Market Access ** Helmut Laux/Robert M. Gillenkirch/Matthias M. Schabel*

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1 Incentive Bidding Compensation Strategies Helmut Laux/Robert M. Gillenkirch/Matthias M. Schabel* Incentive Compensation, Valuation, and Capital Market Access ** Abstract In this paper we consider linear sharing rules for incentive compensation, when principal and agent are potentially restricted from short sales. We derive subjective valuation lines for performance shares, based on graphical analyses, and present results with respect to implementing effort and investment incentives. When the agent is restricted from short sales, he may reject an investment with positive market value, and higher performance shares may not induce proper incentives. Relative Performance Evaluation will improve incentives and risk sharing, but there will generally be no clear definition of a suitable benchmark. JEL-Classification: D86, G11, G32. Keywords: Incentive Compensation; Portfolio Allocation; Principal-Agent-Theory; Short Sales. 1 Introduction Incentive compensation makes a trade-off between the two fundamental goals of risk sharing and motivation. This trade-off, which has been extensively analyzed in the agency literature, is frequently illustrated in a separation of ownership and control setting in which a nondiversified manager is hired by a group of well-diversified investors, who are (approximately) neutral to firm-specific risk. Because the manager is averse to firm-specific risk, efficient motivation implies inefficient risk sharing, and vice versa. In the basic principal-agent model, researchers have analyzed the conflict between motivation and risk sharing by assuming that the agent has no access to the capital market where he could, by trading risky assets, (partially) hedge compensation risk. If the agent and * Helmut Laux, Professor Emeritus, Goethe-University of Frankfurt. Robert M. Gillenkirch, Chair for Management and Accounting, Department of Accounting, University of Osnabrück. Matthias M. Schabel (Corresponding author), Chair for Accounting and Management Information Systems, University of Applied Sciences Frankfurt, Nibelungenplatz 1, Frankfurt am Main, ** We thank an anonymous referee for valuable comments. See Laffont and Martimort (2002), ch. 4, with references. SBR 61 October

2 H. Laux/R. M. Gillenkirch/M. M. Schabel principal had free access to a perfect and complete market, by trading assets, they could always efficiently share risks irrespective of the incentive contract written between them. Recent studies incorporate assumptions about a manager s (restricted) access to financial markets into their analyses of incentives, especially of stock-based compensation. These studies are closely related to the studies that point out the potential ineffectiveness of incentives due to the manager s private capital market transactions. The relation between an agent s capital market access and the potential ineffectiveness of an incentive contract can be best illustrated by assuming that the agent receives stock-based compensation and can freely trade in his firm s stock. In this case, the agent will always hold a share of the firm s stock that maximizes his expected utility from his portfolio, which is, apart from wealth effects, independent of his compensation. This independence makes compensation irrelevant with respect to incentives 2. Complex incentive problems arise when markets are incomplete and imperfect. In such markets, a manager who receives a risky income from incentive compensation will simultaneously plan the firm s investment strategy and his private portfolio holdings. Thus, investors, in designing incentive compensation, will have to anticipate the manager s hedging behavior and his hedging opportunities. Additionally, an investor must account for the change in his own total risky income that results from the change in his residual claim, and his trading opportunities to transform the residual claim s risk when he designs incentive compensation. Much of the debate about executive compensation refers to the question of whether compensation gives managers incentives to maximize value. Value maximization is a goal that all shareholders accept, but only if markets are frictionless and if risk sharing is universally linear between all shareholders. Under these assumptions, an investment is profitable if, and only if, its market value is positive (De Angelo (1981) and Laux (2006)). Unrestricted capital market access, and especially short sales 3, are critical if value maximization is to be accepted by all shareholders. However, capital market participants face various short-sale restrictions, which in the financial crisis have been rigorously extended to make a large number of stocks unshortable in several countries (e.g., the U.S., U.K., and Germany) 4. If short sales are 2 See Neus (1989; 1996); Laux (1990; 1991); Gillenkirch (1999). 3 A short sale of a security means selling the security without owning it. The short seller has to borrow the security from a broker or an institutional investor. At a later date, the short seller will buy the security at the current market price and return it to the lender. In the U.S., when the short seller borrows the security, a margin account will be established, and the short seller has to deposit the proceeds from the short sale in the account. Additionally, the short seller subsequently has to fulfill margin requirements. See, e.g., Finnerty (2005). 4 Private investors can be directly or indirectly restricted with respect to short sales. In Germany, investment funds face short-sale restrictions due to national (Investment Law, 59) and European codes (EU directive Undertakings for Collective Investments in Transferable Securities, 42). Recently, the German Financial Services Supervisory Authority (Bundesanstalt für Finanzdienstleistungsaufsicht), has forbidden short sales for a large list of assets; see Allgemeinverfügung der BaFin vom 19. September Even if a stock can be shorted in general, there may be restrictions on actually selling it. In the U.S., a stock can be sold short only if its last movement was a plus tick ; see Finnerty (2005). 336 SBR 61 October

3 Incentive Compensation restricted, then investors may have subjective values on a firm s investments that deviate from market values. This fact also holds for incentive compensation. When short sales are restricted, neither the manager nor the shareholders will generally value the firm s investment opportunities at their market value. Therefore, incentive compensation design must account explicitly for how both the manager and the shareholders will hedge their respective claims in the firm s risky cash flow, given that they face potential short-sale restrictions or other market imperfections. When the capital market is imperfect, the subjective valuation of incentive compensation and equity shares is fundamentally important for evaluating the efficacy and efficiency of a compensation contract. In this paper we give particular attention to this issue. Thus, individual attitudes to risk and individual initial portfolio positions become relevant; relying on market value analyses in general will be methodologically incorrect. We show how optimal incentive compensation can be derived under the assumption of capital market imperfectness. Furthermore, we show how the optimal contract depends on the risk aversions and private portfolio holdings of the manager (the agent) and the owner (the principal). We focus particularly on concentrated ownership and consider a sole owner as the standard case for the analyses. In this case, not only the agent s, but also the principal s, private portfolio transactions may be impaired by short-sale restrictions, and the firm s value to the principal will thus be generally lower than the market value. The following analyses can be directly applied to the case of the principal being a (representative) shareholder with a small share in the firm, such that individual utility maximization and value maximization are (approximately) equivalent even when the principal faces short-sales restrictions. The paper is organized as follows. In Section 2, we briefly discuss studies that relate to ours. In Section 3 we outline the setting. In Section 4, we will use graphical analyses and illustrations to contrast market valuation and subjective valuation of a risky investment. When the respective investor is short sale restricted, he will value a risky investment at its market value as long as his share of the investments risky cash return is sufficiently small, and if the risky return can be duplicated with traded assets. However, with an increasing share, the investor will be unable to fully hedge the investment s risk, i.e., he cannot entirely sell the duplication portfolio of the investment by reducing his private asset holdings. As a consequence, the investor will subjectively value the investment, and the subjective value will be smaller than the market value. The discount in the subjective value will be the larger, the more structurally distorted the duplication portfolio of the investment is relative to a well diversified portfolio, which the investor would hold without investment. In Section 5, we apply the results of Section 4 to the analysis of the incentive effects of a performance share. Here the standard result with respect to the trade-off between risk sharing and motivation applies to the short sale restricted agent and principal. Pareto-efficient risk sharing maximizes the sum of the subjective values of the agent s and the principal s performance share. This may insufficiently motivate the agent, such that the principal loses value due to the agent s short-sale constraints. If the agent is short sale restricted, then he may value his performance share at a subjective value that is significantly smaller SBR 61 October

4 H. Laux/R. M. Gillenkirch/M. M. Schabel than its market value. The same will hold for the principal unless she holds only a small fraction of the firm s equity. In this case, unanimity between principal and agent is not guaranteed: If the principal is strictly market value oriented, then the agent will potentially underinvest. However, if the principal s short-sale restriction is relevant, inducing the agent to invest may result in overinvestment, as the principal may value the residual negatively. The analysis is extended to differing levels of effort with alternative effects on the distribution of the investment s cash return. If the agent has restricted access to the capital market, incentive compensation generally can be improved by Relative Performance Evaluation (RPE), which we discuss in Section 6. RPE shields the agent from compensation risk that is unrelated to his effort. If RPE is feasible, using the agent s performance as the sole measure in the incentive contract becomes inefficient with respect to risk sharing. That is, risk sharing is improved by introducing a benchmark into performance evaluation, and the difference between the agent s performance and the benchmark serves as the relevant measure for incentive compensation. In RPE, the benchmark will generally be a portfolio of assets (e.g., an index). In this case however, principal and agent could equivalently internally trade the benchmark portfolio. We will consider such trades between principal and agent (or, RPE, equivalently) and show within our framework how they affect the agent s and the principal s subjective investment valuation. Section 7 concludes. 2 Related Studies In this paper we build on the work of Laux and Schabel (2009), who provide portfolio-theoretic analyses of the valuation of risky income streams by a short-sale-restricted investor. Laux and Schabel do not consider risk-sharing or incentive issues. Here, we extend the analyses to the design of performance contingent compensation in an explicit capital market context. Issues of restricted capital market access in incentive compensation have been recently considered in two streams of the literature. First, researchers have examined how costefficient are stock and stock option compensation, once a manager s restriction on capital market access is considered. As we noted above, if a manager is restricted from trading in the market, he will assign a subjective value to the payoff from compensation that will differ from its market value. (In general it will be smaller.) Several papers have examined how large the difference between market value and subjective value can be, and which determinants drive this difference 5. For example, Hall and Murphy (2002), by contrasting the market value of stock options with the subjective value of the same options for an executive, show that the subjective value of the options to a manager may be far smaller than the Black-Scholes fair value of the options, which these authors use as a proxy for the opportunity cost of the options to shareholders. Hall and Murphy, and other authors 6, 5 See Lambert et al. (1991); Meulbroek (2001); Hall and Murphy (2002). 6 See, e.g., Huddart (1994); Ingersoll (2006). 338 SBR 61 October

5 Incentive Compensation also analyze the exercising strategies of executives. Subjective valuation will lead to early exercise, and this fact further contributes to a large discount from fair values under subjective valuation 7. The approach in Hall and Murphy takes a manager s effort and investment decisions as given. In contrast, in this paper we consider linear sharing rules (i.e., stock instead of options), but we explicitly analyze how the subjective valuation of a compensation package by the executive is related to his effort and investment incentives. Second, Relative Performance Evaluation (RPE) has received substantial attention in the literature 8. RPE is closely related to issues of restricted market access, because RPE serves as a substitute for the agent s hedging his compensation risk by private portfolio transactions, and vice versa 9. Maug (2000) and Garvey and Milbourn (2003) consider RPE in a LEN model in which the agent chooses his effort, and where he can privately allocate his wealth to a riskless asset and the risky asset from the market model (the market portfolio). Maug shows that when transaction costs are absent, the agent can realize homemade RPE by trading in the market, so explicit RPE in performance evaluation may become irrelevant. RPE will be present only if the agent is short-sale restricted. Garvey and Milbourn use a similar analysis to derive comparative static hypotheses concerning the pay performance sensitivities of management compensation. In the agency literature, analyses of RPE implicitly or explicitly assume that the principal represents shareholders who are well diversified and not averse to firm-specific risk, and thus value shares at their market values 10. However, if there is a sole owner or if ownership is concentrated, then the principal will have an interest in passing risk to the agent, and RPE affects both the agent s and the principal s valuation of the firm s investment opportunities; see Section 6. 3 Setting To analyze the valuation and incentive effects of compensation in an imperfect market, we use a one-period setting and the LEN-framework (Holmström and Milgrom (1987; 1991); Spremann (1987)). I.e., we make specific assumptions about preferences (negative exponential utilities, E), cash flow distributions (normal, N), and considered sharing rules (linear, L). Capital market imperfections refer to short-sale restrictions for both the agent and the principal 11. Both have homogeneous expectations, but their expectations 7 An extension of Hall and Murphy (2002) is Henderson (2005), who assumes that the executive is restricted from trading in his own stock, but that he can trade in the market portfolio. 8 Theory on RPE goes back to Holmström (1982). For evidence on RPE see Garvey and Milbourn (2003), and Albuquerque (2005), with further references. 9 Moreover, RPE has similar implications with respect to risk sharing as tournament based incentives (Kräkel (1998; 2004); Kräkel and Schauenberg (1994); Kräkel, Schauenberg, and Wilfing (1998)). In fact, extending a partial equilibrium to a total equilibrium model of executive compensation, RPE implies the introduction of tournament based incentives into compensation. 10 Additionally, it is implicitly assumed that management compensation accounts for only a small fraction of the firm s cash returns. However, if compensation is a major cost component for the firm, and if RPE is universally introduced into compensation, then shareholders must bear additional systematic risk. 11 Differences in transaction costs may have similar effects as short-sale restrictions. SBR 61 October

6 H. Laux/R. M. Gillenkirch/M. M. Schabel may differ from third parties expectations, due to (differences in) information costs. The LEN-assumptions allow for a graphical representation of the principal s and the agent s decision problems. Although the L-assumption generally implies that the optimal (linear) contract is third best only, we choose the framework for the convenience of mean-variance representations of utilities 12. An individual investor (the principal) hires a manager (the agent) to manage the firm. The agent has an investment opportunity that requires an initial capital expenditure at time t = 0, x 0, financed by the principal, and yields a risky cash return x 1 at time t = 1. We assume that x 1 is normally distributed. x 0 and x 1 depend on the agent s effort; the relationship is specified in Section 5. In the following, we aggregate x 0 and x 1 to a single measure of performance, residual income. Defining the opportunity cost of capital as the risk-free rate r, the project s residual income amounts to x 1 (1 + r) x The firm s residual income is denoted by y. It amounts to ỹ = x 1 (1 + r) x 0 + ε 1 (1) with the investment and y = ε 1 without. For simplicity, we assume that the risk ε 1 is purely idiosyncratic and has zero mean, and that it has a negligible effect on utilities. Since E( ε 1 ) = 0, E( y) = E( x 1 ) (1 + r) x 0 is the investment s risk premium (in absolute terms). Using residual income for performance evaluation is equivalent to evaluating the agent explicitly at x 0 and x 1 : Any linear compensation with equal explicit shares of x 0 and x 1 can be replicated with a single share of y. (Unequal shares for x 0 and x 1 would violate goal congruence.) Residual income accounts for the investment s capital expenditure by reducing the cash return by the compounded value of x 0. Compounding is riskless, as is x 0. The agent is evaluated at the firm s residual income at time t = 1. The principal gives the agent a share f of y, and principal and agent agree on a transfer payment K from the principal to the agent at time t = 0. K > 0 represents a fixed wage. The agent s income from compensation and the principal s residual claim at time t = 1 are given by: f ỹ + (1 + r) K, (2) (1 f) ỹ (1 + r) K. (3) 12 Hemmer (2004) discusses strengths and weaknesses of the LEN-assumptions. While the model s simplicity allows to avoid technical problems and to focus on economic issues, Hemmer shows that the model produces results that are (seemingly) intuitive, but contradict the results from an analysis that is unrestricted with respect to the contract shape. However, as our focus is on the implications of restricted capital market access for the agent s and the principal s valuation of investment opportunities, allowing for an unrestricted contract shape would prohibit a mean-variance-analysis, but would not change the main results. 13 That is, the capital charge in residual income carries no risk premium. Under general assumptions, this zero risk premium is necessary to achieve goal congruence between principal and agent. For formal analyses, see Velthuis (2004), and Christensen, Feltham, and Wu (2002). 340 SBR 61 October

7 Incentive Compensation Both agent and principal have access to the capital market. Both can borrow and lend at the risk-free rate r, and both can trade in risky assets. The cash return from risky asset i at time t = 1, i.e., its cum-dividend price, is P 1i. We assume that all P 1i, i = 1,..., N, are jointly normally distributed. Expectations with respect to all P 1i are homogeneous. A short sale is defined as selling an asset i at time t = 0, at price P 0i, without actually owning the asset, such that the seller has to borrow the asset to be able to deliver the shorted asset to the buyer. Both principal and agent may be restricted with respect to short sales, which will be specified below 14. Before cooperating, principal and agent have private wealth w 0A and w 0P, respectively. Both principal and agent have negative exponential utilities with coefficients of absolute risk aversion a A and a P, respectively. The agent is effort averse, i.e., effort e causes a disutility to the agent that is equivalent to a private cost c(e) at time t = 0. From the LEN assumptions, it follows that preferences can be represented by the certainty equivalents CE A = E( w 1A ) 1 _ 2 a A Var ( w 1A ) (1 + r) c(e), (4) CE P = E( w 1P ) 1 _ 2 a P Var ( w 1P ). (5) In Equation (4), w 1A and w 1P denote the agent s and the principal s period-1-aggregate wealth from both their performance shares and private portfolio holdings, respectively. Initial wealth levels, preferences, and short-sale restrictions are common knowledge. The principal offers a share and a fixed transfer K to the agent, who accepts the offer if his participation constraint is not violated. Then, the agent chooses his effort level and decides about the investment needed to maximize Equation (4). At time t = 1, cash returns are realized. If we ignore incentive compensation and agency problems, then the assumptions made constitute (a special version of) the CAPM. The reason is that short-sale restrictions have no effect on the market portfolio (Ross (1976)). Consequently, without compensation, both principal and agent will hold a share in the market portfolio, which implies linear risk sharing between principal and agent. 4 Market and Subjective Project Valuation under Short-Sale Restrictions 4.1 Graphical Analysis of the Portfolio Problem To understand the implications of restricted capital market access for the design of incentive compensation, it is useful to first analyze how the agent (the principal) values the performance share (the residual), and how the agent (the principal) simultaneously plans his (her) private portfolio holdings. To do this, we first abstract from the agent s effort and effort costs. 14 Even if a short sale is not forbidden, it may cause large transaction costs. The mechanism for short selling stock is borrowing the stock, and borrowing costs may be prohibitively high. See, e.g., D Avolio (2002). SBR 61 October

8 H. Laux/R. M. Gillenkirch/M. M. Schabel From the assumption of a linear sharing rule it follows that both agent and principal receive a proportional share f and 1 f, respectively, of the investment s residual income. We analyze the agent s valuation of his performance share and of the investment opportunity in two steps: First, we derive the optimal private portfolio allocation ex ante. Second, given realization of the project, we analyze the portfolio reallocation. If the ex post utility is smaller than the ex ante utility, then the project is disadvantageous. By analogy, we can check for unanimity between agent and principal concerning the investment opportunity. We do so by analyzing the principal s portfolio position and utility before (ex ante) and after (ex post) project realization. We analyze both the ex ante and the ex post calculus graphically and use the investor s (the agent s or the principal s) performance share z as a representative for both f and 1 f. Hence, we drop the subscript A or P in w 0 and w 1. In the analysis, we use absolute terms instead of rates of return, such that, for example, an efficient frontier refers to the respective investor s wealth and shifts when wealth increases or decreases. Optimal Portfolio Position Ex Ante Without the investment and ignoring the term ε 1, the investor s income at time t = 1 is given by N w 1 = (1 + r) w 0 + i = 1 q i [ P 1i (1 + r) P 0i ]. (6) q i denotes the number of assets of type i the investor buys at time t = 0 at price P 0i, N is the total number of assets. In a (µ, σ)-diagram with µ on the x axis and σ on the y axis, and with µ = E( w 1 ) and σ = Var( w 1 ), all efficient portfolios lie on a straight line starting at µ = w 0 (1 + r) and σ = 0. We denote this efficient frontier as the Basis Efficient Frontier (BEF ). All portfolios on this frontier are characterized by combinations of the riskless asset and the tangency portfolio and thus have the same composition of risky assets. The ex ante risk premium on asset i is given by E ( P 1i ) (1 + r) P 0i. In the same diagram, the investor s certainty equivalent, CE = E( w 1 ) 1 _ 2 a Var( w 1), (7) has concave indifference curves as graphical equivalents. For any value of σ, these indifference curves will have equal slopes. The larger is the coefficient of risk aversion a, the smaller the slope. The horizontal distance between two indifference curves is the same for every level of σ. The point where an indifference curve starts on the x axis represents the corresponding certainty equivalent, so the distance between two indifference curves on the x axis represents the difference in the corresponding certainty equivalents. The tangency point between an indifference curve and the BEF constitutes the respective investor s (the agent s, the principal s) optimal portfolio. We denote this tangency point for the ex ante portfolio optimization as T ex ante. Without loss of generality, we assume that the tangency portfolio is the market portfolio, i.e., the BEF has an identical slope for both the agent and the principal. The larger is a, the less steep are the investor s indif- 342 SBR 61 October

9 Incentive Compensation ference curves, and the smaller is the amount of wealth which the investor will invest into the market portfolio. Optimal Portfolio Position Ex Post For our ex post analysis of the investor s (the agent s, the principal s) portfolio dispositions and certainty equivalents when the project is realized, we assume that the project s cash return x 1 can be perfectly duplicated in the market, such that, given free access to the market, an investor will decide on realization based on the project s market value MV. We denote the risk premium of the duplication portfolio, i.e., the portfolio of traded assets that duplicates x 1, by RP DP. We can then write the investment s market value net of the capital expenditure as follows: MV = (1 + r) 1 [E( x 1 ) RP DP ] x 0 = (1 + r) 1 [E(ỹ) RP DP ]. (8) That is, the investment s market value is equal to the discounted difference between its risk premium, E(ỹ) = E( x 1 ) (1 + r) x 0, and the risk premium of the duplication portfolio. We assume that both principal and agent have homogeneous expectations for the project s cash return, and thus they agree on the composition of the duplication portfolio. To start the graphical analysis, we assume MV = 0. Now, if the project is realized and the respective investor receives a share z of its residual income ỹ, and if the investor does not adjust his private portfolio holding, then his expected income increases by z RP DP, and the standard deviation of his income increases by z σ, where σ = Var ( i q i P 1i + x 1 ) Var ( q i P 1i ). (9) The investor would reach the same income position if he bought z % of the project s duplication portfolio. In what follows, we assume without loss of generality that all assets are positively correlated with the market portfolio, which is a realistic assumption for stocks and bonds. This assumption ensures positive risk premia for all assets, i.e., each asset s expected return exceeds the riskless rate. By buying (short selling) assets, an investor s expected endof-period wealth strictly increases (decreases), which makes some expositions easier. If the investor could fully hedge his performance share, he would value the project at MV, and he would adjust his portfolio holdings by (short) selling z% of the project s duplication portfolio, and by investing the proceeds into the riskless asset. Therefore, and because we assume that MV = 0, his income would not change in either t = 0 or in t = 1. If we suppose that the investor is restricted from short selling such that he cannot fully hedge his performance share, then we can graphically derive his optimal portfolio position and expected utility as follows (see Figure 1) 15 : 15 With short-sale constraints, the market portfolio is still mean-variance efficient. See Ross (1976). For results on the shape of the efficient frontier of risky assets (without a riskless asset) under short-sale restrictions, see Dybvig (1984). i SBR 61 October

10 H. Laux/R. M. Gillenkirch/M. M. Schabel We start at a point P, where the investor s sole risky income is his share z of the investment s cash return. That is, the investor has sold all his private portfolio holdings at their respective market prices and has invested the proceeds, w 0, entirely in the riskless asset. Thus, the x coordinate of P is (1 + r) w 0 + z RP DP, and its y coordinate is z Var ( x 1 ) = z σ DP. P does not lie below, but generally lies above the investor s BEF. (It will lie on the BEF only if the project s cash return itself represents an efficient portfolio, i.e., has equal composition as the tangency portfolio.) In Figure 1, P lies above the BEF. Thus, the duplication portfolio is inefficiently diversified. The less diversified or the more structurally distorted (relative to the market portfolio) the duplication portfolio is, respectively, the larger will be the distance between P and the BEF. Figure 1: Investor s portfolio positions ex ante and ex post σ TP MEF ex post T z σ DP BEF P ex ante T 0 (1 + r ) w 0 μ z RPDP ex post CE ex ante CE We derive the Modified Efficient Frontier (MEF) for the investor by computing the minimum-variance portfolio of the investor s performance share and private asset holdings for every expected income above (1 + r) w 0 + z RP DP. The MEF merges into the BEF at point TP. At TP, the investor s portfolio is composed such that his performance share is part of the market portfolio, i.e., TP represents the smallest amount of the market portfolio that embodies z % of the project s duplication portfolio. To the upper right of TP, the investor can realize identical portfolios, irrespective of his decision about project realization. The less diversified the duplication portfolio, the larger is the x coordinate of TP (Laux and Schabel (2009, )). The curve between the points P and TP is strictly convex (Laux and Schabel (2009, )). Whether the curve is monotone depends on the correlation between x 1 and the assets in the market portfolio. If x 1 is nonnegatively correlated with every risky asset, then the curve is strictly monotone increasing, as shown in Figure 1. Thus, in Figure 1, the curve between P and TP is the MEF. However, if x 1 were negatively correlated to some assets, then the curve 344 SBR 61 October

11 Incentive Compensation would have a local minimum between P and TP, and the MEF would be the monotone increasing part right from the minimum. Left from point TP, the slope of the MEF is strictly smaller than the slope of the BEF. With increasing expected cash return, the structure of the investor s portfolio more and more approximates the structure of the market portfolio, and the risk premium per unit of risk strictly increases until it reaches the market price of risk, i.e., the risk premium per unit risk of the market portfolio. The investor s optimal portfolio position is characterized by the point of tangency between the MEF and one of his indifference curves. We denote this point by T ex post. The ex post optimal portfolio will be the market portfolio if, and only if, T ex post lies to the upper right of TP. Otherwise, as is the case in Figure 1, the investor will hold an imperfectly diversified portfolio ex post, because short-sale restrictions prevent him from perfectly hedging the cash flow from the investment. Thus, his utility level, or, equivalently, his certainty equivalent, will decrease; see Figure 1. If for some assets short selling is not restricted, then the investor can improve his position by short selling the respective assets. The investor will also be able to partially hedge the investment s risk if he can buy assets that are negatively correlated with x 1. Thus, assets that are negatively correlated with x 1 serve as a partial substitute for short sales. When short sales are feasible, we must interpret point P in Figure 1 as the investor s position after optimal short selling. The more relaxed the investor s short-sale constraints are, the closer will P be to the x coordinate (1 + r) w 0 in Figure 1, and the closer will the MEF be to the BEF. This implies that both the principal s and the agent s portfolio problems and corresponding MEF s may differ considerably if there are differences in short-sale constraints. 4.2 Subjective Project Valuation If T ex post represents an indifference curve with higher utility than T ex ante, then the investor will assign a strictly positive value to the project. This however cannot be the case when the project s market value is zero, as we assumed for the moment. Instead, the project s subjective value to the investor is negative whenever T ex post lies to the lower left of TP. Only if T ex post lies to the upper right of TP will the investor assign the full market value to the investment. Graphically, the time t = 1 subjective value of the investment to the investor is the negative value of the distance from the certainty equivalent ex post (CE ex post ) to the certainty equivalent ex ante (CE ex ante ), i.e., the x coordinate of the indifference curve corresponding to T ex post and the indifference curve corresponding to T ex ante. We denote the corresponding time t = 0 subjective value by SV, i.e., SV = (1 + r) 1 (CE ex post CE ex ante ). (10) The more risk averse the investor, the less steep are his indifference curves, and the more negative will be the project s subjective value to the investor. SBR 61 October

12 H. Laux/R. M. Gillenkirch/M. M. Schabel Of course, whether T ex post lies to the upper right of TP depends not only on the investor s risk aversion, but also, crucially, on z, i.e., the investor s share in the residual income from the investment. Figure 2 repeats the graphical analysis of Figure 1 for three different levels of z, z = 1/3, z = 2/3, and z = 1. Varying z merely changes the scale. In Figure 2, only for the smallest of the three considered levels, z = 1/3, does T ex post lie to the upper right of TP, such that the investor assigns the full market value to the investment. That is, for both z = 2/3, and ex post ex z = 1, the ex post certainty equivalents CE z = 2/3 and CE post z = 1 are smaller than the ex ante certainty equivalent CE ex ante ex post = CE z = 1/3. Figure 2: Investor s portfolio positions ex post for differing levels of the performance share z T ex post z = 1 TP z = 1 P ( z = 1) P ( z = 1/3) P ( z = 2 /3) TP T z = 1/3 ex post z = 1/3 T TP z = 2/3 ex post z = 2/3 0 (1 + r) w 0 C E ex post z = 1 C E ex post z = 2/3 C E ex post z = 1/3 = C E ex ante µ We note that the loss in the certainty equivalent nearly doubles when z is increased from z = 2/3 to z = 1, compared to an increase from z = 1/3 to z = 2/3. This relation implies that SV(z) is a concave function, which we will generally assume in the following. The subjective valuation of a zero-market value investment can easily be generalized to any positive market value of the project. Now, the starting point P for constructing the MEF has the x coordinate (1 + r) (w 0 + z MV ) + z RP DP and the y coordinate z σ DP as in Figures 1 and 2. Correspondingly, the MEF is shifted to the right by the compounded share of the market value, (1 + r) z MV. Due to the investor s constant absolute risk aversion, the subjective time t = 1 project value increases by (1 + r) z MV, and the subjective time t = 0 value increases by z MV. 346 SBR 61 October

13 Incentive Compensation For a given (positive) market value of the investment, there will be a critical performance share z* such that, for all z z*, the investor will subjectively value the project at its market value, since the short-sale constraints do not prevent him from being efficiently diversified. (In Figure 2, z* lies between 1/3 and 2/3.) However, if z > z*, then the subjective value will always be smaller than the market value: SV(z ỹ) = z MV z z* and SV(z ỹ) < z MV z > z*. The smaller is the investor s risk aversion, the larger will be z* and the less restricted the investor is for short sales. Based on the preceding analyses, we can illustrate the investor s subjective valuation of his share of the investment as a function of the investor s risk aversion, the project s market value, and the investor s performance share z. Figure 3 graphs the subjective valuation for an investment with positive market value for varying levels of the performance share. Figure 3: Investor s subjective value of the investment as a function of the performance share z SV MV MVL SVL risk discount 0 z* ẑ 1 z SVL for MV = 0 In Figure 3, the Market Value Line (MVL) displays the market value z MV of the performance share z, and the Subjective Value Line (SVL) represents the subjective value SV(z) of the performance share to the investor. The distance between the MVL and the SVL is zero for all z z*, but strictly positive for all z > z*. As we noted earlier, we assume that the SVL is concave. Depending on the investment s profitability and risk, the SVL may or may not have an interior maximum at a value ˆz < 1. Figure 3 illustrates the case ˆz < 1. For ˆz < 1, there may be two levels of z with identical subjective project values. In addition, in Figure 3, the investment s subjective value to the investor is nonnegative for all levels of z. This need not be the case, either. In general, the more structurally distorted the investment is relative to the market portfolio, the more will SBR 61 October

14 H. Laux/R. M. Gillenkirch/M. M. Schabel short-sale restrictions impair the investment s value. Hence, the subjective value may become negative when z exceeds a critical level. 4.3 Risk Sharing The agent s and the principal s subjective valuation of the investment follow from the preceding analysis by substituting f or 1 f for z, respectively, and by considering the players respective indifference curves in the graphical analysis. Thus, the Subjective Value Lines SVL A and SVL P for the agent and the principal, which represent the subjective values SV A (f y) and SV P ((1 f) y), can be derived and displayed in one graph. In this graph, the agent s SVL starts in f = 0 and ends in f = 1, and the principal s SVL starts in f = 1 and ends in f = 0. Differences in the shapes of the SVLs result from differences in risk attitudes and in individual short-sale restrictions. We define a performance share f to establish pareto-efficient risk sharing, if changing f, in combination with the corresponding adjustments in the agent s and the principal s portfolios, cannot increase one party s utility without decreasing the second party s utility. We denote the share that establishes efficient risk sharing by f PE. f PE depends on the agent s and the principal s risk aversion, but also on the players hedging opportunities. We can use the SVLs to graphically derive pareto-efficient risk sharing. Pareto-efficiency will maximize the sum of the subjective values. We denote this sum by ASV (Aggregate Subjective Value) and the corresponding Aggregate Subjective Value Line by ASVL. Due to the assumption of constant absolute risk aversion, the ASVL is independent of the fixed money transfer K. The maximum of the ASVL is equal to the investment s market value if, and only if, both SVL P and SVL A do not deviate from the respective Market Value Lines (MVLs) at this maximum. Figures 4a and 4b display the individual SVLs and the ASVL for two cases. In the figures, we add the respective MVLs. Figure 4: Subjective Value Lines for Agent and Principal, Aggregate Subjective Value Line, and efficient risk sharing SV MV MVL P ASVL SV MV MVL P ASVL MVL A MVL A SVL P SVL P SVL A SVL A 0 1 z f * 0 z * PE 1 A f * A f f P Fig. 4a Fig. 4b PE f [0, f * A ] 348 SBR 61 October

15 Incentive Compensation In the first case (Figure 4a), we assume that the principal values the investment at its market value for all levels of the performance share. This case represents the assumption of dispersed ownership, in which every shareholder (being part of the group of shareholders that is the principal ) holds only a small fraction of the firm s equity. In Figure 4a, the ASVL starts at a level where the ASV is equal to the market value, i.e., the maximum of the ASVL reaches the investment s market value. For every performance share f f * A, the agent values his performance share at its market value such that the ASV is equal to the market value, and thus every share f f * A is pareto-efficient. In the second case (Figure 4b), the SVL P and the MVL P do not coincide for all levels of f, and the maximum of the ASVL does not reach the market value. Principal and agent are too restricted by their respective short-sale constraints, such that for pareto-efficient risk sharing, both assign a subjective value to the investment that is lower than the market value. In Figure 4b, the agent values his performance share at its market value for all f f * A, and the principal values her residual at its market value for all 1 f 1 f * P f f * P. As f * P > f * A, there is a unique maximum of the ASVL, i.e., there is a unique pareto-efficient performance share f PE. This case indicates that the firm s equity is closely held by one or a few shareholders only. The more risk averse the agent (the principal), the lower (the higher) will be f PE. Additionally, the more restricted the agent (the principal) is with respect to short sales, and the more structurally distorted the duplication portfolio, the lower (the higher) will be f PE. Note that the standard solution for pareto-efficient risk sharing, that is a P f PE =, (11) a P + a A where a P and a A are the Arrow-Pratt measures of absolute risk aversion for the principal and for the agent, respectively [see Equations (4) and (5)], is not relevant in our setting. The reason is that when we explicitly consider the capital market, the principal and agent can either freely trade on the market such that explicit risk sharing by setting the performance share becomes irrelevant, or else they cannot freely trade on the market, but their individual hedging opportunities are relevant for risk sharing, since efficient risk sharing has to account for the agent s and the principal s private portfolio positions. 5 Inducing Effort 5.1 Dichotomous Effort In this section we introduce an incentive problem to the preceding analysis: to realize the project, the agent has to expend effort e. The agent s effort is unobservable to the principal, and the agent is effort averse. His disutility from effort e is equivalent to a personal cost c(e) [see Equation (4)], where c(0) = 0 and c (e) > 0. To motivate the agent and to induce an efficient investment decision, the principal gives the agent a share of residual income according to Equation (2). We first assume that effort is dichotomous, i.e., low (e shirk ) or high (e work ). Only with high effort is the project available. That is, the agent either shirks, or works and realizes the project. Further, we assume that c(e shirk ) = 0 and SBR 61 October

16 H. Laux/R. M. Gillenkirch/M. M. Schabel c(e work ) = c > 0, and that a forcing contract is not available. The incentive compatibility constraint states that the subjective value of the performance share to the agent must be at least c, otherwise he will not realize the investment: SV A (f ỹ) c. (12) We denote the set of performance shares that fulfill the incentive constraint (12) by F, and denote the minimum share that fulfills (12) by f IC min. From Figure 3 it follows that there will generally be a critical level of c for which F = { }, i.e., for which the agent cannot be motivated. In the following, we assume that c is below this critical level. The principal must also account for the agent s participation constraint. We assume that the agent s minimum utility is zero. Hence, the principal s subjective value of the investment, net of compensation, is equal to the aggregate subjective value ASV less the agent s cost of effort, c. Thus, the principal s optimization problem is: Max SV P ((1 f) ỹ) + SV A (f ỹ) c, subject to SV A (f ỹ) c, (13) f and subject to the aggregate subjective value less cost c being nonnegative, which we assume in the following. We characterize the optimal performance share, f opt, as follows: If the performance share f PE that maximizes the ASVL fulfills the incentive constraint, i.e., if f PE F, the maximum of the ASVL constitutes the optimal performance share, and the performance share achieves pareto-efficient risk sharing while motivating the agent to expend the high effort. If not, principal and agent must deviate from efficient risk sharing in order to ensure sufficient motivation. Figure 5 illustrates these two cases. It is based on Figure 4b, i.e., on the assumption that there is a unique solution for pareto-efficient risk sharing. Figures 5a and 5b display the ASVL, the SVL A and the cost of effort c. In Figure 5a, the optimal share efficiently shares risk, since f PE > f IC min. In contrast, in Figure 5b, the incentive constraint is binding, and the optimal share inefficiently shares risk. Figure 5: Agent s subjective value, aggregate subjective value, and optimal performance share SV MV SV MV ASVL ASVL c SVL A c SVL A 0 z 0 IC pt PE f f = f 1 min Fig. 5a PE f f pt = f IC min 1 z Fig. 5b 350 SBR 61 October

17 Incentive Compensation 5.2 Extensions We consider two extensions of the preceding analysis and assume that there exist three levels of effort, e shirk, e L and e H. We write c(e H ) = c H, c(e L ) = c L, where c H > c L, and set c(e shirk ) = 0. We write c for the difference in the cost of effort between high and low effort e H and e L, i.e. c = c H c L. Henceforth, for both the principal and the agent, two alternative Subjective Value Lines, SVL(e H ) and SVL(e L ), are relevant. The principal s problem is: Max f, e* SV P [(1 f) ỹ(e*)] + SV A [f ỹ(e*)] c(e*), subject to e* = arg max SV A [f ỹ(e)] c(e) and e {0, e L, e H }, (14) e and subject to the aggregate subjective value less cost of effort being positive. We denote the difference between the subjective values for a high and a low effort, for a given level of f, by SV (f), i.e. SV (f) = SV(f, e H ) SV(f, e L ). According to Equation (14), for a given performance share f, the agent will choose the high effort only if SV A (f) c and SV A (f, e H ) c H, (15) and he will prefer the low level of effort to shirking only if SV A (f, e L ) c L. (16) We denote the set of performance shares that fulfill the incentive constraint (16) as F (e L ). If SV A (f, e L ) is strictly monotone increasing in f, then the lower bound of F (e L ) fulfills Equation (16) as an equality (if F (e L ) is not empty). We denote this lower bound by f IC min (e L ). If the SVL A is concave and has an interior maximum as in Figures 3, 4, and 5, then the upper bound of F (e L ) may also fulfill Equation (16) as a strict equation. Further, we denote the set of performance shares that fulfill both constraints in Equation (15) by F (e H ), and the lower bound of this set by f IC min (e H ). If, at this lower bound, SV A (f) c in Equation (15) is fulfilled as an equality, then the agent receives the same utility from e H as from e L, and he will (by assumption) choose e H. We analyze two special cases. In case (a), increasing effort to e H adds value to the investment without changing its risk (as is assumed in the standard LEN model). In case (b), e H implies a higher investment volume of the project, i.e., the project is scaled up when the agent expends a high effort e H instead of a low effort e L. Seen individually, each case is very special. However, the following analyses can be extended to consider various combinations of the effects derived below and can account for differing investment opportunity sets and effort-return-relationships. The high level of effort adds riskless value to the investment First, we consider case (a), in which increasing the effort from a low to a high level adds value to the investment without affecting its risk. We denote the increase in the project s market value by MV. Ceteris paribus (i.e., for a given transfer payment K), the agent s SBR 61 October

18 H. Laux/R. M. Gillenkirch/M. M. Schabel subjective value of his performance share increases by f MV : SV A (f) = f MV. Graphically, the Subjective Value Line of the agent for high effort, SVL A (e H ), lies above the line for low effort, SVL A (e L ), at a distance of f MV. Analogously, the Subjective Value Line for the principal for high effort lies above the line for low effort at a distance of (1 f) MV. Therefore, the ASVL is shifted upwards by MV, and the x coordinate of its maximum, i.e., the performance share that efficiently shares risk, f PE, remains unchanged: f PE (e L ) = f PE (e H ). We denote the optimal share for inducing the low effort e L by f opt (e L ). If f PE F(e L ), that is, if the share that efficiently shares risk fulfills the incentive constraint (16), then the principal would choose f PE if she wanted to induce low effort. If, in contrast, f PE F (e L ), then the principal would have to deviate from efficient risk sharing to induce low effort. Whenever f opt (e L ) F (e H ) and f opt (e L ) = f IC min (e L ) > f PE, i.e., whenever the agent can be motivated with f opt (e L ) to exert high effort, but risk sharing is suboptimal, the principal can induce high effort and simultaneously improve risk sharing by lowering the performance share. (The optimal share is the minimum of f PE and f IC min (e H ).) If, in contrast, f opt (e L ) MV < c such that f opt (e L ) cannot induce the high effort, then the principal must choose a larger f, which will impair risk sharing if f opt (e L ) f PE. Figure 6 illustrates the two cases. Here, we assume that there is a unique f PE for efficient risk sharing (as in Figures 4b and 5), and that both the low and high effort can be induced only by deviating from efficient risk sharing, i.e., both f IC min (e H ) and f IC min (e L ) are larger than f PE. In Figure 6a, it is easier to motivate the high effort, such that the minimum performance share that induces the high effort f IC min (e H ) deviates less from f PE than f IC min (e L ). In Figure 6b, the opposite relation holds. Figure 6: Agent s subjective value, aggregate subjective value, and optimal performance share SVA ( f, e ) SVL ( e ) A L SVL ( e ) A H SVA ( f, e ) SVL ( e ) A L SVL ( e ) A H c H c L f MV MV c H c SVA ( f ) = c f MV MV c L 0 IC f f 1 min ( e ) IC min ( ) 0 PE IC f e PE H L fmin ( e ) IC f f L fmin ( e ) H Fig. 6a 1 f Fig. 6b The agent s effort scales up the investment Here, we consider the case (b) in which, for an investment in a given risk class, e H implies a higher investment volume than e L. That is, the agent s effort is a size parameter, and when the agent increases his effort from e L to e H, the investment s capital expenditure, expected 352 SBR 61 October

19 Incentive Compensation cash return, standard deviation, market value, and risk premium increase by the same factor. We denote this factor by λ, where λ > 1. Our analysis follows the same arguments as in case (a), but the SVLs have to be changed. This change has important implications. To analyze the agent s effort choice, we first derive the SVL A (e L ) as shown above. Then we construct the SVL A (e H ) by scaling the graph of the SVL A (e L ). For every performance share f and effort e L, there exists a performance share f f/λ, such that, for effort e H, the agent s risky payoff from the investment remains unchanged (f λ ỹ = (f/λ) λ ỹ = f ỹ). That is, SV A (f, e L ) = SV A (f, e H ). Since we originally defined the SVL A (e L ) only in f [0,1], we can directly apply this scaling only for shares f [0, 1/λ] with 1/λ < 1. For shares f [1/λ,1], we must first derive the SVL A (e L ) for shares f [1, λ] with λ > 1 or, equivalently, directly construct the SVL A (e H ) as in Section 4.2. Note that shares f > 1 are hypothetical; the agent s actual share will not exceed one. If the SVL A (e L ) is a strictly monotone increasing function for all f [0,λ], then the SVL A (e H ) will be strictly monotone increasing in f [0,1]. If, in contrast, the SVL A (e L ) has an interior maximum for a performance share f < λ, then the SVL A (e H ) has its maximum at f/λ = f < 1. The maximum subjective values are identical, SV A ( f, e L ) = SV A ( f, e H ), which implies that there is an upper bound for the subjective value of the investment to the agent, and that this value is independent of the investment s market value. The principal s Subjective Value Line SVL P (e H ) can be derived in the same way as the agent s, by scaling the graph of SVL P (e L ). For every share of the principal, 1 f, and for effort e L, there exists a performance share 1 f (1 f)/λ, such that, for effort e H, the principal s residual remains unchanged. That is, SV P ( 1 f, e L ) = SV P ((1 f ), e H ). From 1 f (1 f)/λ it follows that f 1 (1 f)/λ, and share levels f [0,1] correspond to levels f [1 1/λ, 1], where 1 1/λ > 0. For smaller levels f [0, 1 1/λ] we construct the SVL P (e H ) from SVL P (e L ) by extending the SVL P (e L ) for negative shares f [1 λ, 0]. These negative shares are also purely hypothetical. When the SVL P (e L ) is given for the whole interval [1 λ, 1], we can transform it into the SVL P (e H ) by downscaling, for every point on the SVL P (e L ), the distance between the corresponding x coordinate f [1 λ, 1] and the x coordinate f = 1, whereby the y coordinate is given and the scaling factor is 1/λ. E.g., at f = 1 λ < 0, this distance is 1 (1 λ) = λ, and downscaling by 1/λ gives the y coordinate at f = 0, SV P (f = 1 λ, e L ) = SV P (f = 0, e H ). Figure 7 illustrates the case in which the scale parameter is set to λ = 2. Figure 7 has three parts: Figure 7a displays the agent s Subjective Value Lines for low and high effort, Figure 7b displays the principal s SVLs, and Figure 7c displays the Aggregated Subjective Value Lines. In Figure 7a, the y coordinates of the SVL A (e L ) and SVL A (e H ) are identical for every pair of shares f and f = f/2. For example, the subjective value of the investment, given effort e L and a hypothetical share f = 2, SVL A (e L, f = 2), is equivalent to SVL A (e H, f = 1), the subjective value for the investment given e H and f = 1. By analogy, in Figure 7b, the y coordinates of the SVL P (e L ) and SVL P (e H ) are identical for every pair of shares f and SBR 61 October

20 H. Laux/R. M. Gillenkirch/M. M. Schabel f = 1 (1 f)/2 = (1 + f)/2. For example, the subjective value of the investment to the principal, given e L and the hypothetical share f = 1, SVL P (e L, f = 1), is equal to SVL P (e H, f = 0). In Figure 7c, the Subjective Value Lines are aggregated as in Figures 4b and 5. Here, the lines are reduced to the relevant range f [0, 1]. Figure 7: Agent s subjective value for low and high effort when effort scales up the investment SV A ( f, e ) SVL ( eh ) A SVL ( el ) A c H c L f IC f ( e ) min L SV [(1 f ), e] F ( e H ) Fig. 7a P SVL ( eh ) P SVL ( el ) P f Fig. 7b ASV ( f, e) ( ) ASVL e H ( ) ASVL e L 0 1 f PE PE f ( e L ) f ( e H ) Fig. 7c In Figure 7a, the SVL A (e L ) has a maximum at f 1, and the SVL A (e H ) has a maximum at f/λ 1/2. To induce low effort, the agent s share must not be lower than f IC min (e L ). To induce high effort, the principal must choose the performance share such that SV A (f) c ; SV A (f, e H ) c H is slack. Figure 7a also displays the set F(e H ) of performance shares that fulfill SV A (f) c. In Figure 7b, we assume that for e L given, the principal s subjective value reaches a maximum at a residual that corresponds to a negative share f < 0, or to 1 f > 1. However, the SVL P (e H ) reaches its maximum in the relevant range 0 f 1. According to Figure 7c the performance share f PE (e L ), which maximizes the ASVL for the low effort, is lower than f PE (e H ), the performance share that maximizes the ASVL(e H ). This means that pareto-efficient risk sharing is not independent from the agent s effort, which follows from the effort affecting both cash return and risk. With an 354 SBR 61 October

Review for Exam 2. Instructions: Please read carefully

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