Portfolio Choice When Managers Control Returns

Matsen, Egil Working Paper Portfolio Choice When Managers Control Returns Working Paper, No. 2005/15 Provided in Cooperation with: Norges Bank, Oslo Suggested Citation: Matsen, Egil (2005) : Portfolio Choice When Managers Control Returns, Working Paper, No. 2005/15, ISBN 82-7553-337-6, Norges Bank, Oslo, https://hdl.handle.net/11250/2498433 This Version is available at: https://hdl.handle.net/10419/209862 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. 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Hensikten er blant annet at forfatteren kan motta kommentarer fra kolleger og andre interesserte. Synspunkter og konklusjoner i arbeidene står for forfatternes regning. Portfolio Choice when Managers Control Returns* Egil Matsen** Abstract This paper investigates the allocation decision of an investor with two projects. Separate managers control the mean return from each project, and the investor may or may not observe the managers’ actions. We show that the investor’s risk-return trade-off may be radically different from a standard portfolio choice setting, even if managers’ actions are observable and enforceable. In particular, feedback effects working through optimal contracts and effort levels imply that expected terminal wealth is nonlinear in initial wealth allocation. The optimal portfolio may involve very little diversification, despite projects that are highly symmetric in the underlying model. We also show that moral hazard in one of the projects need not imply lower allocation to that project. Expected returns are generally lower than under the first-best, but the optimal contract shifts more of the idiosyncratic risk in the hidden action project to the manager in charge of it. The minimum-variance position of the investor’s (net) terminal wealth would in most cases involve a portfolio shift towards the hidden action project, and there are plausible cases where this would dominate the overall effect on the second-best optimal portfolio when comparing with the first-best. Keywords: Portfolio choice, diversification, optimal contracts JEL Classifications: D81; D82; G11 * This paper is a heavily revised version of a chapter in my PhD thesis. The revision is based on comments from my dissertation committee, Thore Johnsen, Finn Kydland, Morten Ravn, and Erling Steigum, to whom I am grateful. Remaining errors are solely my responsibility. ** Norwegian University of Science and Technology (NTNU) and Norges Bank. Address: NTNU, Department of Economics, NO-7491 Trondheim, Norway. Email: [email protected] 1. Introduction Investors commonly delegate the management of their resources to several professional managers. Examples include mutual fund investors that spread their holdings between different fund managers, venture capitalists that allocate capital to many entrepreneurs, and direct investments in firms governed by different CEOs. The allocation decision faced by an investor in such cases can be described as a portfolio choice where the return on available assets is, in part, determined by the actions of different agents. Indeed, one may argue that the purpose of delegated investment management is to realize a rate of return that managers, but not the investor, can possibly achieve. As a large literature on optimal contracts makes clear,1 the actions of managers under such investment arrangements are influenced by the compensation schemes offered to them. In a delegated investment setting, a rational investor must thus choose the optimal managerial contracts, in addition to the optimal portfolios. Despite the frequent occurrence of delegated investment arrangements, standard models of portfolio choice usually assume that investors manage their own wealth,2 making them unsuitable for analyzing the type of investment decisions discussed here. Models of optimal contracts have been applied to portfolio choice (Sung, 1995; Dybvig et al., 2001; Ou-Yang, 2003; Westerfield, 2005), but these applications analyze how the portfolio choice of agents (i.e., managers) responds to contractual incentives; they do not discuss how an investor’s portfolio selection interacts with optimal contracts.3 1 See, e.g., Holmstrom and Milgrom (1987), Schättler and Sung (1993, 1997), Sung (1995, 2005), Müller (1998), Ou-Yang (2003), and Westerfield (2005). 2 See Campbell and Viceira (2002) for a synthesis of modern theories of portfolio choice. 3 The contract literature referred to here discusses models with one principal and one agent, which obviously precludes an analysis of how an investor optimally should allocate wealth among several managers. 2 This paper investigates the portfolio decisions of an investor that can allocate her wealth between two projects. The mean rate of return from each project is determined by the actions (effort) of the manager in charge of it. These actions may or may not be observed by the investor, and are affected by the contract between investor and the managers. The model draws on the analysis of dynamic principal–agent problems by Holmstrom and Milgrom (1987), Schättler and Sung (1993, 1997), Sung (1995), and Müller (1998). That is, we explore a continuous-time model where output follows a Brownian motion and both the principals (the investors) and the agents (the managers) have constant absolute risk aversion, defined over terminal wealth4. This is a natural point of departure as, unlike the static principal–agent problem, the continuous-time version admits relatively easily interpreted solutions. Upon presenting the model in Section 2, we analyze the first-best case of observable and enforceable actions in Section 3. We show that, even in this case, the riskreturn trade-off involved in the investor’s portfolio decision is quite different from that in a standard portfolio choice model. First, the investor’s risk aversion is effectively lower than her CARA-coefficient, because some of the terminal wealth risk is carried by the managers, according to the optimal contracts. Second, there are important feedback effects, working through optimal contracts and efforts, that make expected terminal wealth nonlinear in initial wealth allocation. Depending on the shape of managers’ cost functions, the investor may have incentives to choose a highly “nondiversified” portfolio, even if the projects are completely symmetric. A general insight is that, in a principal– agent setting, the return on a given investment is endogenous to the level of investment 4 Williams (2004) and Sannikov (2004) study moral hazard models where the (single) agent can consume continuously. Williams’s model is very general, but in most cases the solution can only be characterized and not solved explicitly. Sannikov assumes a risk neutral principal, making that model less suitable for studying an investor’s portfolio choice. 3 in that project; standard portfolio choice models treat the return (and risk) on available assets as exogenous. In Section 4, we investigate the case where there are hidden actions in one of the projects.5 This introduces an asymmetry where, a priori, one perhaps would expect a tilt in the portfolio away from the hidden action project. We do indeed show that the expected return in this project is generally lower than in the first-best. However, the relevant risk in the project, as seen from the investor’s point of view, may be lower with moral hazard. This is because the investor uses the output from the project where actions can be observed as a signal to be included in the contract with the other manager (Holmstrom and Milgrom, 1987). The optimal contract then implies that the manager with hidden actions must carry more of the idiosyncratic risk associated with her project. Consequently, the minimum-variance allocation of the investor’s wealth is in most cases tilted towards the hidden action project. Even if expected returns generally would be lower in this project, there are therefore plausible cases where investors would in fact invest more in it than if they had full information. Section 5 concludes the paper, and proofs are in the appendix. 2. The principal–agent framework 5 Think of a fund that employs some managers “in-house”, with the possibility of internal monitoring, and delegates some of its assets to external managers, where the possibilities of monitoring are limited. Another example can be taken from the literature on the “home-bias puzzle” (see Lewis, 1999, for a survey), where it is sometimes argued that investors hold the lion’s share of their portfolios domestically because there is asymmetric information across countries. A possible interpretation of the setup in Section 4 is that of an investor who can make direct investments in imperfectly correlated domestic and foreign projects, and where it is easier to monitor the actions of a domestic manager. 4 We investigate the principal–agent relationship on the time interval [0,1]. At time 0, the principal (the investor) decides how to allocate initial wealth W(0) = W0 between two projects, A and B. The investment decisions are assumed to be irreversible; the allocation is fixed until time 1.6 The output from the projects is publicly observable and governed by the processes , (1) () 00 () (), , (), , iii iii dX t f u t X t dt X dz t i A B=+σ= where Xi0 is the amount invested project i, XA0 + XB0 = W0. Furthermore, σi is a diffusion parameter and dzi is a standard Wiener process that represents a project-specific shock. The instantaneous correlation coefficient ρ of these shocks is obtained from dzAdzB = ρdt, ρ ∈ [–1,1). Throughout, we will assume that the production function f is given by B . (2) 00 (, ,) () , , ii i i fu X t utX i AB== The variable ui (later referred to as “effort”) is controlled by the manager in charge of project i (manager i), and may or may not be observed by the investor. In any case, the expected rate of return on invested resources in project i is controlled by manager i. This setup implies that the investor accumulates wealth according to ( ) 00 () () () () ' () AB B dW t u t u t u t W dt W d t ⎡⎤ =ω−++Σ ⎣⎦ wB , (3) where 2 0 1 A BB σ ⎡ ⎤ ⎢ ⎥ Σ= ⎢ ⎥ σρ σ −ρ ⎢ ⎥ ⎣ ⎦ , ω is the fraction of initial wealth invested in project A, , and dh is a standard Wiener process with the property that it is independent of dz '[ 1 ], [ ] A ddzdh≡ω −ω ≡wB' A. 6 This assumption is imposed to obtain tractability, as allowing for continuous reallocation would introduce time-dependent drifts in the processes for XA and XB. Schättler and Sung (1997) show that introducing timedependent drifts of the Brownian motions would destroy the result that sharing rules are linear in output, and therefore the tractability of the model. B 5 At time 0, the investor and the managers individually agree on sharing rules specifying payment from the investor to the managers at time 1. The sharing rules specify salaries SA and SB for manager A and B, respectively, and these are stochastic via dependence on the outcome of the stochastic process for W. The managers’ control variables, u B A ≥ 0 and uBB ≥ 0, can be revised continuously during the time interval [0,1] and may depend on the history of W in [0,t], but not on the future (t,1]. The managers incur costs of effort, assumed, for simplicity, to be quadratic in the level of effort () 2 12 00 (), , () , , ii ii cut X t u tX i AB θ == , where θ ≥ 0. The assumptions of linear output and convex costs in effort are sufficient for well-defined solutions to both the first- and the second-best problems discussed below; see Theorem 4.2 in Schättler and Sung (1993). Observe that cost functions are symmetric: for a given allocation of resources, the cost of effort is equal for the two managers; for given effort, the cost of managing wealth is equal. Finally, both the investor and the managers have exponential utility functions. The investor’s constant coefficient of absolute risk aversion is R while the two managers are equally risk averse with a CARA-coefficient r. 3. Full information We shall first characterize the optimal sharing rules, effort levels, and resource allocation in the first-best setting; that is, when both managers’ controls are observable and can be enforced at no cost. This analysis demonstrates that the trade-offs involved in the diversification decision are quite different from a standard portfolio choice model, even 6 portfolio. The intuition was explained in the last subsection: as θ < 1, the investor can increase expected returns by investing in one project only. 4. Diversification with asymmetric information We now turn to the case where the investor cannot observe the actions of manager B. The investor faces asymmetric information along two dimensions: he knows less about the actions of manager B than does the manager herself, and he knows more about the actions of manager A than of B’s. We are primarily interested in the consequences of these asymmetries for diversification and portfolio choice. In particular, does less information necessarily mean less investment? 4.1. The second-best problem With moral hazard in project B, the investor faces an additional constraint: () { } 1 0 0 arg max exp ( , , ) B BBB u uErScuXt B dt ⎡ ⎤ ⎢ ⎥ ∈−−− ⎢ ⎥ ⎣ ⎦ ∫. This is the familiar incentive compatibility constraint, which says that manager B chooses the uB that is in his best interest. We follow Schättler and Sung (1993), and use the so-called first-order approach to solve the investor’s problem. In this approach, the incentive compatibility constraint in the principal’s problem is relaxed to the first-order necessary condition for optimality in the agent’s problem. B As the investor has full information on the actions of manager A, and as the managers do not control variances of the Brownian processes, the output from project A serves as a signal to be included in the contract between the investor and manager B. In 13 particular, following Holmstrom and Milgrom (1987, pp. 324–25), we assume that the salary to manager B is conditioned on the aggregate performance index () () 0 0 cov ( ), ( ) () () () var ( ) () (). AB BA A BB BA AA XtXt Yt X t X t Xt X Xt Xt X =− ρσ =−σ Using (1), (2), and (3), we can show that this index evolves according to: [ ] 0 () () () ' () B A BB A B dY t X u t u t dt X d t ρσ σ =−+ΣmB 0 , (14) where '[ 1 B A ρσ σ ≡−m] . Now, the problem of manager B is () { } 1 0 0 max exp ( , , ) B BBB uErScuXtdt ⎡ ⎤ ⎢ ⎥ −−− ⎢ ⎥ ⎣ ⎦ ∫, subject to (14), taking as given the actions of manager A and the allocation decision of the investor. Applying the representation given in Schättler and Sung (1993), we can show that the solution to this problem gives the optimal sharing rule: 11 1 2 2 11 22 000 0 00 0 '() ( )'' BBBBB BB SU uXdt uX dt ruX d θθ θ =+ + Σ+ΣΣ ∫∫ ∫ mB m mt ) . (15) The first two terms on the right-hand side of (15) provide manager B with his certainty equivalent plus compensation for the cost he actually incurs. The next term is the compensation error, arising because the investor’s compensation is based on realized outcome rather than the manager’s actual effort. Finally, to compensate manager B for the risk he carries, a risk-premium is paid, given by the last term in (15). The investor’s relaxed problem can then be written as ( { } ,,, max exp (1) AB A AB uuS ERWSS ω ⎡ ⎤ −− −− ⎢ ⎥ ⎣ ⎦, subject to (3), (15), and manager A’s participation constraint. The solution is summarized in the second proposition. 14 Proposition 2: Suppose that manager B’s effort cannot be observed. Manager A’s optimal effort level is still given by condition (6). Manager B’s optimal effort is constant and fulfills: , (16) 1 0 ( BBB uXgX −θ =0 ) where 22 0 022 0 1(1 () (2) 1(1 BB B BB rR X rR gX rr R X rR θ θ +σ−ρ + ≡ + +σ−ρ + ) 1 ) ≤ . (17) The optimal salary contract with manager B is linear in output from each of the projects: 0 0 0 ()(1) (1) BB BB BB A AA X SgXX X X ⎛⎞ ρσ ⎟ ⎜⎟ =κ+− ⎜⎟ ⎜⎟ ⎜⎟ σ ⎝⎠ , (18) where 22222 11 22 00 000 (1 )[ ( )] ( ) [ (1)] BBBBBBB UuX gXXgXEY θ κ≡ ++σ−ρ − is a constant. The salary of manager A is given by () 0 0 0 1 ( ) (1) 1 ( ) (1) BB AA B A B B AA XR SgXXgX rR X ⎡⎤ ⎛⎞ ρσ ⎟ ⎜ ⎢⎥ ⎟ =κ++ +− ⎜⎟ ⎢⎥ ⎜⎟ ⎜⎟ +σ ⎝⎠ ⎢⎥ ⎣⎦ 0 X , (19) with 2 12 00 1ln A AA rRW ru X R rR R θ AB ⎡ ⎤ ⎛⎞ λ⎟ ⎜ ⎢ ⎥ κ≡ − +−κ ⎟ ⎜⎟ ⎜ ⎝⎠ ⎢ ⎥ + ⎣ ⎦ being another constant. Finally, the investor allocates a fraction () () 21 21 2 1200 22 22 0 22 1 2 00 22 0 [2] 2 (1 ) 1 , [2] AB AA BB BAB AB AB AB A rR R BBB BB AB AB uu uX uX rR WRr uX uX W θ− θ− θθ− + ⎛⎞ −−θ − σ−σ + ⎟ ⎜⎟ ⎜ ω=+ ⎟ ⎜⎟ ⎜σ+σ−σ σ+σ−σ ⎟ ⎟ ⎜ ⎝⎠ ⎛⎞ σ−ρ +θ− θ ⎟ ⎜⎟ ⎜ −⎟ ⎜⎟ ⎜σ+σ−σ ⎟ ⎟ ⎜ ⎝⎠ B (20) of initial wealth to project A. Proof: See the appendix. 15 As expected, manager B’s effort is (weakly) lower than A’s for given investments in the respective projects; see equations (16) and (6). The variable g(XB0), defined in (17), can be interpreted as a wedge between the first- and second-best efforts, conditional on the amount invested in B. We can show that ≤ 0, where the equality holds if θ = 0 or ρ = –1. Except in these two special cases, manager B’s effort level will be lower relative to the first-best the more the investor allocates to project B. Whether more investment yields higher effort in absolute terms depends on the sign of 0 '( ) B gX 1 00 0 (1 ) ( ) B BBB B du Xg X gX dX −θ −θ ′ =−θ +0 . (21) As long as < 0, we now need θ  1 for manager B’s effort to respond positively on investment. Compared to the first-best, there is thus smaller range of θ’s for which effort increases with investment. 0 '( ) B gX The sharing rule (18) resembles the relative evaluation scheme discussed by Holmstrom and Milgrom (1987, p. 324), but here the shares of output accruing to the investor and the managers are endogenous to the allocation of initial wealth. Note that (18) can be written as ( ) 0 ( ) (1) [ (1)] BB B SgXYEY=κ+− , where . Manager B is given a constant amount plus a share g of the surprise in the aggregate performance index Y. The third term in the definition of κ [(1)] BB gE Yκ=κ+  B is the risk-premium that is optimally paid to manager B. Note that it depends on the idiosyncratic risk associated with project B. Finally, and show that risk sharing between the two managers is generally imperfect when there is moral hazard. B 22 (1 ) B σ−ρ (18) (19) Turning to portfolio choice, we see that the upper line in (20) bears close resemblance to equation (8). The term in the lower line, however, is new and conceptually different. It occurs because initial wealth allocation affects the risk-premium 16 and the expected value of the performance index (the last two terms in the definition of κB). In turn, this feeds back to the expected value and variability of investors’ net final wealth, and therefore their portfolio decisions. To give a more thorough interpretation of portfolio choice, it is again helpful to recast the portfolio decision within a mean-variance setting. B 4.2. The risk-return trade-off with moral hazard We can use (18), (19), and the performance index Y to express the investor’s net final wealth as (we suppress the argument in g(.) to simplify notation): () () 22 2222 11 22 00 0 (1) (1) [ (1)] (1) (1 ) . AB AA BB B B WSS rWgEYY uXuX rgX ti rR θθ −− ⎡⎤ =+−− +−σ−ρ+ ⎢⎥ ⎣⎦ +p (22) Now, as both managers’ efforts are constant also in the second-best, the processes (3) and (14) both follow arithmetic Brownian motions. It follows from equation (22) that net final wealth is normally distributed over the time interval [0,1]. Maximizing (4) is thus again equivalent to maximize [ ] [ ] 12 (1) var (1) AB AB EW SS R W SS−− − −− . Taking expectations in (22) and substituting from (6) and (17), we find: [ ] () 22222 11 1 22 2 00 0 2222 2 11 1 22 2 00 (1) ( ) (1 )((1 ) ) (1 ) ((1 ) ) ( ) (1 ) ((1 ) ) (1 ) ((1 ) ) , AB B B EW S S rWWggW r gW rR rFggWrgW rR −θ −θ −θ −− ⎡⎤ =+ω+−−ω−σ−ρ−ω ⎣⎦ + ⎡⎤ =ω − − − −ω − σ −ρ −ω ⎣⎦ + 2 0 (23) where the function F(ω) was defined and analyzed in Section 3. Comparing (10) and (23), we see that the link between the initial allocation and expected net final wealth is quite different in the two cases. The last two terms in the lower line of (23) are new relative to the first-best. The first of these shows the loss in expected output net of management 17 costs in project B, compared to the first-best. We can show that this loss is strictly increasing over ω ∈ [0,1], as long as θ ≤ 2. When θ > 2, the loss is not necessarily monotonic over ω ∈ [0,1], but still reaches its minimum of zero at ω = 1. The final term in (23) is the risk-premium paid to manager B. It enters with a minus in front, so it contributes to lowering the investor’s net expected wealth. The risk-premium also has a minimum of zero at ω = 1, and can be shown to be strictly decreasing over ω ∈ [0,1] when θ ≤ 1. These observations allow us to deduce the following on wealth allocation and expected net final wealth in the second-best. Starting with the cases where θ ∉ [1,2], we recall from Section 3 that F(ω) then has maxima at 0 and 1. However, as the last two terms in the lower line of (23) both have a unique maximum at ω = 1, expected net final wealth is greatest when investing in project A only. The cases with θ ∈ {1,2} are analogues; F(ω) is independent of wealth allocation, and hence expected net final wealth with moral hazard is maximized by setting ω = 1. The last possibility is θ ∈ (1,2), where we recall that F(ω) has its maximum at ω = ½. The last term in the upper line of (23) gives an incentive to invest more in project A. However, as the risk-premium to manager B is possibly nonmonotonic in ω for these values of θ, the overall link between initial wealth allocation and expected net final wealth is theoretically ambiguous.9 Turning to the risk of net final wealth, we observe that (22) implies [][] 2 var (1) var (1) (1) AB r WSS WgY rR ⎛⎞ ⎟ ⎜ −− =− ⎟ ⎜⎟ ⎜ ⎝⎠ +. 9 Numerical calculations by the author indicate that, for “plausible” parameters, expected net final wealth is maximized for ω > ½ also when θ ∈ (1,2). 18 The optimal contract with manager B leaves the investor exposed to the risk of gross final wealth less a (endogenous) share g of the aggregate performance index Y. This risk exposition is in turn shared with manager A (the investor carries a share r/(r+R)); hence the above expression. Using equation (14), we have [] () 2 2 22 0 var (1) ' ' (2 ) (1 ) (1 ) AB B r WSS W gg W rR ⎛⎞ 2 0 ⎡ ⎤ ⎟ ⎜ −− =ΣΣ − − − ω σ − ρ ⎟ ⎜ ⎢ ⎥ ⎟ ⎜ ⎣ ⎦ ⎝⎠ +ww . An implication of this equation is that, unlike with full information, the portfolios that minimize gross and net final wealth variance, respectively, are not equal. Interestingly, we can show that the net final wealth minimizing share invested in project A is smaller than 222 ()( 2 BABAB AB σ−σ σ+σ−σ ) unless θ  2. The most plausible cases of the model thus imply that the relevant minimum-variance position is tilted towards project B when there is moral hazard. The implications for portfolio choice of the above discussion are straightforward. Relative to the first-best, expected returns can generally be increased by tilting the portfolio towards project A. On the other hand, the minimum-variance position would in most cases be tilted towards project B. The combined effect on portfolio choice is therefore generally ambiguous. Interestingly, we cannot rule out that the investor should optimally increase allocation to the project with moral hazard compared with the case of full information. We close this section by illustrating the latter point using an example with θ = 0, the only case where the portfolio rule can be expressed in closed form. For simplicity, we set σA = σB = σ (implying an optimal first-best rule ω = ½; see the discussion in Section 3.3), and assume that ρ = 0. When θ = 0, the wedge g is a constant given by B 2 2 () rRrR grRrrRR ++σ =++ + σ. 19 Substituting into (20), we can show that the optimal share invested in A is () ( ) () () 2 222 2 2 222 2 32 33 22 22 rR r r R rR r R rR r r R rR r R r σσ−−+σ− + ω=σσ−−+σ− +−σ . It is straightforward to show that this share can be less than half for plausible parameter values. 5. Conclusions We have studied the resource allocation of investors that delegate the management of their wealth to two different managers. Managers’ effort levels determine the expected return from the projects that they govern, but these effort levels are affected by the contracts offered by investors. We show that even when managers’ actions are observable and enforceable, the investors’ diversification decision involves trade-offs other than a standard portfolio problem. This is because, in a principal-agent setting, expected returns are endogenous to the allocation of initial wealth. Depending on the shape of managers’ cost functions, the expected final wealth net of managerial compensation can exhibit both increasing and decreasing returns to invested wealth. In the former case, optimal portfolio holdings may be highly undiversified despite the symmetry between projects in the underlying model. If managers’ actions cannot be observed, additional mechanisms come into play. We explore the case with moral hazard in one of the projects. Relative to the first-best, expected returns net of managerial costs can generally be increased by tilting the portfolio away from the moral hazard project. However, the optimal contract with the manager of this project transfers more of the idiosyncratic risk associated with the 20 project to the manager. As a consequence, the minimum-variance position of the investors’ net final wealth would in most cases be tilted towards the moral hazard project. There are plausible cases of the model where the allocation to the project with hidden actions is higher than in the first-best. 21 Appendix A.1. Proof of Proposition 1 We start by deriving the optimal sharing rules in terms of the optimal controls uA and uB, following Müller (1998). Net compensation to manager i is B 12 1200() ii i i yS X utd θ =−∫t . Then, integrating and inserting the result in imply that the investor’s problem can be expressed as (3) (4) ( { ) } , 00 0 {,}, , 122 11 22 00 0 0 0 max exp ' ( ) ( ) ( )( ) ( )(1 ) ( )((1 ) ) , AB AB AB uu yy AA B B ERWW yy utWutWut Wut Wdt ω θθ ⎡−− +Σ− −− ⎢⎣ ⎤ ⎡⎤ ⎥ +ω− ω +−ω − −ω ⎣⎦ ⎥⎦ ∫ 1 wBB subject to (5). Pointwise maximization with respect to yA and yB gives the first-order conditions B () 122 12 00 0 0 1ln (1) ( ) ((1 ) ) i iAB rRR yWWuWuWd rR R rR rR θθ ⎛⎞ λ⎡⎤ ⎟ ⎜ =+−− ω +−ω − ⎢⎥ ⎟ ⎜⎟ ⎜ ⎝⎠ ++ ⎣⎦ j ty + ∫ , where i,j = A,B, i ≠ j, and λi is the Lagrange-multiplier associated with (5). Solving these two equations for yA and yB and using the two participation constraints to demonstrate that λ B A = λBB, we find () 122 12 00 0 1ln 2 (1) ( ) ((1 ) ) 2 A AB AA r yyrR R RWW uWu Wd rR θθ ⎛⎞ λ⎟ ⎜ == ⎟ ⎜⎟ ⎜ ⎝⎠ + 0 t ⎡ ⎤ +−− ω +−ω ⎣ ⎦ + ∫ . (A.1) The optimal sharing rules are () 1 0 0(), , , iii Sy cutXdtiAB=+ = ∫ , where y is given in (A.1). Next, we substitute the optimal salary functions into (4). The investor’s problem can then be simplified to 22 29 2003/6 Harald Moen Nåverdien av statens investeringer i og støtte til norske banker Avdeling for finansiell analyse og struktur, 24 s 2003/7 Geir H.Bjønnes, Dagfinn Rime and Haakon O.Aa. Solheim Volume and volatility in the FX market: Does it matter who you are? Research Department, 24 p 2003/8 Olaf Gresvik and Grete Øwre Costs and Income in the Norwegian Payment System 2001. 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