THE OUTCOME
18.3.2 Hurdle Model with Economy-wide and Firm-specific Risks
well-diversified shareholders and the agent so that the agent has no incentive to do any additional trading in economy-wide event claims.
At first glance it may seem surprising that the no-trading constraint is not binding when the agent has additively separable exponential utility since this utility function exhibits wealth effects. However, recall that the agent has only one consumption date. Hence, his trading only affects the variation in his con- sumption across the economy-wide events and not the level of consumption. If the agent has an initial consumption date (at the contracting date) so that he can shift the level of consumption between consumption dates, the no-trading con- straint will be binding for the additively separable exponential utility function.
In that case there will be a tension between the intertemporal allocation of con- sumption and optimal incentives (which we explore in Chapter 24). However, the no-trading constraint will still be non-binding for the multiplicatively sepa- rable exponential utility function with multiple consumption dates since the level of consumption has no impact on action choices for this utility function.
In general, we expect the performance measure y to be correlated with the economy-wide event, and to be influenced by the agent's action. Consequently, the economy-wide event is expected to be insurance informative. For example, knowledge of the economy-wide event can be helpful in making inferences about whether a good outcome is due to the agent working hard or to favorable market conditions. In such cases, there will be tension between the sharing of economy-wide risk, the agent's trading, and optimal incentives. Intuitively, if the agent can trade in economy-wide event claims, the principal cannot as efficiently allocate incentive bonuses and penalties across the economy-wide events as would be possible if the agent could not trade in these claims. When the agent can trade in these claims, he will have an incentive to "insure" (i.e.,
"smooth") these bonuses and penalties through his trading. We illustrate this in the following section using the hurdle model.
18.3.2 Hurdle Model with Economy-wide and Firm-specific
whereas the probability of the bad outcome given a is ^(x^ \a) = (1 - ci)(p{9^ +
vie,).
The good event is associated with "large" aggregate consumption compared to the bad event. Hence, the risk-adjusted probability for the good event is less than or equal to the original probability for the good event, i.e.,^(^ ) < (p{6\
and vice versa for the bad event.
In the following we assume that the agent has an additively separable expo- nential utility function and consider the optimal contract for inducing a = Vi.
We use the following data:
u{c) = - exp[-c]; v(a) = Aal{\ - a ) ; ^ = - 1;
(p{9^) = f{e,) = V2.
Risk Neutral Shareholders and No Agent Trading
Note that the outcome is only informative about the agent's action in the good event - the bad outcome obtains with certainty in the bad event. In order to illustrate the impact of the differences in information content for the two events, we assume initially that the shareholders are risk neutral so that the risk-adjusted probabilities are equal to the original probabilities for the two economy-wide events. Furthermore, the agent is exogenously precluded from trading. The optimal contract is shown in Table 18.2 along with the agent's expected mar- ginal utilities conditional on the economy-wide events.
c\a) 0.155
u%c,a\e;)
c(x^,e^)
0.542
c(xt„Og) -0.323 0.982
c(xi„0b) 0.200 0.818 Table 18.2: Optimal contract for inducing a = V2 with risk
neutral shareholders and agent exogenously precluded from trading.
We can view compensation as imposing two types of risk on the agent: out- come risk and event risk. In this example, the outcome risk only occurs if the good event occurs, and is required to induce the agent to select a = V2. Event risk is imposed if the agent's expected marginal utility in the good event differs from his expected marginal utility in the bad event. Since the shareholders are risk neutral, there are no risk sharing reasons for imposing event risk. However, Table 18.2 reveals that event risk is imposed. The reason for this is that with additive utility, the outcome risk premium required to induce a given action a can be reduced if the compensation in the good event is reduced (see Appendix
17C). Of course, this reduction must be offset by an increased compensation in the bad event (so the participation constraint is satisfied), which creates event risk for the agent. The greater the reduction in the good event compensation, the lower is the outcome risk premium, but the greater is the event risk pre- mium. The contract in Table 18.2 makes an optimal tradeoff between these two types of risk premia.^^ If event claims are available, the compensation contract is such that the agent has an incentive to buy claims for the good event and sell claims for the bad event (since the conditional expected marginal utility is higher in the good event than in the bad event).^^
Risk Averse Shareholders and No Agent Trading
Now consider the setting in which the shareholders are risk averse and, there- fore, require a risk premium for taking economy-wide risk. This is depicted as the risk-adjusted probability for the good event being less than the original probability for that event. For the purpose of our numerical example we set (p{0 ) = A (< (p(0 ) = VT). Suppose again that the agent is exogenously pre- cluded from trading. Table 18.3 shows the optimal contract along with the agent's expected marginal utilities conditional on the economy-wide events times the ratio of the original and risk-adjusted event probabilities.
c\a){c\a)) 0.146(0.173)
t/>,a|^,)x^(^.)V(^.)
c{x^,eg) 0.832
c(Xh,eg)
-0.211 0.835 x.5/.4 = 1.044
c(Xi,6'i)
0.036 0.965 X.5/.6 =0.804 Table 18.3: Optimal contract for inducing a ^Vi with risk averse share-
holders ((p(0 ) = A) and agent exogenously precluded from trading.
^^ Chapter 25 uses a similar argument in an intertemporal setting where utility levels are shifted between multiple consumption dates.
^^ It can be shown that if the agent's action does not affect the probabilities for the outcome in the bad event, the following relation holds between the optimal compensations in the good and the bad event (see Christensen and Frimor, 1998),
M{c{x„0,)) = E[M(c)\a,e^].
Since M(-) = \/u'(') and 1/w' is a convex function, Jensen's inequality implies that U:ic,a\d,) < ir:(c,a\d,X
SO that the agent has incentive to buy claims for the good event in return for selling claims for the bad event.
Note that the agent is paid more in the good event and less in the bad event compared to the setting in which there is no risk adjustment of the probabilities for the economy-wide events (see equations (18.19) and (18.20)). This is a con- sequence of the fact that paying compensation in the good event has a lower market value than paying the same amount in the bad event - the expected com- pensation (0.173) is higher, but the market value of the compensation contract in Table 18.3 (0.146) is lower than that of the compensation contract in Table 18.2 (0.164). However, there is a tradeoff between shifting compensation (and, thus, utilities) from the "expensive" bad event to the "less costly" good event and a higher risk premium paid in the good event to induce the agent to jump.
This tradeoff is such that the agent's marginal utility conditional on the events is now lower in the good event than in the bad event. However, the agent still has an incentive to buy claims for the good event and sell claims for the bad event since the claim for the good event is relatively cheap, i.e. the conditional expected marginal utility times the ratio of the original and risk-adjusted proba- bilities is higher for the good than for the bad event.^^
Risk Averse Shareholders and Agent Trading
Now consider the setting in which the shareholders are risk averse and the agent can trade in claims for the two economy-wide events. Without loss of gene- rality the optimal contract is determined such that the agent has no incentive to trade, i.e., subject to the constraint (18.3p"). Table 18.4 shows the optimal con- tract along with the agent's expected marginal utilities conditional on the econ- omy-wide events times the ratio of original and risk-adjusted event probabilities.
In order to eliminate the agent's incentive to trade, the agent's conditional expected marginal utility must be reduced for the good event and increased for the bad event compared to the contract in Table 18.4. This is achieved by paying the agent less in the bad event, and more in the good event but with a higher variability (to maintain inducement of a = Vi). This tends to further increase the
^^ As in the risk neutral shareholder setting (see footnote 28), it can now be shown that M(c(x„d,))xf{d,)l<p(d,) = E[M(c)|a,0Jx^'(0p/^(^p.
Jensen's inequality now implies that
U:(c,a\0,)>^<p(e,)l<^(9,) < U:(c,a\d,)x<P(d,V<P(0,),
SO that the agent has incentive to buy (sell) claims for the good (bad) event. More generally, the optimal contract is such that the agent has incentives to buy claims for the event in which the out- come is most informative about the agent's action, and sell claims for the other event.
outcome risk premium paid in the good event to induce the agent to jump.^^
Hence, the agent's trading opportunities make it more costly to induce him to jump because of the higher risk premium he must be paid in the good event (where his utility level is relatively higher than without trading opportunities).
c\a){c\a)) 0.159(0.218)
\u:{c,a\O)^<p{9)l(p{0)
c(x^,dg) 1.139
4xt,0g) -0.113 0.720x.5/.4 = 0.900