Last week we launched an article series aimed at discovering the types of behavioural biases that plague investors and the ways in which they can recognise and seek to control them. Today, we take a detailed look at a now famous descriptive model for how people make choices between risky and riskless alternatives. The theory was first proposed in 1979 by Daniel Kahneman and Amos Tversky in a paper entitled 'Prospect Theory'. A basic understanding of this model should provide a rational framework for how to decide between investing with active or passive vehicles.

In our previous article, we showed how expected utility theory, one of the underlying assumptions of modern economic theory, cannot explain all choices made by individuals. Expected utility theory asserts that individual actors (you and me) are believed to act rationally when presented with uncertain or risky outcomes. However, in dozens of studies since 1979, it has been shown that individuals exhibit a tendency to place a premium on certainty. Most individuals would be willing to pay some amount to gain the assurance of a payoff as opposed to taking the risk associated with potentially receiving a higher payoff. Willingness to pay for certainty will vary from individual to individual, but the mere fact that a willingness to pay for greater certainty exists suggests that expected utility theory cannot fully describe our decision making processes.

As an alternative, Prospect Theory states that instead of weighting outcomes by their probabilities (à la expected utility theory), outcomes should be weighed by a "decision weight." In this context, "decision weights" are essentially individuals' preferences about the observed probabilities of certain outcomes. For those of you who are maths-inclined people, the formal statement of prospect theory is stated as the expected value of a riskless outcome and the expected gain from a risky outcome weighed by a decision weight:

V(x,p,y,q) = v(y) + p(p)[v(x)-v(y)] ; where y = riskless outcome, x = risky outcome, p = decision weight, p = probability of x

Listed above, it is somewhat easier to visualise the relationship between decision weights, outcomes, and probabilities. Decision weights are functions of probabilities and not necessarily equal to the probabilities themselves. Put another way: decision weights are how people feel about their chances. Like most functions, decision weights have several properties. They are increasing functions of probabilities, do not have to equal the probability, are usually smaller than the probability, and are typically not linear. The first three properties do not require an overly lengthy explanation. As your chances of realising a gain increase, you tend to feel better about your chances (property #1). Decision weights do not typically equal probabilities as evidenced by our example in the previous article (property #2). Thirdly, decision weights are typically smaller than the value of the probability due to risk aversion (property #3). As a sidenote, Kahneman and Tversky have found evidence that property #3 does not hold for very small probabilities as individuals become risk seeking. Examples of this behaviour include purchasing lottery tickets and paying insurance premiums. In these cases, decision weights tend to be larger than the associated probabilities.

The fourth property of decision weights refers to their tendency to be nonlinear. Nonlinearity means that for a 1% increase in the likelihood of an event (p) one may or may not increase our decision weight (p) by 1%. Here, a concrete example is useful. Consider a game of Russian roulette. How much would you be willing to pay to reduce the number of bullets in the chamber of a loaded gun from 4/6 to 3/6 versus 1/6 to zero? The majority of us would prefer and pay more for the latter scenario despite the fact that the decrease in probabilities is equal. Expanding on this insight, Kahneman and Tversky showed that individuals prefer (by a very wide margin) a decrease in probability of an adverse scenario from 1% to 0% than from 2.1% to 1%. Think about that for one second. Most individuals would prefer a 1% reduction of risk over a 1.1% reduction depending on where the initial reference point lies within the probability spectrum. Again, we see evidence that individuals do not always make rational decisions and prefer to pay for certainty.

We can apply prospect theory to excahnge-traded products (ETPs) pretty simply. Here the riskless outcome labelled "y" is the market return or what we would achieve by investing in an index-tracking ETP. It is important to note that in this context I say "riskless" to mean that one can always achieve the market return (less fees) via an index-linked vehicle, not that the market return itself is without risk. Obviously, index-linked investments have elements of market risk and can earn negative or positive returns over a given period. However, as an investor looks at their choices to achieve a certain exposure, they can choose the “riskless” (again, meant in the context of relative performance) outcome of earning the market return or can take a “risk” and try to beat the market in question using an active strategy. Taking the “risk” of attempting to beat the market is defined in the latter part of the value equation as the expected gain (or loss) one would achieve by choosing an active manager. Depending on the likelihood of that manager's success in beating the market (p) and the magnitude of his/her outperformance (x), one would assign a decision weight of p(p) to assess the expected value of trying to beat the market. Below, I have rewritten the value function with these investing terms substituted for variables:

V(x,p,y,q) = market return + p(p)[expected outperformance of active strategy] ; where p = decision weight, p = probability of outperformance

Since investing involves two actionable steps: deciding to buy and deciding whether to continue to hold, the value equation will need to be consistently referred to when the active manager strategy "x" is chosen in preference to the market return "y". However, this is not the case when the market return "y" is chosen in preference to the active manager strategy "x". For instance, a poorly performing ETP in terms of absolute performance does not mean it is no longer the riskless outcome. For an investment in an ETP, the market return remains the market return and no re-evaluation is necessary. On the other hand, the risky outcome "x" originally chosen for its apparent ability to beat the market (y) will need to be re-evaluated consistently to confirm our expectation that it will continue to outperform in the future. If one chooses to invest in an active fund in hopes that it will beat the market, they then need to constantly reassess this thesis anew. For purposes of this article series, I label this process of continually re-evaluating your investment choices as critical reassessment.

Now that we've got a little bit of Prospect Theory under our belt, let's look at some recent research that offers some insight into decision weights. Morningstar's Vice President of Research, John Rekenthaler, has __shown data__ indicating that two-thirds of active managers tend to underperform their benchmarks for longer periods (5-15 years). For the longer term investor, active managers have proven to be successful in delivering alpha only one-third of the time on average across all categories. Now, those chances of success can change across categories and vary in the short-run. But Prospect Theory is useful as a framework for analysing these sorts of decisions. It suggests that when you know the odds and the expected magnitude of success, you should apply a decision weight to your assessment. In this context, your choice of decision weight will be similar to your appetite for risk. Risk adverse individuals will tend to favour decision weights lower than the probabilities of success. In the scenario where you have a low decision weight, Prospect Theory would suggest that the expected relative outperformance must be substantial (or your level of conviction in a manager's abilities quite high) in order to prefer an active strategy over a passive one.

As New Year's resolutions go, thinking through these concepts might feel akin to a 15km jaunt on a treadmill. My hope is that readers can see how, as investors, we should be framing our investing decisions relative to a reference point not described by profit or loss but rather by a “riskless” alternative (e.g. the benchmark index). Depending on our confidence in the size and likelihood of deviation from these “riskless” alternatives, we can better judge whether or not an active strategy should be pursued. In some cases, the answer may be an overwhelming 'yes' and in others, a definite 'no'. The choice really depends on investors' individual circumstances, preferences, and judgment.

Next week, we look at the loss aversion bias and how it may be investors' worst enemy.

*Lee Davidson is an ETF analyst with Morningstar.*