BACKGROUND RISK IN THE PRINCIPAL-AGENT MODEL. James A. Ligon * University of Alabama. and. Paul D. Thistle University of Nevada Las Vegas

mhbr\brpam.v10d 7-17-07 BACKGROUND RISK IN THE PRINCIPAL-AGENT MODEL James A. Ligon * University of Alabama and Paul D. Thistle University of Nevada Las Vegas Thistle s research was supported by a grant from Michael Shustek and Vestin Mortgage and by the Nevada Insurance Education Foundation. * Ligon: Department of Economics, Finance and Legal Studies, University of Alabama, Box 870224, Tuscaloosa, AL 35487-0224, Phone: 205-348-6313, Fax: 205-348-0590, Email: jligon@cba.ua.edu Thistle (corresponding author): Department of Finance, University of Nevada Las Vegas, 4505 Maryland Parkway, LasVegas, NV 89154, Phone: 702-895-3856, Fax: 702-895-4650, Email: paul.thistle@unlv.edu

BACKGROUND RISK IN THE PRINCIPAL-AGENT MODEL ABSTRACT We examine the effect of background risk in the standard two-state, two-action principalagent model. We analyze situations where the background risk is environmental (always present) and where the background risk is contractual (only present if the contract is accepted). With contractual background risk, expected wages always rise and the incentive scheme is flatter if the agent s preferences satisfy weak DARA. With environmental background risk, the optimal incentive scheme becomes flatter if the agent is weakly prudent. We provide conditions under which the environmental background risk decreases the agent s expected wage. JEL Classification: D81, D82 Keywords: risk aversion, moral hazard, incentives, contract

BACKGROUND RISK IN THE PRINCIPAL-AGENT MODEL 1. INTRODUCTION. The effects of exogenous uninsurable background risk on optimal investment, hedging, and insurance decisions has 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, e.g., Eeckhoudt and Gollier (2000) and Gollier (2001) 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. Holmstrom s (1979) informativeness principle states that the optimal contract between a principal and an agent should be based on sufficient statistics for the agent s action; that is, the contract should use all of the data that is informative about the agent s behavior. If the background risk is exogenous, independent, and free of the agent s actions, then it is not informative about the agent s behavior. Then the informativeness principle, applied without sufficient thought, would seem to suggest that the contract does not depend on the background risk. However, the optimal contract is also affected by the agent s attitude toward risk. If background risk affects the agent s attitude toward 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 analyze the effect of a fair background risk on the design of compensation contracts in a simple 1

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 (e.g., Salanie, 1997, Laffont and Martimort, 2002). 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 contracting environment and is present whether or not the contract is accepted. Previous research examining the effect of background risk on optimal contracts in the principal-agent model is limited. Ligon and Thistle (2007) 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 prudent (exhibits weak DARA), then the introduction of an environmental (contractual) 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 only if prudence is no more than three times absolute risk aversion and the risk premium is weakly concave 2

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. 2. 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 (a = 1) and not working (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 + w) a when the wage is w and the agent s outside income is y. The principal s objective is to maximize expected profit net of the agent s wages while the agent s objective is to maximize 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 u(y + w S ) = U + θ S u(y + w F ) = U θ F, (1a) (1b) where U is the agent s reservation utility when outside income is not risky (i.e. U = u(y 0 ) = u 0 ), 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 U, which in turn is larger than the right-hand side of (1b), it follows directly 3

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. We now assume that the agent s outside income is risky, so that it can be written as y = y 0 + z, where z is a zero mean random variable distributed independently of the outcome. The Arrow-Pratt risk premium, π(y 0 ), is defined by u(y 0 π) = Eu(y 0 + z) and depends on total income. 1 The precautionary premium ψ(y 0 ) is defined by u (y 0 ψ) = Eu (y 0 + z) (Kimball, 1990). 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 (y 0 ) = Eu(y 0 + z) (Nachman 1982, Kihlstom, Romer and Williams, 1981). If u is increasing, risk averse and prudent, then so is û, and, if u is weakly prudent, then û (y 0 ) u (y 0 ). When the agent s outside wealth is risky, the contract for the agent must satisfy Eu(y 0 + z + v S ) = U * + θ S Eu(y 0 + z + v F ) = U * θ F, (2a) (2b) where U * is the agent s reservation utility in the presence of background risk. 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* = u(y 0 ). 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 U* = Eu(y 0 + z) when environmental background risk is present. 1 For notational convenience, we suppress the dependence of π on z. 4

We first need to show the incentive scheme is increasing. Since the right-hand side of (2a) is larger than U*, 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. 2 Then solving (1) yields w S = u 1 (U + θ S ) y 0 and w F = u 1 (U θ F ) y 0. Set Eu(y 0 + z + v j ) = u(y 0 π j + v j ), where π j = π(y 0 + v j ), j = S, F. Then solving (2) yields v S = u 1 (U + θ S ) y 0 + π S = w S + π S v F = u 1 (U θ F ) y 0 + π F = w F + π F. (3a) (3b) If the reservation utility is the same in both cases, then 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. If the background risk 2 The case of the contractual background risk could be modeled 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 (2005) 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 where the return on the stock is a function both of the agent s effort and random noise. In such an interpretation, v S and v F are interpreted as the expected value of the stock compensation in the success and failure states, respectively. Treating the random component of the stock return as a background risk may provide additional insight into optimal agent selection and contract design. 5

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 power, if, and only if, π S π F. This holds if, and only if, the agent s preferences satisfy weak DARA, since DARA implies that the risk premium decreases in wealth. 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 = u(y 0 ) = u 0 in the absence of background risk and U * = Eu(y 0 + z) = u(y 0 π 0 ), where π 0 = π(y 0 ), in the presence of background risk. Let v * S and v * F denote the wages in this case. Substituting into (1), solving, and taking Taylor expansions yields v S * u 1 (u 0 + θ S ) y 0 + π S * u 1 (u 0 + θ S )u (y 0 )π 0 v F * u 1 (u 0 θ F ) y 0 + π F * u 1 (u 0 θ F )u (y 0 )π 0, (4a) (4b) where π j * = π(y 0 + v j * ), j = S, F. The last term captures the effect of the reservation utility background risk on the contract. The principal must still compensate the agent for the background risk born if the contract is accepted, but this is at least partially offset by the background risk in the agent s reservation utility. Thus, expected compensation with an environmental background risk is lower than with a contractual background risk. We now compare contracts where the background risk is environmental to the no background risk case. Proposition 1: Suppose background risk affects reservation utility, U* = Eu(y 0 + 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 6

û (y 0 + v S * ) = û (y 0 ) + θ S û (y 0 + v F * ) = û (y 0 ) θ F (5a) (5b) Then, from (1a) and (5a), u(y 0 + w S ) u(y 0 ) = θ S = û (y 0 + v * S ) û (y 0 ) (6) v S * = u ˆ'( y + s ds 0 vs * 0 ) u '( y + s ds = u(y 0 + v S *) u(y 0 ), 0 0 ) since weak prudence implies û u. Therefore, w S v * S. Similarly, from (1b) and (5b), u(y 0 ) u(y 0 + w F ) = θ F = û (y 0 ) û (y 0 + v * F ). (7) By the same argument, weak prudence implies w F v * F. Combining results, we have w S v * S > v * F w F and v S * - v F * < w S - w F. 3 Prudence implies that the addition of an undesirable risk reduces expected utility by more as wealth increases. With our normalization, w S > 0 > w F. 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 *. 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 3 Proposition 1 also holds if the background risk is multiplicative, y = zy 0, with Ez = 1. If u is monotonic, risk averse and prudent, then so is the derived utility function u ~ (y 0 ) = Eu(zy 0 ). In particular, weak prudence implies u ~ u. See Franke, Schelsinger and Stapleton (2006) for a discussion of multiplicative background risk. 7

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 to 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. Proposition 2: Suppose background risk affects the agent s reservation utility, U* = Eu(y 0 + z). If the risk premium is weakly concave and P 3A, then background risk reduces the agent s expected compensation, V * < W. Proof: Write the left-hand side of (2) in terms of the risk premium, u(y 0 π j + v * j ). Adding and subtracting π 0 in the solution to (2), we can write v * j = w * j + (π * j π 0 ), j = S, F, where w * j is the wage for an agent with no background risk and income y 0 π 0. Let W* = p H w S * + (1 p H )w F *. Then the difference in expected wages is V* W = W* W + [p H π S * + (1 p H )π F * π 0 ] (8) Theile and Wambach (1999, Proposition 1) 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 the term in brackets is non-positive. Consequently, V * < W. If it affects reservation utility, the introduction of background risk has the same effect as if the agent s outside income is reduced by π 0. As Thiele and Wambach 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, (π * j π 0 ). This also tends to reduce expected compensation if the risk premium is concave. 8

Observe that Proposition 2 does not assume prudence, so background risk may make the incentive scheme either steeper or flatter. Proposition 2 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 it is satisfied by hyperbolic absolute risk aversion (HARA) utility functions of the form u(y) = (y + a) 1 γ /(1 γ) for γ ½. Weak concavity of the risk premium imposes the restriction 2A 3-3A 2 P + APT < 0, where T = u (4) /u is the coefficient of termperance, on the first four derivatives of the utility function. 4 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 1 and 2 imply that the introduction of background risk makes the incentive scheme flatter and decreases the agent s expected compensation. If the agent has a HARA utility function, Propositions 1 and 2 imply the introduction of environmental background risk makes the incentive scheme flatter but may increase or decrease expected compensation. We should point out that an important simplification in our analysis is the assumption of two outcomes. The results of Propositions 1 and 2 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 minimizing 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 4 Weak concavity of the risk premium is not inconsistent with the conditions P < 3A. For example, if P = 3A, then both conditions simultaneously hold for T < P/3. Note since v S * > 0 > v F * the wealth level associated with y 0 lies between y 0 + v S * and y 0 + v F * which implies that the concavity of π insures that the bracketed term is non-positive. Gollier (2001, p. 323) suggests that there are some reasons to believe the risk premium is convex in wealth, which would leave the change in expected compensation ambiguous. 9

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 * = u(y 0 ) (resp. U * = û (y 0 )). If the action a is implementable, then Propositions 1 and 2 both continue to hold. 3. 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 risk premium is concave and the coefficient of absolute prudence is no more than three times the coefficient of absolute risk aversion. 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 10

lines of business. Applied to compensation contracts, the background risk might arise from factors such as investment income or a spouse s earnings. 5 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 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, 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 2 and reduce the variability of compensation if they satisfy the conditions of Proposition 1. Thiele and Wambach (1999) 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. 5 These background risk are not immutable in the long run. Policyholders can invest in different types of human capital, regulated firms can enter or exit unregulated markets and businesses, and employees can change their investment portfolios and spouse s income. 11

Consider health expenditure risk as an environmental background risk that can add to or reduce income. If agents satisfy both the conditions of Proposition 1 and Proposition 2, 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 1 but not Proposition 2, then the background risk increases expected wage costs but reduces their variability. In such a situation large employers, with essentially risk neutral objective functions, might be willing to provide health insurance since expected wages would fall, but small employers, with a risk averse objective function, would find the lower variation in wages in the presence of the background risk a benefit and, at the margin, be less likely to do so. Employers may also have an interest in the overall risk level of society. Consider, for example, the threat of globalization. If agents satisfy the conditions of Proposition 2 then, the threat of globalization may not only be helpful in curbing high wage demands directly, it may also reduce wages through the higher environmental background risk perceived by the workers. 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 12

the background risk through the equity investment may lower expected compensation and its variability if the conditions of Propositions 1 and 2 are met. Our results are thus consistent with the large body of literature that suggests that the performance sensitivity of an executive s contract should depend on the other risks he/she faces (see Prendergast, 1999 for a review). Finally, it is interesting to compare the results here to the results for the standard portfolio problem. In the standard portfolio problem the agent solves max α E{u(y 0 + αr)}, where α is the proportion of wealth invested in the risky asset and r is the return on the risky asset. In order for the introduction of background risk to reduce the optimal level of investment, conditions such as standard risk aversion (Kimball, 1993) or risk vulnerability (Gollier and Pratt, 1996) 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 analyzed here, the effect of background risk depends on how it affects the level of utility, so that weaker restrictions on preferences are sufficient. 13

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