Economics 2010a Fall 2003 Edward L. Glaeser Lecture 2 8. Financial Markets a. Insurance Markets b. Moral Hazard c. Adverse Selection with one price contracts d. Simple Financial Markets—Comparative Statics e. Option Pricing and Redundant Assets Insurance– assume that there is some probability ? that a negative shock Z occurs. ?1 ? ??U?Y? ? ?U?Y ? Z? Assume further that insurance exists at price “p”. In that case, individuals solve: max I ?1 ? ??U?Y ? pI? ? ?U?Y ? Z ? I ? pI? This yields F.O.C. ?1 ? ??pU ? ?Y ? pI? ? ??1 ? p?U ? ?Y ? Z ? I ? pI U ? ?Y ? pI? U ? ?Y ? Z ? I ? pI? ? ? 1 ? ? 1 ? p p Differentiation yields: ?I ?p ? ??1???U ? ?Y?pI???1???pIU ?? ?Y?pI???U ? ?Y?Z?I?pI????1?p?IU ?? ?Y?Z?I?pI? ??1???p 2 U ?? ?Y?pI????1?p? 2 U ?? ?Y?Z?I?pI? Wecansignthisintwoways: (1) some assumption about concavity or (2) looking at the effect of price coming from perfect insurance. Assume that there is free entry in supply but it costs c to process each dollar of insurance (is this reasonable?); then we must have that pI ? ?I ? cI ? 0 for any insurance level I,orp ? ? ? c (“arbitrarily fair insurance”). Using this the F.O.C becomes U ? ?Y ? ?? ? c?I? U ? ?Y ? Z ? ?1 ? ? ? c?I? ? 1 ? c ?1 ? ???? ? c? For c ? 0, U ? ?Y ? pI? ? U ? ?Y ? Z ? I ? pI? or I ? Z and Y ? pI ? Y ? ?I ? Y ? Z ? I ? pI. That is, consumption/income is equal across states. Hence, for c ? 0, people perfectly insure against the shock. (What if c ? 0?) The derivative of insurance w.r.t. c or p,for c ? 0 becomes ?I ?p ? ?U ? ?Y ? ?I? ???1 ? ??p 2 ? ??1 ? p? 2 ?U ?? ?Y ? ?I? ? ?U ? ?Y ? ?I? ??p 2 ? ? ? 2p??U ?? ?Y ? ?I? which is negative and inversely proportional to the coefficient of absolute risk aversion at Y ? ?I. Alternately, go back to the denominator of the ugly expression we had before: ??1 ? ??U ? ?Y ? pI? ? ?1 ? ??pIU ?? ?Y ? pI? ??U ? ?Y ? Z ? I ? pI? ? ??1 ? p?IU ?? ?Y ? Z ? I ? We know that p ? ? ? c ? ? or ?1 ? ??p ? ??1 ? p? so as long as U ?? ?Y ? pI? ? U ?? ?Y ? Z ? I ? pI? we’re done– this would require what to hold? Alternately, as long as U ? ?Y ? Z ? I ? pI? ? ??1 ? p?IU ?? ?Y ? Z ? I ? pI we’re done – what would ensure that? Now assume that the problem is to maximize: In that case, individuals maximize: ?1 ? ??U?Y ? pI,0? ? ?U?Y ? pI,L? Where L indicates some sort of loss: In this case, the first order condition is: ?1 ? ??pU 1 ?Y ? pI,0? ? ?pU 1 ?Y ? pI,L? In the case that c ? 0, this implies that: U 1 ?Y ? pI,0? ? U 1 ?Y ? pI,L? What does this imply about the value of I? What does this tell you about betting on sports events? How about insuring the death of child, parent, etc.? Moral Hazard– Self Protection Assume in this case that ? can be changed with effort, i.e. there exists a function ??d? which represents lowering the probability at a cost d. Now the consumer maximizes w.r.t. both I and d: ?1 ? ??d??U?Y ? pI ? d? ? ??d?U?Y ? Z ? I ? pI ? This yields F.O.C. ?1 ? ??pU ? ?Y ? pI ? d? ? ??1 ? p?U ? ?Y ? Z ? I ? and ? ? ?d??U?Y ? pI ? d? ? U?Y ? Z ? I ? pI ? d?? ? ?1 ? ??U ? ?Y ? pI ? d? ? ?U ? ?Y ? Z ? I ? pI ? d?? If perfect insurance is available, what does this imply about the level of c? This is moral hazard– because of the availability of insurance, the only equilibrium has far too little self protection. How can we construct a more efficient contract that still leaves zero profits for the insurer? Cap the insurance payout at I, and then maximize over I allowing p to equal ??d?I?? ??1 ? ??pU ? ?Y ? pI ? d? ? ??1 ? p?U?Y ? Z ? I ? ? ?d ?I ?? ? ?d??U?Y ? pI ? d? ? U?Y ? Z ? I ? pI ? d?? ? ?1 ? ??U?Y ? pI ? d? ? ?U?Y ? Z ? I ? pI ? d??? ?d ?I ?1 ? ????? ? ?d?I?U ? ?Y ? pI ? d? ? ?d ?I ???? ? ?d?I?U ? ?Y ? Z ? I ? pI ? d? At the point where this is equal to zero, you have an optimal insurance cap. Using the first order condition for self-protection, this reduces to: U ? ?Y ? pI ? d? ? U ? ?Y ? Z ? I ? pI ? d? ? ? ?d ?I ? ? ?d?I? 1 ? U ? ?Y ? pI ? d? ? 1 1?? U ? ?Y ? Z ? I ? As long as ?d ?I ? 0 (self-protection declines with insurance) then we should expect to see some limits on the completeness of insurance. These will in turn be a function of the degree to which self-protection matters and the concavity of the utility function. Why might this solution break down? Adverse Selection– No asymmetric information, but a fixed price. Now assume a distribution of people with different values of ?. Characterized by a density g??? which is continuous on the interval ?0,1?. Assume any fixed price p, and assume (for simplicity) that the only available insurance contract perfectly insures (i.e. I ? Z). Then everyone for whom: U?Y ? pZ? ? ?1 ? ??U?Y? ? ?U?Y ? Z?, will insure. As ?1 ? ??U?Y? ? ?U?Y ? Z? is monotonically decreasing in ?, and as U?Y? ? U?Y ? pZ? ? U?Y ? Z? for any positive Z,p there exists a marginal consumer, denoted with ? ? for whom U?Y ? pZ? ? ?1 ? ? ? ?U?Y? ? ? ? U?Y ? Z? So everyone with more risk than that takes the insurance, and everyone with less risk does not. This is adverse selection. To close the model, we need to incorporate supply. Again assume zero profits, which in this case means that: ? ??? ? 1 g????? ? p?Zd? ? 0 or p ? ? ??? ? 1 g????d? ? ??? ? 1 g???d? Will there be an insurance market at all? We know that one equilibrium sets p ? 1. Does there exist another equilibrium? This requires the existence of a ? ? ? 1 at which: U Y ? ? ??? ? 1 g????d? ? ??? ? 1 g???d? Z ? ?1 ? ? ? ?U?Y? ? ? ? U To build intuition, take a Taylor series expansion for U?.? around Y, and use the notation ? ? ? ? ??? ? 1 g????d? ? ??? ? 1 g???d? or U?Y? ? U?Y ? ? ?Z? ? ? ?ZU ? ?Y ? ? ?Z? ? 1 2 ? ? 2 Z 2 U ?? ?Y ? ? ?Z? U?Y ? Z? ? U?Y ? ? ?Z? ? ?1 ? ? ??ZU ? ?Y ? ? ?Z? ? 1 2 ?1 ? ? ?? 2 Z 2 U ?? ?Y ? ? ?Z? In which case, the equation becomes: 0 ? ? ? ? ? ??ZU ? ?Y ? ? ?Z? ? 1 2 ? ? ? 2 ? ?1 ? ? ?? 2 ?Z 2 U or ? ? ? ? ?? ? ?U ?? ?Y? ? ?Z? 2U ? ?Y? ? ?Z? ?1 ? 2 ? ? ? 2 ? ? 2 ? Does there exist a value ? ? at which this holds? It depends on the coefficient of absolute risk aversion. If this is close to zero (i.e. utility is close to linear), this can never hold. As the coefficient of relative risk aversion goes to infinity almost everywhere. In the case where g?.? is uniform on the unit interval, we can actually derive even clearer results. In that case ? ? ?.5?1 ? ?? or 2 1?? 1?? 2 ? ?U ?? ?Y? ? ?Z? U ? ?Y? ? ?Z? You can actually solve this using the quadratic equation. In this case, there will generally exist some sufficiently high level of risk where people actually do ensure. How would you think about solving this problem? We often talk about their being a “risk-return frontier”. We are generally interested in deriving the relationship between asset prices and risk characteristics. One useful utility function for showing this is quadratic utility u?x? ? x ? 1 2 ?x 2 We use that fact that E?x ? E?x?? 2 ? E?x 2 ? ? ?E?x?? 2 to get E?u?x?? ? E?x? ? 1 2 ??E?x ? E?x?? 2 ? ?E?x?? 2 ? ? E?x? ? 1 2 ??E?x?? 2 ? 1 2 ?Var?x? Of course, the problem quadratic utility is that you need x ? 1/? to avoid utility falling with income. If you start with income y, and buy z units of an asset with price one, mean return 1 ? r and variance ? 2 , your income have a mean value of y ? rz and a variance of z 2 ? 2 So E?u?x?? ? y ? rz ? 1 2 ??y ? rz? 2 ? 1 2 ?z 2 ? 2 In this case, the optimal choice of the risky asset gives us: r ? ??y ? rz? ? ?z? 2 ? 0 or z ? r??y ?? 2 ??r What happens if there are multiple independent assets? Another way to think about it is that there is fixed amount of risky asset to be allocated– this implies an r,whichis increasing in risk, concavity and decreasing in wealth. The pure mean-variance utility function is exponential, i.e. u?x? ? ?e ??x and if x is normally distributed, i.e. f?x? ? 1 ? 2 2? e ??x??? 2 2? 2 then E?u?x?? ? 1 ? 2 2? ??e ??x e ??x??? 2 2? 2 dx ? 1 ? 2 2? ??e ?2? 2 ?x??x?? 2? 2 E?u?x?? ? 1 ? 2 2? ??e ? 2 ? 2 ?2?? 2 ?? x???? 2 ? 2 2? 2 dx ? ?e ???? ? 2 ? 2 2 1 ? 2 2? ?e ? x????? 2 ?? 2 2? 2 dx But 1 ? 2 2? ?e ? x????? 2 ?? 2 2? 2 dx ? 1,so E?U?x?? ? ?e ???? ? 2 ? 2 2 And maximizing this is just equivalent to maximizing ? ? ? 2 ? 2 Options– sometimes there are redundant assets. Take for example a three state world. The numeraire asset, pays off 1 in each state, we think of the price of this asset as one. Asset 1 pays off 1 dollar in state 1. The price of this asset is p 1 Asset 2 pays off 1 dollar in state 2. The price of this asset is p 2 Any additional asset can be priced as a function of these two assets. For example an asset that pays of 1 in state 3 must cost 1 ? p 1 ? p 2 An asset that pays off in states 1 ? 2 should cost p 1 ? p 2 Imagine a company whose stock is worth 1 instate1,2instate2and3instate3. The price of that company’s stock ex ante should be p 1 ? 2p 2 ? 3?1 ? p 1 ? p 2 ? Now imagine that we are pricing an option to buy the company’s stock at a price of 1. How much should that option go for? In state one it is worth zero. In state 2 it is worth 1 and in state 3 it is worth 2. Thus the option is worth p 2 ? 2?1 ? p 1 ? p 2 ? Imagine the stock gets riskier and pays zeroinstate1and4instate3. In this case, the option is worth p 2 ? 3?1 ? p 1 ? p 2 ? This suggests that option prices (or at least the gap between option prices and stock prices) are increasing in the amount of risk. To get this point, consider a risk-neutral investor considering an option to buy a stock price at one. Imagine the the stock price is uniformly distributed on the interval ?1 ? a,1? a?. Then the expected value of the option is a/4, which is increasing in risk as measured by a. More generally, consider a continuous state space, indexed by x, and assume that assets span so that there exists a price for each state of p?x?. A stock has value x in each state I can always define the states so that this is true. Consider the option to buy a stock at price c The price of that option will be ? x?c p?x??x ? c?dF?x? As the option is a concave function– mean preserving spreads will increase the value of the option.