Introduction

The effects of exogenous uninsurable background risk on optimal investment, hedging and insurance decisions have been the subject of considerable recent research. This research has largely focused on determining the conditions under which the addition of, or an increase in, an exogenous background risk will reduce the demand for risky assets, see, for example, Eeckhoudt and Gollier and GollierFootnote 1 for reviews. Given the complexity of the problem, the literature has consistently assumed that the distributions of the risks faced are given and not affected by the decision maker's behavior; that is, there is no moral hazard.

The optimal contract is affected by the agent's attitude towards risk. If background risk affects the agent's attitude towards risk, then it affects the design of the contract. The objective of this paper is to examine the effect of background risk on optimal incentive contracts when there is moral hazard. For concreteness, we analyse the effect of a fair background risk on the design of compensation contracts in a simple principal-agent model in which there are two outcomes (success and failure) and two actions (high and low effort). This is a standard and widely used model.Footnote 2 We consider two cases, contractual background risk and environmental background risk. A contractual background risk is a background risk that is present only if the contract is accepted. An environmental background risk is part of the contracting environment and is present whether or not the contract is accepted.

The principal-agent model has many applications and background risk arises naturally in a number of these applications. Applied to insurance, the principal is a monopoly insurer, the agent is the policyholder and background risk might arise from risks to the agent's human capital. Applied to regulation, the principal is the regulator, the agent is the regulated firm and the background risk might arise from the firm's other lines of business. Applied to compensation contracts, the background risk might arise from factors such as investment income or a spouse's earnings.Footnote 3

The case of the contractual background risk could be modelled simply by treating differing payoffs as separate states of the world. In this approach, high and low effort give rise to a distribution of payoffs. However, where the randomness of the payoff structure is independent of the agent's choice, additional insight may be gained by separating the risk associated with the agent's choice of action from the background risk. See Ligon and Thistle,Footnote 4 who treat mutual insurance as equivalent to a conventional insurance policy plus a background risk. Consumers are exposed to this risk only if they purchase the mutual policy. An example in the principal-agent context is where the agent is compensated in the form of corporate stock and where the return on the stock is a function both of the agent's effort and of random noise. In such an interpretation, high and low wages can be interpreted as the expected value of the stock compensation in the success and failure states, respectively, and the random component of the stock return can be interpreted as a background risk.

Previous research examining the effect of background risk on optimal contracts in the principal-agent model is limited. Ligon and ThistleFootnote 5 examine the effect of background risk on insurance markets subject to moral hazard. The model of competitive insurance markets with moral hazard can also be regarded as a principal-agent model where the agent/policyholder has all of the bargaining power. In competitive insurance markets, the solution lies on the principal's participation constraint (i.e., insurers make zero expected profits). Ligon and Thistle show that if policyholders are prudent then background risk increases policyholders’ loss prevention effort and expands the set of insurance policies that earn non-negative expected profit. However, the effect on equilibrium premiums and coverage levels is indeterminate.

In the standard principal-agent model, the principal is assumed to have all of the bargaining power and the resulting contract lies on the agent's participation constraint. We show that if the agent is weakly decreasing absolute risk averse (weakly prudent), then the introduction of a contractual (environmental) background risk makes the optimal incentive scheme flatter. The contractual background risk increases the agent's expected wage for all risk averters. Environmental background risk decreases expected wages if prudence is no more than three times the absolute risk aversion and the risk premium is weakly decreasing and weakly concave in wealth. Our results imply that principals may prefer agents with certain characteristics and may, depending upon the characteristics of agents in the society, have an interest in either increasing or decreasing the level of background risk in society. Our results also have implications for the design of optimal contracts for agents and principals of different types.

The Principal-Agent Model

A risk-neutral principal must design an incentive contract for a risk-averse agent. The principal cannot observe the agent's action, but can observe whether the outcome is success, x S , or failure, x F (x S >x F ). The agent can choose between working (with cost a=1) and not working (with cost a=0). The probability of success is p H if the agent works and p L if she does not (0<p L <p H <1). The agent's utility is u(y 0 +w)−a when the wage is w and the agent's fixed outside income is y0. The principal's objective is to maximise the expected profit net of the agent's wages while the agent's objective is to maximize the expected utility.

The interesting case is when the principal wants the agent to work. The principal must then pay wages w S and w F in the case of success and failure, respectively, such that the agent accepts the contract and is induced to work. It is easy to show that when the incentive compatibility and participation constraints bind this implies

where is the agent's reservation utility when outside income is not risky (i.e., =u(y0)), θ S =(1−p L )/(p H −p L ) and θ F =p L /(p H −p L ). Since the right-hand side of (1a) is larger than e , which in turn is larger than the right-hand side of (1b), it follows directly that w S >0>w F , that is, the agent is rewarded for success and punished for failure. The incentive scheme is said to be ‘flatter’ or ‘less high powered’ as the difference w S −w F becomes smaller.

Contractual and environmental background risk

We now assume that the agent's outside income is risky, so that it can be written as y0+z, where z is a zero mean random variable distributed independently of the outcome. The Arrow-Pratt risk premium, π(y0), is defined by u(y0−π)=Eu(y0+z) and depends on the total income. The precautionary premium ψ(y0) is defined by u′(y0−ψ)=Eu′(y0+z).Footnote 6,Footnote 7 The agent is (weakly) prudent if ψ⩾0, or equivalently, if u′ is weakly convex. Weak decreasing absolute risk aversion (DARA) requires u′′′⩾(u″)2/u′, so prudence is necessary but not sufficient for DARA. The derived utility function is defined by û(y0)=Eu(y0+z).Footnote 8 If u is increasing, risk averse and prudent, then so is û, and if u is weakly prudent, then û′(y0)⩾u′(y0).Footnote 9

Now suppose that the agent's outside wealth is risky. Let e denote the agent's reservation utility in the presence of background risk. If the incentive compatibility and participation constraints are binding, the contract for the agent must satisfy

where v S and v F are the agent's wages when the outcomes are success and failure, respectively. We will consider two cases. In the case of contractual background risk, the agent is exposed to the background risk only if the contract is accepted. The agent's reservation utility is

=u(y0). In the case of environmental background risk, the agent is exposed to background risk whether or not the contract is accepted. The agent's reservation utility is

=Eu(y0+z) when environmental background risk is present.

We first need to show that the incentive scheme is increasing. Since the right-hand side of (2a) is larger than , which is in turn larger than the right-hand side of (2b), it follows directly that v S >0>v F . This is true whether or not the background risk affects the agent's reservation utility.

Now we want to show that the background risk makes the incentive scheme flatter. First, consider the case of contractual background risk. That is, the background risk and the contract are bundled or, equivalently, it is acceptance of the contract that exposes the agent to the background risk. Then solving (1) yields and . Set Eu(y0+z+v j )=u(y0−π j +v j ), where π j =π(y0+v j ), j=S, F. Then solving (2) yields

Since the reservation utility is the same in both cases, wages increase in both states of the world. This is because the agent is only exposed to the background risk if the contract is accepted, and therefore a compensating wage differential must be paid for this risk by the principal. Hence, the wage must rise in both states to cover the agent's risk premium. Note that w S and w F do not depend on z, so that the effect of background risk on wages comes solely through the effect on the risk premium. Again, since the contractual background risk does not affect reservation utility, it follows directly from (3) that v S –v F ⩽w S –w F , or the incentive scheme is less high powered if and only if π S ⩽π F . This holds if, and only if, the risk premium does not increase with wealth; that is, if the agent's preferences satisfy weak DARA.

This proves the following result:

Proposition 1

  • If background risk is contractual ( =u(y0)) and the agent's preferences satisfy weak DARA, then the introduction of background risk makes the incentive scheme flatter, v S −v F ⩽w S –w F .

Another possibility is that the background risk is environmental; that is, the agent is exposed to the background risk whether or not the contract is accepted. That is, assume reservation utility is =u(y0) in the absence of background risk and =Eu(y0+z)=u(y0−π0), where π0=π(y0), in the presence of background risk. Let v S * and v F * denote the wages in this case. We now compare contracts where the background risk is environmental to the no background risk case.

Proposition 2

  • If background risk is environmental ( =Eu(y0+z)) and the agent's preferences satisfy weak prudence, then w S ⩾v S *>v F *⩾w F . Hence, the introduction of background risk makes the incentive scheme flatter, v S *−v F *⩽w S –w F .

Proof Using the derived utility function, the contract (v S *, v F *) satisfies □

Then, from (1a) and (4a),

since weak prudence implies û′⩾u. Therefore, w S ⩾v S *. Similarly, from (1b) and (4b),

By the same argument, weak prudence implies w F ⩽v F *. Combining the results, we have w S ⩾v S *>v F *⩾w F and v S *−v F *⩽w S −w F .Footnote 10 ∣∣

Prudence implies that the introduction of an undesirable risk reduces expected utility by more as wealth increases. Reservation utility declines with the introduction of the environmental background risk, but utility in the success state declines by less while utility in the failure state declines by more. This leads to w S >v S * and w F <v F *.

Expected compensation

In this section, we examine the effects of background risk on the agent's expected compensation. When there is no background risk, the agent's expected compensation is W=p H w S +(1−p H )w F . When the background risk is contractual and does not affect the agent's reservation utility then the agent's expected compensation is p H v S +(1−p H )v F , which is larger than when background risk is absent. When the background risk is environmental and therefore does affect the agent's reservation utility, then the agent's expected compensation is V*=p H v S *+(1−p H )v F *. In this case, in general, expected compensation may increase or decrease compared with the situation in which there is no background risk. Let A=−u′′/u′ be the coefficient of absolute risk aversion and let P=−u′′′/u′′ be the coefficient of absolute prudence.

Since it affects reservation utility, the introduction of environmental background risk has the same effect ‘as if’ the agent's outside income is reduced by π0. As Theile and WambachFootnote 11 show, a decrease in the agent's outside wealth reduces the agent's expected compensation if P⩽3A. The agent must also be compensated for the background risk borne under the contract, net of the reservation utility background risk. This also tends to reduce the expected compensation if the risk premium is non-increasing and concave. Observe that weak DARA implies weak prudence, so that if environmental background risk decreases expected compensation then it also makes the incentive scheme flatter.

Proposition 3

  • If the risk premium is weakly concave and A⩽P⩽3A, then the introduction of environmental background risk ( =Eu(y0+z)) reduces the agent's expected compensation, V*<W.

Proof Since the agent's outside wealth is risky, the contract must satisfy □

where =Eu(y0+z) includes the environmental background risk and π j *=π(y0+v j *), j=S, F. Adding and subtracting π0 in the solution to (7), we can write the wages as v j *=w j *+(π j *−π0), j=S, F, where w j * is the wage for an agent with no background risk and income y0–π0. Let W*=p H w S *+(1−p H )w F *. Then the difference in expected wages is

Theile and WambachFootnote 12 show that if P⩽3A, then the agent's expected compensation is increasing in outside income, so W*<W. If the risk premium is weakly concave, then

Adding and subtracting π0 and rearranging yields

Weak DARA (A⩽P) implies π0−π(y0+V*)⩾0, which in turn implies that the term in brackets must be non-positive. Consequently, expected compensation falls, V*<W.∣∣

Proposition 3 assumes that the agent's preferences satisfy P⩽3A. This condition is trivially satisfied by constant absolute risk aversion (CARA) utility functions since P=A and it is easy to show that it is satisfied by hyperbolic absolute risk aversion (HARA) utility functions of the form u(y)=(y+a)1−γ/(1−γ) for γ⩾1/2. Weak concavity of the risk premium imposes the restriction 2A3−3A2P+APT<0, where T=−u(4)/u′′′ is the coefficient of temperance, on the first four derivatives of the utility function.Footnote 13 The CARA utility function satisfies this condition, but the risk premium for the HARA utility function is convex. If the agent has a CARA utility function, then Propositions 2 and 3 imply that the introduction of environmental background risk makes the incentive scheme flatter and decreases the agent's expected compensation. If the agent has a HARA utility function, Propositions 2 and 3 imply that the introduction of environmental background risk makes the incentive scheme flatter but may increase or decrease the expected compensation.

We should point out that an important simplification in our analysis is the assumption of two outcomes. However, the assumption of two actions is not. The results of the Propositions can however be extended to models in which the agent can choose from more than two actions. Suppose, for example, the agent can choose a∈[0, 1] and the probability of success is p(a) with p′>0, p′′<0. The agent's utility function is u(y)−ϕ(a) with ϕ′, ϕ′′>0. Now consider the principal's problem of minimising the cost of implementing some action a. If there is no background risk then the cost-minimizing incentive scheme (w S , w F ) satisfies (1), where now θ S =ϕ(a)+(1−p(a))ϕ′(a)/p′(a) and θ F =ϕ(a)–p(a)ϕ′(a)/p′(a). If background risk does not (resp. does) affect the agent's reservation utility then the agent's wages are again given by the solution to (2) with reservation utility =u(y0) (resp. =û(y0)). If the action a is implementable, then Propositions 1, 2 and 3 all continue to hold.

Conclusion

We examine the effect of background risk in the standard two-state, two-action principal-agent model. Intuitively, we expect that the introduction of background risk will make agents want to reduce their exposure to risks that they can control; in this case, the risk inherent in the incentive scheme. We show that if the background risk is contractual (does not affect the agent's reservation utility) then expected compensation rises and the background risk makes the optimal incentive scheme flatter if the agent's preferences satisfy weak DARA. If the background risk is environmental and therefore present whether or not the contract is accepted, then background risk makes the optimal incentive scheme flatter if the agent is weakly prudent. In this situation, the background risk reduces the agent's expected compensation if the agent is weakly DARA, if the risk premium is concave and if the coefficient of absolute prudence is no more than three times the coefficient of absolute risk aversion.

It is interesting to compare the results here with the results for the standard portfolio problem. In order for the introduction of background risk to reduce the optimal level of investment, conditions such as standard risk aversionFootnote 14 or risk vulnerabilityFootnote 15 must be invoked. These require additional restrictions beyond prudence. In the portfolio problem, the effect of background risk depends on how it affects marginal utility and not on how it affects the level of utility. In the principal-agent problem analysed here, the effect of background risk depends on how it affects the level of utility, so that weaker restrictions on preferences are sufficient.

Our results suggest that employers increase their costs when acceptance of the employment contract effectively introduces a background risk for the agent, although the incentive scheme is still flatter than in the case where acceptance of the contract involves no background risk if agents exhibit DARA. For larger employers who effectively maximize the net profit, this is an unambiguous negative. They dislike paying higher wages and they do not consider the flatter incentive scheme a benefit. However, for smaller employers who are risk averse, the flatter incentive scheme creates an offsetting benefit to the higher expected wage. Taken together, this might suggest that large employers in dangerous occupations (e.g., mining, railroads) would support health and safety measures designed to reduce the risks in their industries. Smaller employers may be somewhat less willing to support such measures, however, if this introduces greater variability into the compensation schedule. Thus, small employers in risky occupations may be the least likely to purchase health or disability insurance for their employees.

However, agents who face pre-existing background risks can actually lower the expected cost of the contract to the principal if the agent's risk preferences satisfy the conditions of Proposition 3 and reduce the variability of compensation if they satisfy the conditions of Proposition 2. Theile and WambachFootnote 16 show that the wealth level of the agent may matter to the principal. We show that the background risks faced by the agent and his or her risk preferences may be relevant as well.

Consider health expenditure risk as an environmental background risk that can add to or reduce the income. If agents satisfy the conditions of Proposition 3, then principals would find that providing health insurance as part of the compensation contract would increase both expected wages and variation in wages. However, if agents satisfied the conditions of Proposition 2 but not Proposition 3, then the background risk could increase expected wage costs but reduces their variability.

Our results may also be relevant to the structure of executive compensation. The results suggest that not only may the wealth level of the executive be relevant to the optimal incentive contract, the structure of the executive's investment portfolio may be relevant as well. A wealthy agent who invests in the risk-free asset requires a different contract structure than a wealthy agent who invests in a risky portfolio. If a firm faces non-systematic risk, then equity-related compensation introduces a background risk that the executive may be prohibited by the terms of the employment contract from rebalancing (e.g., stock options, restricted stock grants). Forcing the executive to retain the background risk through the equity investment may lower the expected compensation and its variability if the conditions of Proposition 3 are met. Our results are thus consistent with the large body of literature (see PrendergastFootnote 17 for a review) that suggests that the performance sensitivity of an executive's contract should depend on the other risks he or she faces.