 20 25 30 35 10 15 20 25 30 35 0 5 0 5 $0 $0 $110 $143 $231 $286 $352 $440 $462 $583 $572 $693 $726 $814 $880 $924 Uniform Bimodal Payoff Payoff $1,023 $1,034 $1,166 $1,155 $1,276 $1,309 $1,386 $1,452 $1,496 $1,606 $1,617 $1,749 $1,738 $1,848 $1,892 $1,958 $2,046 $2,079 $2,200 $2,200 350 Appendix A Long Shot 35 30 25 20 15 Probability (%) Probability 10 5 0 $0 $539 $1,078 $1,606 $2,145 $2,684 $3,212 $3,751 $4,290 $4,829 Payoff Figure A.1 This figure displays the shapes of the six probability density functions (pdfs) Lopes used in her experiments. The pdfs are respectively labelled Risk Floor (RF), Peaked (PK), Short Shot (SS), Uniform (U), Bimodal (BM), and Long Shot (LS). Table A.1 Payoff means and standard deviations for the six Lopes risky alternatives Mean Stdev Risk Floor $1,097 $354 Short Shot $1,103 $354 Peaked $1,093 $442 Uniform $1,097 $666 Bimodal $1,095 $851 Long Shot $1,100 $1,185 Table A.1 provides payoff means and standard deviations for the six alterna- tives, with lower standard deviations closer to the top. In her experiments, Lopes asked her subjects to rank order all possible pairs of alternatives, through the use of pairwise comparisons. In doing so, she found that RF most frequently emerged as the most-favored choice, and LS most frequently emerged as the least-favored top choice. She reports that the order of attractiveness is RF, PK, SS, U, BM, and LS. For complete- ness, I should mention that Lopes used lower stakes than those displayed in Figure A.1 , with displayed payoffs ranging between $0 and $348. In my own research, I have used experiments based on the six Lopes alter- natives displayed in Figure A.1 . My sample size was 315, and included risk managers, portfolio managers, business executives, PhD students in finance and economics, and undergraduate finance majors. Beth and Larry from Appendix A 351 Chapter 2 , as real people now, participated as subjects, and their responses in the short story are the identical responses they provided as subjects in the experiment. Basics of How Lopes’s Model Works This section describes the general features of how Lopes structured her model, along with a description of how the model can be implemented.1 The discus- sion below explains how SP/A theory would capture how someone like Larry might rank PK over RF. The explanation proceeds in steps, where we imagine Larry’s thought process to be as follows. In the first step, Larry examines the expected payoffs for both alternatives, which happen to be $1.1 billion. In the second step, Larry considers the range of outcomes from both alter- natives and asks himself how much he fears the occurrence of outcomes that he regards as unfavorable but possible. He then expresses his fear by being pessimistic and allowing his mind to increase the probabilities of unfavorable outcomes relative to favorable outcomes. In doing so, he scales up the prob- abilities of very unfavorable events and scales down the probabilities of very favorable events. The result of his emotional rescaling will be to reduce the expected payoffs from the two alternatives. The more he fears an outcome, the higher he rescales the probability of that outcome. The more fear he feels, the lower the rescaled-expected payoff. Notably, this property underlies the risk neutral probability approach used to price options and other securities. In the third step, Larry considers the range of outcomes from both alterna- tives and asks himself how much hope he attaches to outcomes that he regards as favorable (and, of course, possible). He then expresses his hope by being optimistic and allowing his mind to increase the probabilities of favorable out- comes, arrived at in the second step above, relative to unfavorable outcomes. He makes the adjustment by multiplying the probabilities of extremely favor- able events by a factor greater than unity, while simultaneously multiplying some other probabilities by a factor less than unity. In doing so, he increases the expected payoffs for the alternatives, from his step two computations. The more he hopes for an outcome, the more his mind increases the probability attached to that outcome. The more hope he feels, the higher the recomputed expected payoffs. In this regard, Lopes calls the net recomputed expected pay- off “ SP ” because it reflects the relative needs for security and potential. In the fourth step, Larry considers whether there is some specific outcome that he would label as “borderline successful,” so that he regards all outcomes at least as good as borderline successful to qualify as “success.” He then ascer- tains the success probabilities for both risky alternatives based on the original values before adjustment in steps 1 and 2. Lopes uses the symbol “A ” (for aspiration) to denote success probability. 352 Appendix A In the fifth step, Larry boils down his choice between the two alterna- tives by comparing their SP- recomputed expected payoffs and the A suc- cess probabilities. If one alternative has both a higher SP and a higher success probability A , then Larry’s choice is easy because he values higher values of both. However, there will be some situations in which Larry will have to consider trading a lower success probability A for a higher SP - recomputed expected payoff. In such situations, Larry’s choice will depend on how strong his need for success is relative to his needs for security and potential. As we saw in the previous section, Larry attaches a very high weight to chance of success, which would incline him to rank PK over RF, unless RF offered so much more SP than PK that it would tip the balance. Beth, who is less ambitious than Larry, places more weight on SP relative to A . Therefore, even if she were similar to Larry in respect to fear and hope, she would be more inclined to rank RF over PK. SP/A Theory: Mathematical Structure This section presents the formal structure of SP/A theory. Consider one of the risky alternatives, such as PK, whose pdf is portrayed in Figure A.1 . Let x denote a random payoff and p(x) the probability with which this payoff will occur under PK. PK is a discrete random variable, and the set of payoffs ( x ) having positive probability ( p(x) ) is finite. The decumulative probability D(y) is the probability under PK that the payoff will exceed y . That is, D(y) = Σp(x), where the summation is over x ≥ y . In order to recover p from D , rank Σ order the outcomes x as x 1 , x 2, . , x n , where x 1 <x n , p(x i )>0 and p(x i ) = 1, where the summation is from i = 1 to n .
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