In my last post I made the argument that time does not necessarily reduce the risk of an investment since the standard deviation of total return actually increases with time. I discussed how you have to take into account the direction and magnitude of losses and gains, while considering your own wealth. This leads us to Utility Theory. Today I will briefly review Utility Theory and how it relates to time diversification.

Review of Utility Theory

Utility Theory starts with the idea that most investors are risk adverse. This does not mean that most investors don’t want to take risk. It means that for a given level of risk (in this case risk is defined as the probability of a good or bad outcome) an investment must have a sufficient level of expected return.

To measure an investor’s risk aversion you use utility functions. Utility functions measure an investor’s relative preference for different levels of total wealth. To be a valid utility function it must have two characteristics. First, more wealth is always preferable to less wealth and the investor is never satisfied with his current wealth (the first derivative is greater than zero, i.e. upward sloping). Second, an investor’s marginal utility of wealth decreases as wealth increases (the second derivative is less than zero, i.e. concave). Think of someone with $1,000,000 receiving an extra dollar verses someone with $5 receiving an extra dollar. Obviously the person with $5 gains more use (increase in utility) from the extra dollar than the person with $1,000,000.

Below is a simple utility function U_w=√w. It is a valid utility function since 〖U’〗_w=〖0.5w〗^(-0.5)>0 and 〖U”〗_w=〖-0.25w〗^(-1.5 )<0

Utility Function

Once we have chosen a utility function for an investor we can decide how they should act when given an investment opportunity based on their current wealth and the nature of the investment.

How does this relate to time diversification?

Paul Samuelson is one of the greatest contributors to utility theory and won the Nobel Prize in 1970. In a 1963 paper “Risk and Uncertainty: A Fallacy of Large Numbers,” he recalls an encounter with a colleague who refused to take a bet on a single coin flip with favorable odds.  He would, however, take the bet if it could be a series of 100 coin flips at the same odds. At first this made sense to Samuelson, but upon further thought he felt this thinking was irrational. If someone finds the risk of one bet unacceptable why would they find a series of the exact same bet acceptable? In his paper he proves that the decrease in the probably of a loss is exactly offset by the increase in the potential magnitude of loss. In other words, the impact of flipping a coin 99 more times is neutral to overall expected return.

Below is a chart that shows this idea using utility curves. It shows that as utility increases at a slower rate (concave) the risk of loss decreases over time. What this shows is that the benefit from the decreasing risk of loss is exactly offset by the declining benefit of having more wealth. Mathematically speaking, the slopes of utility and risk of loss sum to zero in each time period.

Utility, Risk and Wealth

He concludes by stating that an investor should choose an asset allocation based upon the amount of risk they are willing to take a much shorter time horizon (i.e. one year). Adding a longer time horizon to the thought process of asset allocation adds no real value.

Another way to use the concept of utility to demonstrate that extending the time horizon does not elevate risk is by looking at some hypothetical investors and see how they would invest over some fixed time period (i.e. 5 years).

First, let’s say we have an investor who prefers to take on more risk when he is poor because he feels he will need much more money in the future. When he is rich he invests much more conservatively because he wants to protect what he has. This person is said to have increasing relative risk aversion.

Another investor decides that she will be more conservative when she is poor, since she doesn’t want to lose what little she has. When she is rich she is more aggressive, since she has enough to live on and can take risk with the excess. This person is said to have decreasing relative risk aversion.

These are both rational positions. Both investors are risk adverse; they only differ in their patterns of risk and wealth.

A third example is someone in the middle who invests the same regardless of wealth. This is referred to as someone with constant relative risk aversion or iso-elastic utility.

Now let’s examine this investor over variable investment horizons. If time makes an investment less risky we would expect ALL investors, even our investor with iso-elastic utility, to increase their risk with time horizon. This investor would prefer, for example, a 50/50 stock/bond portfolio if he was investing over a 5-year period regardless of wealth. If time elevates risk we would expect him to prefer something like an 80/20 portfolio over 20 years regardless of wealth, since a riskier portfolio would become less risky over time. An interesting thing happens when we do the math using calculus and probability theory. We find in addition to wealth, relative attitudes towards risk are also independent of time horizon! In this particular case our investor’s optimal portfolio would be say 50/50 stocks/bonds over 1 year, 5 years or even 20 years regardless of wealth.

While this result is for a specific case, it definitely shoots down the idea that time always reduces the risk of an investment.

In my next post I will look at option pricing theory and its correlation to time diversification.