“Fear of harm ought to be proportional not merely to the gravity of harm, but also to the probability of the event”. Ars Cogitandi, 1668. (Hacking, Ian 1975).

The common methodology used to determine an investor’s utility (Von Neumann – Morgenstern or variations) begins by assuming a series of universal behaviors (axioms) (Utility Review). It then adds a sufficient number of constraints to uniquely specify an individual’s investment choice. The typical result is a single factor non-linear function that defines decision-making behavior under uncertainty.

There is a major problem with this approach. One must build a utility function for the investor. This is extremely difficult to do, since it involves defining a host of trade-offs.

What are the investor’s goals and constraints?

In utility theory, goals are implicitly embedded into the utility function. But this makes life difficult since most people have a much better understanding of their goals and constraints than they do of a series of seemingly disconnected trade-offs. The result is the utility function is frequently structured as a couple of vague assumptions that investors prefer more to less and that they prefer less uncertainty to more.

When converted into a practical decision making model we end up with a generic equation (typically quadratic which is equivalent to a mean-variance utility function). This means that the only difference between one investor and the next is their sensitivity to the variance of annual returns.

There are a lot of problems with this approach. It is too simplistic, too much information is lost and it becomes disconnected from the investor’s intuitive understanding of their circumstances.


Our Approach

We will approach the problem from the other direction. Instead of defining the basic principles of decision-making and explicitly constructing a utility function, we will start by defining an investor’s perception of risk and then determine their optimal portfolio. This approach does not involve explicitly determining a utility function, and only makes a limited number of hypotheses on investor behavior.

We will make the following assumptions:

1. People seek to maximize their utility.
2. Investor utility maximization involves maximizing expected return while minimizing risk.
3. Risk is the possibility and consequences of failing to achieve one's goals.

Our first assumption is consistent with basic economic theory.

Our second assumption substantially narrows the field. We are no longer dealing with trade-offs between apples and oranges, or cars and vacations. Rather we are dealing solely with money. This assumption contains what is called first order stochastic dominance, which simply means that investors prefer more to less. By seeking to maximize expected return, we are assuming that investors prefer more to less and that they can obtain more by engaging in behaviors that maximize expected return.

The other element of our second assumption, the minimization of risk is connected to our third assumption. In most models, risk aversion is usually treated as second order stochastic dominance. This means defining risk aversion as preferring a higher probability event to a lower probability event. In the case of a normal distribution this would involve preferring the investment with the lower variance (if their expected return were the same).

Instead, we define risk as the possibility and consequences of failing to achieve one's goals. We are now capturing more information in our definition of risk then simple uncertainty. Mathematically, we view volatility as a subset of uncertainty and uncertainty as a subset of risk.

Risk Uncertainty Volatility

By including goals in our definition of risk, we are able to directly connect the fundamental objectives of each investor with the key parameters of our model. This becomes a very powerful method in not just building an optimal portfolio, but also understanding what perceptions and misperceptions are driving investor behavior.