Utility refers to the satisfaction, happiness or well-being that an individual derives from the consumption of a good or service. It is a nebulous term that constantly changes. Not only will one person’s utility will be quite different from another’s, but their utility will change as their circumstances change. For example, the utility of chocolate versus vanilla ice cream will change with different people, just as the utility of ice cream on a hot day will likely be different from ice cream on a cold day for the same individual.

In 1738, Daniel Bernoulli suggested his expected utility hypothesis which arose from his solution to the St. Petersburg Paradox (St. Petersburg Paradox). The Paradox challenges the idea that people value random ventures according to their expected return. Bernoulli suggested that instead of valuing a risky venture based on its expected return, people value it based on the venture’s expected utility. Therefore, we do not seek to maximize return but rather we seek to maximize utility.

Bernoulli's expected utility hypothesis was ignored for almost two hundred years. With only a handful of exceptions it was never really picked up until John von Neumann and Oskar Morgenstern (1944) published their seminal work on the Theory of Games and Economic Behavior. While Von Neumann – Morgenstern’s (V-M) expected utility hypothesis is essentially identical to Bernoulli’s, it is constructed in a mathematically rigorous manner. The major impact of their effort was their use of axioms in building a hypothesis of agents' preferences over different ventures with random prospects. These preferences were constructed over what can be called lotteries. (Game Theory) The result was that an expected utility theorem could be developed from four basic axioms concerning investor behavior. Those axioms are Comparability, Transitivity, Independence and Certainty Equivalence.

There are a number of advantages to deriving utility from a series of axioms on investor behavior. Intrinsic to the methodology is a mathematically rigorous approach. Experiments can then be constructed to test each axiom and insure that it is correct.

A major problem, however, is whether the axioms describe reality. There is no room for minor exceptions. This rigidity can be a problem in the hard sciences, let alone when discussing human behavior. When it comes to decision making, which is essentially behavioral based, the narrow focus of a handful of axioms may break down.

In fact, researchers have examined the various axioms under different circumstances and have encountered a number of contradictions, the most famous being the Allais Paradox. (Allais Paradox). These studies have reinforced the need to rethink much of the theory. The result has been a number of attempts to alter the theory’s axioms, such as weighted expected utility, rank-dependent expected, non-linear expected utility and regret theory. (Alternative Expected Utility).

The V-M approach is fundamentally an objective approach in determining decision making behavior. It involves deriving an expected utility by imposing objective probabilities. Another method is the “subjectivist” approach by Leonard Savage and expanded by F.J. Anscombe and R.J. Aumann. This involves inferring the investor’s utility from the bets they make. In some regards, the Savage-Anscome-Aumann (SAA) Subjective approach is more "general" than the V-M concept. (Subjective Expected Utility)

Another "subjectivist" approach was initiated with the “state-preference” approach to uncertainty of Kenneth J. Arrow and Gerard Debreu. Although not necessarily "opposed" to the expected utility hypothesis, the state-preference approach does not involve the assignment of mathematical probabilities, whether objective or subjective. (State Preference Approach). While the (SAA) and “state-preference” methods have the advantages of being more general, there has been very little practical application in for investment decision making. Because of this, we will focus on the V-M approach, which most practitioners (who use quantitative techniques) use.

If the investor acts such that they obey the general V-M postulates (the constraints) on their behavior, then their investment choices will be the same as those predicted under the expected utility theorem. However, we must still determine which specific utility function the investor implicitly uses in arriving at their investment decisions. We do so by having the investor make choices between a series of simple investments.

A major difficulty is that we must ask the investor a large number of questions (many of which appear unrelated from the investor’s perspective) each time we try to construct a function. Firms that have tried this approach have been unsuccessful. Many investors do not obey all the postulates (constraints) set down in the theorem, or change their reasoning when moving from the simple to the more complex scenarios.

Since we can’t ask thousands of questions (and thereby develop a proper lottery set to determine the preference function) each time we try to construct a function, we explicitly build a utility that we hope properly reflects investor behavior.



Explicit Utility Function

We assume that whatever function we create will meet the four basic V-M constraints. Mathematically, we then have the following equation that describes our expected utility:

where U is utility, W is terminal wealth and P(W) is the probability of the outcome W. Since we are utility maximizers, our optimal portfolio will that which maximizes E(U).

In order for utility analysis to be at all practical we must simplify the process. We can’t ask thousands of questions (and thereby develop a proper lottery set to determine the preference function) each time we try to determine an individual’s utility. Instead we construct a utility function explicitly that approximates most people’s behavior.

Our expected utility equation is still very general, since we have not defined the function U(W). Therefore, we will try to narrow down the number of possible functions by putting forward a few postulates (constraints).

Since we are dealing with money (instead of some service like haircuts or a commodity like ice cream) we can add the constraint that people prefer more to less. For the utility function U(W), where W is total wealth this becomes:
Constraint: People prefer more money to less money, defined as

This simply means that all else being equal, we always prefer an extra dollar. What this entails is that if one is faced with two identical investments with the exception that one has a higher return than the other, we will select the investment with the higher return.

The second constraint is that we will make an assumption about an investor’s attitude toward risk. (This should be towards uncertainty as opposed to “risk” since we have not properly defined risk. Unfortunately, the literature is full of this confusion, and we will address this issue elsewhere.) There are three possibilities: the investor is risk averse, risk neutral or is risk seeking. An investor is risk averse when they reject a fair gamble and is risk seeking whenthey accept a fair gamble. A simple fair gamble would be to flip a coin where with heads one receives a dollar and with tails one must pay a dollar.

For the utility function U(W), risk aversion occurs when the second derivative is less than zero, and risk seeking when it is greater than zero.

Attitude	Definition	Property
Risk Aversion	Reject fair gamble
Risk Neutral	Indifferent to fair gamble
Risk Seeking	Accept fair gamble

We then set a second constraint.

Constraint: People are “Risk Averse”, defined as

This constraint is not as innocuous as it first appears. It rules out gambling at a casino, playing the lotto and arguably, a large number of venture capital and entrepreneurial activities.

We could either continue to add constraints or move straight to a function that implicitly contains additional constraints. Before moving to a function however, we will briefly define two terms that describe changes in investor behavior with changes in wealth. (Absolute & Relative Risk Aversion) The first of these is a measure of their attitude to Absolute Risk Aversion (ARA). As an investor’s wealth increases, if they hold fewer dollars in a risky asset then they exhibit increasing ARA, and if they choose to hold more dollars in a risky asset then they exhibit decreasing ARA. Mathematically, the ARA is described as:

and their attitude is the first derivative.

The second of these measures is called Relative Risk Aversion (RRA). As an investor’s wealth increases, if they hold a smaller percentage of assets in a risky asset then they exhibit increasing RRA, while if they hold a larger percentage of assets in a risky asset then they exhibit decreasing RRA. Mathematically, RRA is described as:

and their attitude is the first derivative.

(For a more comprehensive discussion of risk aversion see: Arrow-Pratt)

We now select a function that not only implicitly defines these two constraints, but is also sufficient to determine an exact solution.


Quadratic Utility Function

The most common utility function used is the quadratic. It is defined as:

where W is the investor’s level of wealth.


The derivatives of this function are:

The Absolute Risk Aversion then becomes:

and the Relative Risk Aversion is:

From our first constraint, we assumed that the investor prefers more to less and therefore the first derivative must be positive. From our second constraint, we assumed the investor was risk averse and therefore the second derivative must be negative and b positive (from the first constraint). Since b is positive and assuming positive wealth, the requirement that the first derivative is positive means that b <1/(2W). Therefore, inserting into the Absolute and Relative Risk Aversions and taking their first derivative, we find that:

and

This means that our investor with a quadratic utility function has an increasing Absolute Risk aversion and an increasing Relative Risk Aversion.

The assumption of a quadratic utility function also leads to mean variance analysis being the optimal solution. (Quadratic leads to MV). The use of mean variance optimizations is extremely common and is a very simple and straightforward method. (Mean Variance)

Therefore, if we assume the investor follows the following constriants:

• 4 axioms
• Prefers more to less
• Risk averse
• Quadratic Utility

Then we have the investor exhibits the following behavior:

• Increasing Absolute Risk Aversion
• Increasing Relative Risk Aversion
• Mean-Variance is optimal

These constraints are quite severe. They rule out a significantly large body of investor behavior. Some of these behaviors are:

• Regret is eliminated since it violates V-M
• Any form of risk taking (gambling, lotto etc.) is eliminated since investors are assumed to be risk averse.
• As the investor’s wealth increases it is assumed that they hold a greater proportion of their assets in risky assets. This contradicts a lot of investor behavior.

Also, certain investments are implicitly eliminated by the mean-variance approach. Everything can now be described by the mean and variance, since the solution is a function of the two variables. The problem is that this vastly simplifies the probability distributions available. The mean is the first moment and the variance the second (Moments). The third moment is skewness and the fourth kurtosis. The normal distribution can be described with just the mean and variance (the first two moments). Unfortunately, stock price movements have been shown not to follow the normal/lognormal distribution. Rather the “tails” are fatter ("leptokurtic"). Secondly, investments such as insurance and derivatives or risky bets like venture capital etc. will demonstrate a skewness in their return expectations. By assuming mean-variance we are eliminating all of these investment behaviors. Since the optimization will not recognize the skewness and kurtosis these investments will be improperly recognized.

So, the quadratic (or mean-variant) utility function not only places unrealistic constraints on investor behavior, but it also implicitly eliminates (or improperly recognizes) many investment options available to the investor.

Mean-variance analysis is not the only method of portfolio selection. Other portfolio selection models relax the quadratic utility and return normality assumptions. Here, we consider three alternative portfolio selection criteria: stochastic dominance, geometric mean and safety first. In general, these criteria lead to optimal portfolios that are different from each other, and different from the optimal portfolio of mean variance analysis. However, imposing certain additional assumptions can reconcile the different models.

Stochastic dominance

An alternative to mean-variance analysis is called Stochastic Dominance. In its most general form, it makes no assumptions about the form of the probability distribution of returns. It is also not necessary to assume a specific form of a utility function. The approach is very similar to our explanation for deriving the quadratic utility function.

Stochastic dominance is built by defining efficient sets under various assumptions or constraints. There are three progressively stronger assumptions about investor behavior; first order, second order and third order. Each involves increasing restrictions on the form of investors’ utility functions.

1st order – investors prefer more to less

2nd order – additionally, investors are risk (uncertainty) averse

3rd order – additionally, investors have decreasing absolute risk aversion

Notice how similar this is to the approach we just described for quadratic utility. Mean variance is simply 2nd order stochastic dominance with a normal distribution (or quadratic utility function).

Associated with each level of stochastic dominance, there is a theorem that allows us to eliminate sub-optimal portfolios. The main advantage to stochastic dominance is that we can apply it to non-normal distributions.

The major disadvantage is that the elimination of sub-optimal portfolios may not lead to a unique optimal portfolio. If we cannot find a unique portfolio we are left with requiring pairwise comparisons for the remaining alternatives. We are then essentially back to the decision method used by VM.

Geometric Mean
Another alternative to mean-variance is to select the portfolio that has the highest expected geometric mean return. This, in effect, maximizes the expected value of terminal wealth.

The geometric mean is defined as:

where R_ij is the i^th possible return on the j^th portfolio and each outcome is equally likely.

If the likelihood of each outcome is different and P_ij is the probability of the i^th outcome for the j^th portfolio, then

and can be written as,

The resulting portfolio is usually very well diversified and extreme values have a tendency to be eliminated. If a strategy has a probability of bankruptcy then the whole product will become zero.

The geometric mean is a measure of central tendency, just like a median. It is different from the traditional mean (which we sometimes call the arithmetic mean) because it uses multiplication rather than addition to summarize data values. Geometric means are often useful summaries for highly skewed data.

The geometric mean for any time period is less than or equal to the arithmetic mean. The two means are equal only for a return series that is constant (i.e., the same return in every period). For a non-constant series, the difference between the two is positively related to the variability or standard deviation of the returns.

The main problem with this method is that it does not differentiate between investors and thereby does not explicitly refer to risk. If our expected return forecasts were the same, then every investor, irrespective of their circumstances, would hold the same portfolio.

Arguably, this method could be used by a mutual fund that has a broadly diversified group of investors. It is very quick and easy to use.

Maximizing the geometric mean is equivalent to maximizing the expected value of a log utility function.

Safety First and Value at Risk

Investors may be unwilling to perform calculations that are necessary for selecting the mean-variance optimal portfolio. One argument is that they use much simpler decision rules that concentrate on the worst possible outcomes of an investment. These decision rules are called “safety first” portfolio selection criteria.

There are three safety first criteria: Roy’s criterion, Kataoka’s criterion and Telser’s criterion.

Roy’s criterion

Choose the portfolio that minimizes the probability of the portfolio return being lower than some minimum acceptable return. If R_P is the return on the portfolio and R_L is the level below which the investor does not wish returns to fall, Roy’s criterion can be mathematically described as:

Minimize Prob(R_P < R_L)

If returns are normally distributed, then the optimum portfolio would be the one where R_L is the maximum number of standard deviations away from the mean. This normality assumption imposes a strong constraint and Roy’s criteria becomes:

Reverse the sign and assume the maximum value is K. Therefore:

Which becomes:

and is simply the equation of a straight line with slope K in mean variance space. The optimal portfolio then becomes the intercept between this line and the efficient set curve. Graphically, this can be represented as:

Any line that does not intersect the curve (lies above the curve) involves a portfolio that does not exist.

The normality assumption can be relaxed slightly. If we use Tchebyshev’s Inequality we can see that this technique still holds. However, the more the return distribution deviates from one which can be completely described by the first two moments, the more Tchebyshev’s Inequality is likely to be too conservative and therefore selects either an inefficient portfolio, or determines that no optimal portfolio exists where in fact one does.

Kataoka’s criterion

Another safety first criterion was suggested by Kataoka. This involves choosing the portfolio that maximizes the return below which the portfolio return will fall only with a certain predetermined probability. If a is the probability then we would have:

                                                Maximize          R_L

            Subject to the constraint:           Prob(R_P < R_L) £ a

Assuming normality in the return distribution, we can also convert this criterion into a straight line. This becomes:

where F(a) is the critical value of the standard normal distribution associated with the probability level alpha. For example, if our probability a was 10%, then F(a) = 1.28. Kataoka’s criterion can be described graphically as:

The same caveats hold for Kataoka’s criterion as they do for Roy’s. Notice however that in our graphical description of Roy’s criterion, the intercept RL stays at a fixed point while the slope is adjusted. In Kataoka’s case, we predetermine the probability so the slope remains constant. Since we are maximizing the lower limit RL, we parallel shift the line and find that the optimal portfolio is tangential to the efficient curve.

Telser’s criterion

Our third “safety-first” criterion is more general than Kataoka’s. In this case we maximize the expected return of a portfolio subject to the probability that the actual return is less than some minimum acceptable return being lower than some predetermined level. This can be expressed as:

Maximize          E(R_L)

            Subject to the constraint:           Prob(R_P £ R_L) £ a

As in the case with Kataoka’s criterion, assuming normality in the return distribution, we can also convert Telser’s criterion into a straight line. This becomes:

where F(a) is the critical value of the standard normal distribution associated with theprobability level alpha. Since we are maximizing expected return instead of the lower limit we have:

In the case of Telser’s criterion, we set the lower limit and the slope. We then find that the optimal portfolio is the point at which this line intersects the efficient curve. The method is more general in that if the line does not intersect the efficient curve, then we can either adjust our lower limit (parallel shift of the line) or our tolerance for risk (changing the slope of the line).

A common use of a variation of the safety first criteria is in Value at Risk (VaR).

Value at Risk

Value at Risk (VaR) was first used in the 1980s by companies to measure the risk of their trading portfolios. Its attraction is that it provides a quick single value to summarize what can be an extremely complex investment position. VaR is defined as the maximum loss on a portfolio that can be expected over a certain time interval with a certain degree of confidence, and is given as the solution to:

In 1996, the Basle Committee on Banking Supervision (BCBS) required banks to set their capital requirements according to internal models. They recommended that banks hold capital equal to three times the 99 percentile 10-day VaR. As one would expect, once the BCBS began “suggesting” the use of VaR, every major bank in the world now uses it.

It is fairly easy to equate our three safety first criterion to VaR. The equivalences are:

• Roy’s criterion is equivalent to choosing the portfolio with the highest VaR confidence level for a given level of VaR.
• Kataoka’s criterion is equivalent to choosing the portfolio with the lowest VaR at a given confidence level.
• Telser’s criterion is equivalent to choosing the portfolio that has the highest expected return subject to some maximum VaR level at a particular confidence level.

It is interesting to note that banks with trillions of dollars at risk, who are the leaders in managing risk (it’s their business) use a safety first like measurement in the form of VaR. These methods have the advantage of being adaptable to non-normal return distributions.