Behavioral Portfolio Theory 2 – SP/A Theory

Last time we began our discussion of Behavioral Portfolio Theory with a look at Safety-First Portfolio Theory (SFPT), which basically posits investor motivation to be to avoid ruin.  An extension to SFPT was introduced in 1987 by Lopes, named SP/A Theory. The letters in its title can be summarized:

S = security, a general concern about avoiding low levels of wealth

P = potential, reflects the general desire to maximize wealth

A = aspiration, the desire to reach a specific goal, such as achieving no less than the subsistence level s.

SFPT is a psychological theory of choice under uncertainty. In Lopes’ framework, risk-taking is balanced between fear and hope. Lopes posits that fear is such a strong factor because fearful people overweight the probability of the worst outcomes, underweight those for the best outcomes. This leads individuals to understate the probability of achieving the highest level of expected wealth E(W).  In other words, fearful individuals are pessimistic.

Hope has the inverse effect on individuals – optimism causes hopeful investors to overstate the probability of achieving the highest level of expected wealth.

For those of you who like symbols, wealth W is actually a series of n different levels of wealth, ordered from lowest to highest, designated as W1, W2, W3, … Wn-1, Wn. Lopes denotes a decumulative distribution function[1] D(x) as the likelihood that the wealth W of a portfolio will be greater than the investor’s subsistence level s:

D(x) = Prob {W >= s)

where 0 < x <= n different levels of wealth.

Expected wealth E(W) is the weighted sum of each level of wealth, the weight being the probability pi of achieving a particular level of wealth Wi:

E(W) = ∑piWi

Lopes notes that expected wealth, E(W) = ∑piWi, can be expressed as the sum of the decumulative probabilities ∑Di (Wi – Wi -1), where the summation is from i = 1 to n and W0 is zero. In this expression for E(W),the individual receives W1 with certainty (note that decumulative probability D1 = 1), receives the increment W2 – W1 (that is, an amount over W1) with probability D2, receives the further increment W3 – W2 with probability D3, and so on. Lopes contends that fear operates through an overweighting of the probabilities attached to the worst outcomes (W1). The way that works is that fearful investors set the p1 too high, and pn too low. The reverse is true for hopeful investors.

Lopes concludes that the emotions of fear and hope reside within all individuals, and that each emotion serves to modify the decumulative weighting function D. She suggests that the final shape of the decumulative transformation function h is a convex combination (shaped like a smile) of hs (for fearful investors worried about security) and hp (for hopeful investors looking for maximum potential), reflecting the relative strength of each.

The qualitative lesson from SP/A theory is that you should establish a portfolio in which you have reigned in your emotions of fear and hope, so that neither dominates. Unfortunately, it doesn’t tell you how to accomplish this. Still, not a bad lesson for a hedge fund trader.

[1] The decumulative distribution function is just (1 – cumulative distribution function).  Basically it measures the curve to the right (instead of the left) of a given point.