Concave utility functions
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1 Meeting 9: Addendum
2 Concave utility functions This functional form of the utility function characterizes a risk avoider. Why is it so?
3 Consider the following bet (better numbers than those used at Meeting 8, the numbers do not matter that much, but their graphical representation must be accurate): Win SEK 500 with probability 0.7 and lose SEK 700 with probability 0.3 There are two actions: a 1 = take the bet and a 2 = do not bet Focus on the range of money defined by the bet:
4 The straight line corresponds with a utility function that is linear in payoff values along the line can be interpreted as payoff expressed in the same scale as U(R). Now, the expected payoff of the bet is ER which is a convex combination of 500 and 700. Since any convex combination of two points v 1 and v 2, i.e. any v = p v 1 + (1 p) v 1 where 0 < p < 1 lies on the segment joining v 1 and v 2 (in one dimension this means v 1 < v < v 2 ) we can represent the expected payoff as a point on a straight line joining the points ( 700, U( 700)) and (500, U(500)): To clarify: a point in this sense is a point on the curve defining the utility function. Hence, v 1 and v 2 both need to be on the curve, which then automatically defines the coordinates of the points.
5 Now, it also holds that the expected utility (for money) of taking the bet is EU bet U U This is also a convex combination, but here of two utilities. This must be represented by the same point as was the corresponding convex combination of payoffs, but now we should view it from the utility perspective Using the mathematical function that was used to produce the curve we can calculate EU 0.802
6 Now, it becomes clearer why the utility function of a risk avoider is concave Plotting R as a function of U(R) we can see which cash equivalent (expressed on the same scale as U(R) ) corresponds with which utility. Here we can see that the cash equivalents for the utility of taking this bet are all lower than or equal to the payoff of taking the bet. In particular, the value of EU is equal to the utility of a cash equivalent satisfying R 1 EU U EU where U -1 is the inverse function of U(R) restricted to R (-700, 500) Here, again using the mathematical function behind the curve, we can calculate R(EU) 167.
7 Hence, the decision maker appreciates the expected utility of taking the bet to be equivalent to a payoff of SEK 167 while the expected payoff of taking the bet is SEK 140. Now, the expected payoff of not taking the bet is (always) SEK 0. The expected utility of not taking the bet must be equal to the utility when R = 0, i.e. U(0). This can again be calculated using the mathematical function behind U(0) 0.87
8 Now, the very decision problem is about taking the bet corresponding with lottery 2 or not. The expected utility of taking the lottery is and the expected utility of not taking the bet is the utility corresponding with a payoff of SEK 0, which is Thus the optimal decision with the EU-criterion is to not take the bet. R(EU) is called the certainty equivalent of the decision-maker. This is the lowest value of a certain payment that the decision-maker would prefer versus taking the bet. Apparently this value can be negative which would render a cost and not a return for the decision-maker. Alternatively expressed, the decision-maker is in this case indifferent between 1. Obtaining SEK 167 for certain and 2. Obtain SEK 500 with probability 0.7 and losing SEK 700 with probability 0.3
9 Risk premium When a risk avoider is exposed to a bet, he or she will always have an expected utility of taking the bet that is lower than (or at highest equal to) the expected utility that is linear in payoff. The payoff equivalent to the expected utility of this risk avoider is their certainty equivalent, CE, and the difference between the expected payoff, ER, and the certainty equivalent is called their risk premium, RP. RP ER CE In the above example the risk premium of the decision-maker is then approximately SEK 140 ( 167) = 307 Notice (again) that all functions and specific quantities are personal to the decision-maker!
10 What is then the difference between the certainty equivalent and the risk premium? In the example above, the expected payoff was positive (ER = 140) while the certainty equivalent was negative (CE = 167). The certainty equivalent is what the decision maker considers to be the expected utility of taking the bet. Hence they will never consider taking a bet with a negative certainty equivalent, but it would not generally suffice with a positive certain equivalent either. The risk premium tells how much money must at least be additionally paid to the decision maker for making them take the bet. In the example above that amount was SEK 307. RP
11 It is the shape of the utility function that implies that the decision-maker does not become indifferent between 1. Obtaining SEK x for certain and 2. Obtain SEK 500 with probability 0.7 and losing SEK 700 with probability 0.3 RP until x = RP. Hence, the higher the certainty equivalent the lower the risk premium. If the expected payoff of a bet is 0, the bet is said to be a fair bet. In that case the risk premium will be equal to the certainty equivalent in absolute sense since RP = 0 CE.
12 Now, if the decision-maker is risk neutral the expected utility for money of a bet coincides with the expected payoff (the utility is linear in payoff). This means that the certainty equivalent is always 0 and so is the risk premium. For a risk taker the situation is the opposite to the risk avoider. The utility function of a risk avoider is convex, e.g.
13 This function is here made such as the utilities for R = 700 and R = 500 are the same as with the previous utility function of a risk avoider. Hence with the same bet as before, i.e. Win SEK 500 with probability 0.7 and lose SEK 700 with probability 0.3 we can graphically illustrate this as
14 The value of EU for taking the bet is the same here as before, i.e If we as above plot R against U(R) we obtain: and the certainty equivalent here becomes CE = R(EU) 372
15 The expected utility of not taking the bet is calculated as U(0) 0.51 Thus the optimal decision with the EU-criterion is to take the bet, since U(0) < EU. The risk premium with this utility function becomes RP ER CE (SEK) hence negative!
16 For a risk taker the certainty equivalent is always higher than (or at least equal to) the expected payoff. Hence, the risk premium for taking a bet is always negative for a risk taker. This means that a risk taker is willing to pay a certain amount for taking the bet The application to insurances When it comes to deciding the premium for an insurance, which one (if any) of the insurance provider and the insurance taker is (closest to) being a risk avoider? a risk taker?
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