Question

Draw (by hand is fine) a graph depicting the utility function of a risk-seeking individual.Suppose she is endowed with initial wealth level $w, and is offered an actuarially-fair gamble G wherein she may win or lose $h. For simplicity, assume $w = $h.

Answer 1

Risk Aversion

Do human beings seek out risk or avoid it? How does risk affect behavior and what are the consequences for business and investment decisions? The answers to these questions lie at the heart of any discussion about risk. Individuals may be averse to risk but they are also attracted to it and different people respond differently to the same risk stimuli.

In this chapter, we will begin by looking at the attraction that risk holds to human beings and how it affects behavior. We will then consider what we mean by risk aversion and why it matters for risk management. We will follow up and consider how best to measure risk aversion, looking at a range of techniques that have been developed in economics. In the final section, we will consider the consequences of risk aversion for corporate finance, investments and valuation.

The Duality of Risk

            In a world where people sky dive and bungee jump for pleasure, and gambling is a multi-billion dollar business, it is clear that human beings collectively are sometimes attracted to risk and that some are more susceptible to its attraction than others. While psychoanalysts at the beginning of the twentieth century considered risk-taking behavior to be a disease, the fact that it is so widespread suggests that it is part of human nature to be attracted to risk, even when there is no rational payoff to being exposed to risk. The seeds, it coud be argued, may have been planted in our hunter-gatherer days when survival mandated taking risks and there were no play it safeoptions.

            At the same time, though, there is evidence that human beings try to avoid risk in both physical and financial pursuits. The same person who puts his life at risk climbing mountains may refuse to drive a car without his seat belt on or to invest in stocks, because he considers them to be too risky. As we will see in the next chapter, some people are risk takers on small bets but become more risk averse on bets with larger economic consequences, and risk-taking behavior can change as people age, become wealthier and have families.  In general, understanding what risk is and how we deal with it is the first step to effectively managing that risk.

I am rich but am I happy? Utility and Wealth

            While we can talk intuitively about risk and how human beings react to it, economists have used utility functions to capture how we react to at least economic risk.  Individuals, they argue, make choices to maximize not wealth but expected utility. We can disagree with some of the assumptions underlying this view of risk, but it is as good a staring point as any for the analysis of risk. In this section, we will begin by presenting the origins of expected utility theory in a famous experiment and then consider possible special cases and issues that arise out of the theory.

The St. Petersburg Paradox and Expected Utility: The Bernoulli Contribution

            Consider a simple experiment. I will flip a coin once and will pay you a dollar if the coin came up tails on the first flip; the experiment will stop if it came up heads. If you win the dollar on the first flip, though, you will be offered a second flip where you could double your winnings if the coin came up tails again. The game will thus continue, with the prize doubling at each stage, until you come up heads. How much would you be willing to pay to partake in this gamble?

            This is the experiment that Nicholas Bernoulli proposed almost three hundred years ago, and he did so for a reason. This gamble, called the St. Petersburg Paradox, has an expected value of infinity but most of us would pay only a few dollars to play this game. It was to resolve this paradox that his cousin, Daniel Bernoulli, proposed the following distinction between price and utility. the value of an item must not be based upon its price, but rather on the utility it yields. The price of the item is dependent only on the thing itself and is equal for everyone; the utility, however, is dependent on the particular circumstances of the person making the estimate.

Bernoulli had two insights that continue to animate how we think about risk today. First, he noted that the value attached to this gamble would vary across individuals, with some individuals willing to pay more than others, with the difference a function of their risk aversion. His second was that the utility from gaining an additional dollar would decrease with wealth; he argued that one thousand ducats is more significant to a pauper than to a rich man though both gain the same amount. He was making an argument that the marginal utility of wealth decreases as wealth increases, a view that is at the core of most conventional economic theory today. Technically, diminishing marginal utility implies that utility increases as wealth increases and at a declining rate.[2] Another way of presenting this notion is to graph total utility against wealth; Figure 2.1 presents the utility function for an investor who follows Bernoullis dictums, and contrasts it with utility functions for  investors who do not.

If we accept the notion of diminishing marginal utility of wealth, it follows that a persons utility will decrease more with a loss of $ 1 in wealth than it would increase with a gain of $ 1. Thus, the foundations for risk aversion are laid since a rational human being with these characteristics will then reject a fair wager (a 50% chance of a gain of $ 100 and a 50% chance of a loss of $100) because she will be worse off in terms of utility. Daniel Bernoullis conclusion, based upon his particular views on the relationship between utility and wealth, is that an individual would pay only about $ 2 to partake in the experiment proposed in the St. Petersburg paradox.[3]

            While the argument for diminishing marginal utility seems eminently reasonable, it is possible that utility could increase in lock step with wealth (constant marginal utility) for some investors or even increase at an increasing rate (increasing marginal utility) for others. The classic risk lover, used to illustrate bromides about the evils of gambling and speculation, would fall into the latter category. The relationship between utility and wealth lies at the heart of whether we should manage risk, and if so, how. After all, in a world of risk neutral individuals, there would be little demand for insurance, in particular, and risk hedging, in general. It is precisely because investors are risk averse that they care about risk, and the choices they make will reflect their risk aversion. Simplistic though it may seem in hindsight, Bernoullis experiment was the opening salvo in the scientific analysis of risk.

Mathematics meets Economics: Von Neumann and Morgenstern

            In the bets presented by Bernoulli and others, success and failure were equally likely though the outcomes varied, a reasonable assumption for a coin flip but not one that applies generally across all gambles. While Bernoullis insight was critical to linking utility to wealth, Von Neumann and Morgenstern shifted the discussion of utility from outcomes to probabilities. Rather than think in terms of what it would take an individual to partake a specific gamble, they presented the individual with multiple gambles or lotteries with the intention of making him choose between them. They argued that the expected utility to individuals from a lottery can be specified in terms of both outcomes and the probabilities of those outcomes, and that individuals pick one gamble over another based upon maximizing expected utility.

            The Von-Neumann-Morgenstern arguments for utility are based upon what they called the basic axioms of choice. The first of these axioms, titled comparability or completeness, requires that the alternative gambles or choices be comparable and that individuals be able to specify their preferences for each one. The second, termed transitivity, requires that if an individual prefers A to B and B to C, she has to prefer A to C. The third, referred to as the independence axiom specifies that the outcomes in each lottery or gamble are independent of each other. This is perhaps the most important and the most controversial of the choice axioms. Essentially, we are assuming that the preference between two lotteries will be unaffected, if they are combined in the same way with a third lottery. In other words, if we prefer lottery A to lottery B, we are assuming that combining both lotteries with a third lottery C will not alter our preferences. The fourth axiom, measurability, requires that the probability of different outcomes within each gamble be measurable with a probability. Finally, the ranking axiom, presupposes that if an individual ranks outcomes B and C between A and D, the probabilities that would yield gambles on which he would indifferent (between B and A&D and C and A&D) have to be consistent with the rankings. What these axioms allowed Von Neumann and Morgenstern to do was to derive expected utility functions for gambles that were linear functions of the probabilities of the expected utility of the individual outcomes. In short, the expected utility of a gamble with outcomes of $ 10 and $ 100 with equal probabilities can be written as follows:

E(U) = 0.5 U(10) + 0.5 U(100)

Extending this approach, we can estimate the expected utility of any gamble, as long as we can specify the potential outcomes and the probabilities of each one. As we will see later in this chapter, it is disagreements about the appropriateness of these axioms that have animated the discussion of risk aversion for the last few decades.

The importance of what Von Neumann and Morgenstern did in advancing our understanding and analysis of risk cannot be under estimated. By extending the discussion from whether an individual should accept a gamble or not to how he or she should choose between different gambles, they laid the foundations for modern portfolio theory and risk management. After all, investors have to choose between risky asset classes (stocks versus real estate) and assets within each risk class (Google versus Coca Cola) and the Von Neumann-Morgenstern approach allows for such choices. In the context of risk management, the expected utility proposition has allowed us to not only develop a theory of how individuals and businesses should deal with risk, but also to follow up by measuring the payoff to risk management. When we use betas to estimate expected returns for stocks or Value at Risk (VAR) to measure risk exposure, we are working with extensions of Von Neumann-Morgensterns original propositions.