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Mathematics and Financial Economics

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01-09-2014

On managerial risk-taking incentives when compensation may be hedged against

Authors: Jakša Cvitanić, Vicky Henderson, Ali Lazrak

Published in: Mathematics and Financial Economics | Issue 4/2014

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Abstract

We consider a continuous time principal-agent model where the principal/firm compensates an agent/manager who controls the output’s exposure to risk and its expected return. Both the firm and the manager have exponential utility and can trade in a frictionless market. When the firm observes the manager’s choice of effort and volatility, there is an optimal contract that induces the manager to not hedge. In a two factor specification of the model where an index and a bond are traded, the optimal contract is linear in output and the log return of the index. We also consider a manager who receives exogenous share or option compensation and illustrate how risk taking depends on the relative size of the systematic and firm-specific risk premia of the output and index. Whilst in most cases, options induce greater risk taking than shares, we find that there are also situations under which the hedging manager may take less risk than the non-hedging manager.

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Other papers considering the effect of specific contracts on portfolio managers, typically without possibility of hedging, include [2, 3, 8, 14], and [16]. Grinblatt and Titman [10], Henderson [13] and Hodder and Jackwerth [15] include hedging possibilities, again only in frameworks with specific contracts.

Whilst [15] obtain numerical results showing that the manager prefers low total volatility when compensated with shares, and higher volatility when compensated with calls, they do not consider separately controlling systematic and firm-specific risks, and do not delineate any dependence on corresponding risk premia. [13] does consider separately systematic and firm-specific risk but only in the case where the output $X$ is traded and $\alpha _x = \lambda $.

Panageas and Westerfield [23] show that even the risk-neutral managers need not behave aggressively when paid with high water mark contracts, if the time-horizon of their compensation is not fixed.

Alternatively, we could assume that the firm and the manager enjoy utility from discounted values.

These technical conditions are typically either satisfied, or they are satisfied if instead of $\mathcal{Z}$ we consider its closure in an appropriate topology. In particular, the condition $V(H_0)=\bar{V}(H_0)$ means that there is no duality gap between the primal problem of portfolio optimization and the dual problem of finding the optimal dual state-price density. Economically, it means that the standard marginal utility expression holds for the agent’s problem at the optimum: $U'_A(P_T)= yY_T$ for the agent’s final total wealth $P_T$, for some constant $y$ and some SDF $Y_T$ (or $Y_T$ in the closure of the set $\{ \mathcal Z \}$ of SDF’s). For details, see the references mentioned in the proof.

If we modeled $S$ as a Brownian motion with drift rather than a geometric Brownian motion, the optimal contract would be linear in $S$, not in $\log (S)$, in the case of CARA preferences.

Alternatively, we could assume that the market trades a risk-free asset with the interest rate independent of all the risk-neutral densities in the market.

The Excel spreadsheet is available at http://www.hss.caltech.edu/~cvitanic/PAPERS/Excelhedge.xls. In order to interpret this, write now

$$\begin{aligned} \tilde{W}=\rho W +\sqrt{1-\rho ^2} M \end{aligned}$$

(4.23)

for a Brownian Motion $M$ independent of $W$. Note that we have

$$\begin{aligned} dS/S=[r+\sigma \lambda ] dt +\sigma /\rho d\tilde{W}_t-\sigma \sqrt{1-\rho ^2}/\rho dM_t=rdt+\sigma /\rho dW^Q-\sigma \sqrt{1-\rho ^2}/\rho dM_t \end{aligned}$$

(4.24)

Thus, $Q$ is a risk-neutral measure for $S$. It is actually the projection on $\tilde{F}$ of the measure which would be the risk-neutral measure if $S$ was the only traded asset.

[11] consider the case of non-CARA preferences with no hedging, and with linear contracts.

Acharya, V., Bisin, A.: Managerial hedging, equity ownership, and firm value. RAND J. Econ. 40(1), 47–77 (2009)CrossRef

Basak, S., Pavlova, A., Shapiro, A.: Optimal asset allocation and risk shifting in money management. Rev. Financ. Stud. 20(5), 1583–1621 (2007)CrossRef

Basak, S., Shapiro, A., Tepla, L.: Risk management with benchmarking. Manag. Sci. 54, 542–557 (2006)CrossRef

Bettis, J.C., Bizjak, J.M.: Managerial ownership, incentive contracting, and the use of zero-cost collars and equity swaps by corporate insiders. J. Financ. Quant. Anal. 36, 345–370 (2001)CrossRef

Bisin, A., Gottardi, P.: Managerial hedging and portfolio monitoring. J. Eur. Econ. Assoc. 6(1), 158–209 (2008)CrossRef

Cadenillas, A., Cvitanić, Zapatero, F.: Optimal risk-sharing with effort and project choice. J. Econ. Theory 133, 403–440 (2007)CrossRefMATH

Carpenter, J.N.: Does option compensation increase managerial risk appetite? J. Financ. 55, 2311–2331 (2000)CrossRef

Cuoco, D., Kaniel, R.: Equilibrium prices in the presence of delegated portfolio management. J. Financ. Econ. 101, 264–296 (2011)CrossRef

Garvey, G., Milbourn, T.: Incentive compensation when executives can hedge the market: evidence of relative performance evaluation in the cross section. J. Financ. 58, 1557–1581 (2003)CrossRef

10.

Grinblatt, M., Titman, S.: Adverse risk incentives and the design of performance based contracts. Manag. Sci. 35(7), 807–822 (1989)CrossRef

11.

Guo, M., Ou-Yang, H.: Incentives and performance in the presence of wealth effects and endogenous risk. J. Econ. Theory 129, 150–191 (2006)CrossRefMATHMathSciNet

12.

Henderson, V.: Valuation of claims on non-traded assets using utility maximization. Math. Financ. 12(4), 351–373 (2002)CrossRefMATH

13.

Henderson, V.: The impact of the market portfolio on the valuation, incentives and optimality of executive stock options. Quant. Financ. 5(1), 1–13 (2005)CrossRef

14.

Hodder, J.E., Jackwerth, J.C.: Incentive contracts and hedge fund management. J. Financ. Quant. Anal. 42(4), 811–826 (2007)CrossRef

15.

Hodder, J.E., Jackwerth, J.C.: Managerial responses to incentives: control of firm-risk, derivative pricing implications, and outside wealth management. J. Bank. Financ. 35(6), 1507–1518 (2011)CrossRef

16.

Hugonnier, J., Kaniel, R.: Mutual fund portfolio choice in the presence of dynamic flows. Math. Financ. 20(2), 187–227 (2010)CrossRefMATHMathSciNet

17.

Holmstrom, B.: Moral hazard in teams. Bell J. Econ. 19, 324–340 (1982)CrossRef

18.

Jin, L.: CEO compensation, diversification and incentives: theory and empirical results. J. Financ. Econ. 66, 29–63 (2002)CrossRef

19.

Karatzas, I., Shreve, S.: Methods Math. Financ. Springer-Verlag, New York (1998)CrossRef

20.

Kramkov, D., Schachermayer, W.: A condition on the asymptotic elasticity of utility functions and optimal investment in incomplete markets. Ann. Appl. Probab. 9, 904–950 (1999)CrossRefMATHMathSciNet

21.

Ofek, E., Yermack, D.: Taking stock: equity-based compensation and the evolution of managerial ownership. J. Financ. 55, 1367–1384 (2000)CrossRef

22.

Ozerturk, S.: Financial innovations and managerial incentive contracting. Can. J. Econ. 39, 434–454 (2006)CrossRef

23.

Panageas, S., Westerfield, M.M.: High-water marks: high risk appetites? J. Financ. 64(1), 1–36 (2009)CrossRef

24.

Ross, S.A.: Compensation, incentives, and the duality of risk aversion and riskiness. J. Financ. 59, 207–225 (2004)CrossRef

25.

Tehranchi, M.: Explicit solutions of some utility maximization problems in incomplete markets. Stoch. Process. Appl. 114, 109–125 (2004)CrossRefMATHMathSciNet

Title: On managerial risk-taking incentives when compensation may be hedged against
Authors: Jakša Cvitanić
Vicky Henderson
Ali Lazrak
Publication date: 01-09-2014
Publisher: Springer Berlin Heidelberg
Published in: Mathematics and Financial Economics / Issue 4/2014
Print ISSN: 1862-9679
Electronic ISSN: 1862-9660
DOI: https://doi.org/10.1007/s11579-014-0123-3

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