So far, when dealing with a decision problem under risk, we have used expected profit or expected cost as the criterion of choice. We did so, e.g., for the newsvendor problem,2 as well as for the decision trees of the previous section. But does this actually make sense? The following examples show that this need not be the case.
Example 13.4 (A single bet vs. many repeated bets) Consider the following offer by a professor. A fair coin is flipped: If it lands tail, you win €10, otherwise you lose €5. When offered this lottery, most students would be willing to play. The reasoning is that the expected win (€2.5) is positive. But then, the same should apply if we make things more interesting by scaling the lottery up: You may win €10 million or lose €5 million. No sensible person would play this game, unless he could afford losing €5 million without changing his lifestyle. However, if one could play the game many times, settling the score only at the end, one should probably accept. Of course, playing the game a lot of times reduces variance; in the limit, the law of large numbers applies and risk disappears. However, if you can be thrown out of business in the short term, after a couple of losses, bright long-term prospects will be little solace, if any.
Example 13.5 (Putting all of your eggs in one basket) Consider an investor who must allocate her wealth to n assets. The return of each asset, indexed by i = 1, …, n, is a random variable Ri with expected value μi = E[Ri]. We have introduced this kind of choice in Example 12.5, and we know that asset allocation decisions may be expressed by decision variables ωi representing the fraction of wealth invested in asset i. If we rule out short-selling, these decision variables are naturally bounded by 0 ≤ ωi ≤ 1. If we assume that the investor should just maximize expected return, she should solve the problem
However, its solution is quite trivial; she should simply pick the asset with maximum expected return, i* = arg maxi=1…,n μi, and set ωi* = 1. It is easy to see that this concentrated portfolio is a very dangerous bet. In practice, portfolios are diversified, which means that decisions depend on something beyond expected values. Furthermore, one would also add some additional constraints on portfolio composition, bounding exposure to certain geographic areas or types of industry, and they would render the trivial solution above infeasible. However, it may be necessary to add many such additional constraints to find a sensible solution; this means that the solution is basically shaped by the user who enforces these bounds. Incidentally, if shortselling is allowed, the decision variables are unrestricted, and the expected value of future wealth goes to infinity. In fact, one would short-sell assets with low expected return, to make money to be invested in the most promising asset. This is clearly unreasonable.
Example 13.6 (St. Petersburg paradox) Consider the following proposition. You are offered a lottery, whose outcome is determined by flipping a fair and memory less coin. The coin is flipped until it lands tail. Let k be the number of times the coin lands head; then, the payoff you get is $2k. Now, how much should you be willing to pay for this lottery? We may consider this as an asset pricing problem, and set the expected value of the payoff as the fair price for this rather peculiar asset. The probability of winning $2k is the probability of having k consecutive heads followed by one tail, which stops the game, after k + 1 flips of the coin. Given the independence of events, the probability of this sequence is l/2k+1, i.e., the product of individual event probabilities. Then, the expected value of the payoff is
This game looks so beautiful that we should be willing to pay any amount of money to play it! No one would probably do so. Again, we see that expected values do not tell the whole story.
These examples should suffice to convince us that considering expected monetary values, whether costs or profits, is not enough to fully address decision making under risk. We should find a way to account for the natural tendency to avoid unnecessary or excessive risk. Sometimes, ad hoc tools are used. For instance, when evaluating an investment, cash flows are discounted by a rate accounting for risk.3 In this section we try to find a more general framework, which is provided by utility theory. This is a standard approach in classical economics, fraught with many difficulties and shortcomings; yet, it is a useful conceptual tool. Later, we consider more practical ways to account for and measure risk.
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