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 **W _{1}, W_{2}, W_{3}, … W_{n-1}, W_{n}**. Lopes denotes a

**decumulative distribution function**as the likelihood that the wealth

**[1]**D(x)**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 p _{i}** of achieving a particular level of wealth

**W**:

_{i}**E(W) = ∑p _{i}W_{i}**

Lopes notes that expected wealth,** E(W) = ∑p _{i}W_{i},** can be expressed as

**the sum of the decumulative probabilities ∑D**, where the summation is from

_{i}(W_{i }– W_{i -1})**i**= 1 to

**n**and

**W**

_{0}is zero. In this expression for

**E(W)**,the individual receives

**W**with certainty (note that

_{1}**decumulative probability**

**D**), receives the increment

_{1}= 1**W**(that is, an amount over

_{2}– W_{1}**W**) with probability

_{1}**D**, receives the further increment

_{2}**W**with probability D

_{3}– W_{2}_{3}, and so on. Lopes contends that

*fear*operates through an

*overweighting*

*of the probabilities attached to the worst outcomes (*. The way that works is that fearful investors set the

**W**)_{1}**p**too high, and

_{1}**p**too low. The reverse is true for hopeful investors.

_{n}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 **h _{s} **(for fearful investors worried about

**ecurity) and**

*s***h**(for hopeful investors looking for maximum

_{p}**otential), reflecting the relative strength of each.**

*p*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.

Copyright 2011 Eric Bank, Freelance Writer