September 5 The market opportunity line depicts in the plane the different combinations of outcomes and that are available to the individual at the prevailing market prices, depending on how much of an asset he chooses to purchase. A market opportunity line is a fair odds line if it can be written in the form +( ) = for some constant, where denotes the actual probability of occurrence of and is the probability of occurrence of. Campbell is defining this term as a prelude to graphically depicting optimal investments and insurance. Let s go back to Problem above. An individual has a utility-of-wealth function () =ln( +)and a current wealth of $0. How much of this wealth will the person use to purchase an asset that yields zero with probability and with probability returns $4 for every dollar invested? Again, we ll use for the amount that the individual chooses to invest. If the asset yields zero, the individual receives =0 and if the asset succeeds, he receives =0 +4 =0+3 Solve each euation for : 0 = = 0 3 60 3 = 0 80 = 3 + We can check that this is correct by considering =0(which produces = =0)and =0(which produces =0and =80). The market opportunity line thus depicts the range of risky positions that the individual can take on given the asset available for purchase. This is not a fair odds line because it cannot be rewritten in the form + = for some constant. Example 6.6, p. 0. An individual with current wealth of $00 will have 70% of it stolen with a probability of 0.3. He can purchase insurance for 40 cents per dollar of coverage. Determine the market opportunity line. Let 70 be the amount of coverage that he buys. If he is not robbed, his wealth is = 00 04 If he is robbed, then his wealth is =30 (04) + =30+(06) Solving for in both euations produces 00 04 = = 30 06 47
Figure : Cross-multiplying produces 06(00 ) = 04( 30) 60 + = (04) +(06) 7 = (04) +(06) Let s check our answer by considering a few points on the line. With =0,we have =30and = 00. This satisfies the euation. With =70we have = = 00 (04)70 = 7, which also satisfies the euation. Notice that this is not a fair odds line because doesn t occur with probability 070. Notice that the probability of the low outcome occurring plays no role in the definition of the market opportunity line. The market opportunity line purely reflects the opportunities for purchases available in the market (e.g., from an insurance company or a stock broker). Of course, the market prices may reflect the probability of loss, but the market opportunity line is purely defined by the market prices themselves. 0.0.9 Section 7 Insurance There are two aspects of the market for insurance that we will address later in the course:. moral hazard: Individuals may take preventative action to diminish the likelihood of losses, and having insurance that covers losses may make one less diligent in taking such actions.. adverse selection: Different individuals may face different probabilities of losses, and 48
posting prices for insurance attracts those who are most likely to benefit from coverage (i.e., those with the greatest likelihood of losses). This is an adverse selection for an insurance company from the entire pool of individuals who may benefit from coverage. We re going to ignore both of these issues in this chapter for simplicity and to establish the following benchmark result. Theorem (Complete Insurance Theorem) A risk averse individual faced with fair odds will maximize his expected utility at the point on the fair odds line at which =. Proof. A risk aversion person, faced with the option to buy insurance at prices that reflect the true probabilities of the outcome, will choose to insure himself against losses completely by opting for a point = (i.e., the individual bears no risk because his wealth is the same regardless of the outcome). Let 0 denote the probability of a loss (i.e., the bad or low outcome) and the probability of no loss (the good or high outcome). The assumption is that the market opportunity line is a fair odds line, i.e., it can be rewritten in the form +( ) = where is a constant. The value of is determined by the individual s wealth, the market prices, etc.. We don t need to do this derivation here; we ll just take it as a constant. The individual s problem is to maximize ( ) =()+( ) () subject to the constraint +( ) = and 0. We can use the fair odds line to substitute for in terms of and reduce this to a problem in the single variable : µ () =()+( ) I ve added the notation () to simplify the discussion below. We have the first order condition µ µ 0= 0 () = 0 ()+( ) 0 µ = 0 () 0 = 0 () 0 () Risk aversion implies 00 0, i.e., 0 is strictly decreasing. The last euation therefore reuires that = in order to satisfy the first order condition. Finally, 00 () = 00 () 00 () 0 and so the second derivative test therefore implies that this point is indeed the maximizer of expected utility. 49
It is also possible to prove this result by forming the Lagrangian ( ) =()+( ) () [ +( ) ] Campbell uses the techniue of substituting for with to keep the math as elementary as possible. Example 7., p. 7. Insurance without fair odds. An individual s utility function is () = and she faces the following situation: state probability wealth no accident 07 00 accident 03 30 She can buy a dollar s worth of coverage for $040. How much insurance does she buy? We figured out the market opportunity line above: (04) +(06) =7 The number 04, 06 sum to but do not reflect the true probabilities of 03 and 07. This market opportunity line is not a fair odds line. We therefore cannot conclude that the individual purchases coverage of =70, i.e., = =7. What is the optimal coverage for this person? Her expected utility is () = (03) +(07) = (03) p 30 + (06) +(07) p 00 (04) Taking the derivative and setting it eual to zero produces 0=(03) (06) p +(07) (04) p 30 + (06) 00 (04) 08 08 = p p 30 + (06) 00 (04) 08 p 00 (04) =08 p 30 + (06) µ 08 (00 (04)) =30+(06) 08 µ 9 (00 (04)) =30+(06) 4 µ 8 96 (00 (04)) =30+(06) 800 34 = 5880 + 76 0 = 50 48 = Going back above, we can see that the second derivative of expected utility is (03) µ (06) (30 + (06)) 3 (06)+ (07) µ ( 04) (00 (04)) 3 ( 04) 0 50
and so =48 is indeed the maximal value. We can then demonstrate that 6= by calculating =30+(06) (48) = 3888 and = 00 (04)(48) = 9408 There is another way to solve this problem using the market opportunity line. The individual wishes to maximize (03) +(07) subject to the constraint that and are on the market opportunity line (04) +(06) =7 We substitute for in the objective s (03) 7 (04) +(07) (06) Set the derivative with respect to eual to 0: 0=(03) Cancel the and simplify: 0=(03) (07) (07) 7 (04) (06) (04) (06) 7 (04) (06) 3 7 (04) (06) = 9 4 Suaring each side produces (06) 7 (04) = 8 96 583 34 = 76 583 = 50 3888 = This is the same value of that we obtained above. From the market opportunity line we can calculate (04) 3888 + (06) =7 =9408 The premium paid is 04 = 00 9408 =48 5