Implicit Incentives for Fund Managers with Partial Information
Flavio Angelini, Katia Colaneri, Stefano Herzel, Marco Nicolosi
,, IMPLICIT INCENTIVES FOR FUND MANAGERS WITHPARTIAL INFORMATION
FLAVIO ANGELINI, KATIA COLANERI, STEFANO HERZEL,AND MARCO NICOLOSI
Abstract.
We study the optimal asset allocation problem for a fundmanager whose compensation depends on the performance of her port-folio with respect to a benchmark. The objective of the manager is tomaximise the expected utility of her final wealth. The manager observesthe prices but not the values of the market price of risk that drives theexpected returns. The estimates of the market price of risk get moreprecise as more observations are available. We formulate the problemas an optimization under partial information. The particular structureof the incentives makes the objective function not concave. We solvethe problem via the martingale method and, with a concavification pro-cedure, we obtain the optimal wealth and the investment strategy. Anumerical example shows the effect of learning on the optimal strategy. Introduction
The reward of a fund manager usually increases when the Asset UnderManagement (AUM) grows, while it decreases when the AUM shrinks. TheAUM may grow either because of a higher value of the assets or becauseof new money flowing into the fund. Good performances of the fund withrespect to its relative benchmark are likely to attract new investors. There-fore, contracts based on the AUM create an implicit incentive for the man-ager to beat the benchmark. We study the problem of a portfolio managerwhose compensation depends on the AUM modelled through the relativeperformances with respect to a benchmark. This framework generalizes thesetting of Basak, et al. (2007)[7], by considering a market model with onerisk-free and one risky asset whose expected returns depend on an unob-servable stochastic process, the “market price of risk”. We introduce therealistic assumption that the manager has a limited knowledge on the mar-ket, she can only observe stock prices and estimates the market price of riskfrom them. Therefore, the manager is facing an optimization problem underpartial information.
Flavio Angelini and Marco Nicolosi, Department of Economics, Universityof Perugia, Via A. Pascoli 20, 06123 Perugia, IT.Katia Colaneri and Stefano Herzel, Department of Economics and Fi-nance, University of Rome Tor Vergata, Via Columbia 2, 00133 Roma, IT.
E-mail address : [email protected], [email protected],[email protected], [email protected] . a r X i v : . [ q -f i n . P M ] N ov IMPLICIT INCENTIVES AND PARTIAL INFORMATION
Optimization problem under partial information are usually solved in twosteps: the first step, called reduction, consists of deriving the conditionaldistribution of the market price of risk with respect to the observed informa-tion flow; the second step solves the equivalent problem under the observedinformation. An important feature of our setting is that, while the market ofclaims contingent to the knowledge of the market price of risk is incomplete,the market restricted to those claims contingent only to stock prices is in-stead complete. We will exploit this fact to solve the optimization problemapplying a martingale approach with the unique equivalent martingale mea-sure (under the restricted setting) and then using a concavification argumentto determine the unique optimal solution.Although many papers have been written about the issues of relative in-centives and of optimization under partial information (see the literaturereview provided in Section 1.1), this one is, to the best of our knowledge, thefirst one to analyse the combined effect of such issues on the optimal strat-egy of a portfolio manager. We contribute to the literature by providingthe solution to the optimization problem in semi-closed form and we presentone example where we show that the optimal strategy depends on the riskaversion of the manager and on the economic situation of the market. Whenthe risk aversion of the manager is larger (lower) than that of a managerequipped with a logarithmic utility, she will tend to decrease (increase) herinvestment in the risky asset to hedge against the future adjustment in theestimates of the unknown parameter.The paper is organized as follows. After a literature review (Section 1.1),Section 2 presents the market model and the portfolio optimization problemfaced by the manager. In Section 3 we solve the optimization problem intwo steps. First, we derive the dynamics of the filtered estimate of marketprice of risk, in order to reduce the problem to a common information flow.This procedure allows to obtain market dynamics driven by a unique sourceof randomness and hence the market model under partial information turnsout to be complete. Second, we apply the martingale method along withconcavification to characterize the optimal final wealth and the optimal in-vestments strategy. Section 4 contains a numerical illustration of our results.Conclusions and comments are provided in Section 5. We relegated proofsand calculations to Appendix A–C.1.1.
Literature review.
The structure of portfolio managers’ compensa-tion is studied for instance in Ma et al. (2019)[34], who show that perfor-mance based incentives represent the main form of compensation for portfoliomanagers in the US mutual fund industry. Of course, this is not the onlytype of incentive for fund managers. Option-like incentives of different nature(as for example management fees, investor’s redemption options or fundingoptions by prime brokers) apply in fund managers compensation contract,and influence manager’s leverage decisions (see, e.g. Lan et al. (2013)[30]and Buraschi et al. (2014)[13])
MPLICIT INCENTIVES AND PARTIAL INFORMATION 3
Basak et al. (2007)[7] compute the optimal strategy followed by the man-ager under the assumption that she knows exactly the parameters drivingthe asset price process. They show that, when at an intermediate date thereturn of the fund is either very low or very large compared to the bench-mark, the manager forgets abut the implicit incentives determined by thefund-flows and reverts to the normal strategy, that is the one determined byMerton (1971)[35]. However, when the current return is closer to the bench-mark, the manager tilts her strategy from the Merton level to try to beatthe benchmark. Nicolosi et al. (2018)[36] extend their framework to considermean-reversion either in the market price of risk or in the volatility. Basaket al. (2008)[8] introduce additional restrictions on the set of admissiblestrategies to contrast the tendency of managers to increase riskiness whentheir portfolio under-performs the benchmark, in order to align managers’scope to that of investors. The optimal allocation problem for institutionalinvestors concerned about their performance with respect to a benchmarkindex is studied in Basak and Pavlova (2013)[6]. The objective there is toshow how incentives influence the prices of the assets hold by institutionalinvestors. In particular the authors found that, differently from standardinvestors, institutions tend to form portfolios of stocks that compose thebenchmark index, they push up prices of stocks in the benchmark indexby generating excess demand for index stocks and induce excess correlationamong these stocks. Carpenter (2000)[14] analyses the optimal investmentproblem of a risk adverse manager who is compensated with a call option onthe asset under management. In this paper the market model is assumed tobe complete and the non-concavity of the objective function is addressed byintroducing a concavification argument and showing that the optimal solu-tion takes values on a set where the original non-concave objective functionis equal to the minimal concave function dominating it. An explicit solutionto this problem in the Black-Scholes setting is provided in Nicolosi (2018)[25]while Herzel and Nicolosi (2019)[26] extend the solution to the case of mean-reverting processes. The impact of commonly observed incentive contractson managers’ decisions is also studied in Chen and Pennacchi (2009)[17],where the authors found that for particular compensation structure, whena fund is performing poorly, the deviation from the benchmark portfolio islarger than in case of good performance.Other important contributions on the literature of delegated portfoliomanagement problem include Cuoco and Kaniel (2011)[21], who investi-gate the case where managers receive a direct compensation, related to theperformance, from investors and discuss asset price implications in equilib-rium. Different compensation schemes have been considered, for instance,in Barucci and Marazzina (2016)[4] in a portfolio optimization problem fora manager who is remunerated through a High Water Mark incentive feeand a management fee and in Barucci et al. (2019)[5] where a penalty onthe remuneration is applied if the fund value falls below a fixed threshold,namely a minimum guarantee.
IMPLICIT INCENTIVES AND PARTIAL INFORMATION
Optimal asset allocation under partial information has been widely stud-ied in the literature. Brendle (2006) [11] considered the optimal investmentproblem for a partially informed investor endowed with bounded CRRA pref-erences in a market model driven by an unobservable market price of riskvia the HJB approach. Hata et al. (2018) [24] also included consumption.A more general setting, not necessarily Markovian, has been analysed forinstance in Björk et al. (2010) [10] and Lindensjo (2016)[32], under the as-sumption of market completeness. The optimization problem in these papersis solved using the Martingale approach.The partial information case in a delegated portfolio management hasbeen considered in the recent literature by Barucci and Marazzina (2015)[3]in a slightly different setting compared to ours, where market is subject totwo regimes, modelled via a continuous time two-state Markov chain andin Huang et al. (2012)[27] where investment learning is studied under aBayesian approach.Other contributions in the case where prices are modelled as diffusions areLackner (1995, 1998)[28, 29]. Brennan (1998)[12] and Xia (2001)[37] studythe effect of learning on the portfolio choices, and Colaneri et al. (2020)[19] address the problem of computing the price that a partially informedinvestor would pay to access to a better information flow on the marketprice of risk. Investment problems in a market with cointegrated assetsunder partial information are studied in some recent works as for instanceLee and Papanicolaou (2016) [31] and Altay et al. (2018, 2019) [1, 2].2.
Market model and the portfolio optimization problem
We fix a probability space (Ω , F , P ) . Let F = {F t , t ≥ } be a completeand right continuous filtration representing the global information. We con-sider a market model with one risky asset with price S t , the stock , and onerisk-free asset with price B t . We assume that the price of the risk-free assetfollows dB t B t = rdt with the constant r > representing the constant interest rate. The riskyasset price is modelled by a geometric diffusion dS t S t = µ t dt + σdZ St where Z St is a one dimensional standard Brownian motion, σ > is theconstant volatility and the drift is the process µ t = r + σX t , which depends linearly on the market price of risk X t . The process X t satisfies dX t = − λ ( X t − ¯ X ) dt + σ X dZ Xt , MPLICIT INCENTIVES AND PARTIAL INFORMATION 5 where λ > is a constant representing the strength of attraction toward thelong term expected mean ¯ X , σ X > is the volatility of the market price ofrisk and Z X is a one-dimensional standard Brownian motion correlated with Z S with correlation ρ ∈ [ − , .We assume that the market price of risk is a latent variable that is notdirectly observed, and its value can only be derived through the observationof S t . That means that the available information is given by the filtration F S := {F St , t ∈ [0 , T ] } , generated by the process S . Let us note that, sincethere are two risk factors Z S and Z X , but only one traded asset besides themoney market account, this market model is incomplete.We study the problem of a fund manager who trades the two assets, S t and B t , continuously in time on [0 , T ] , starting from an initial capital w .We assume that the stock does not pay dividends before time T . We de-scribe the trading strategy of the manager by a process θ = { θ t , t ∈ [0 , T ] } representing the fraction of wealth invested in the risky asset at any time t ∈ [0 , T ] . We only consider trading strategies that are self-financing andbased on the available information, hence defining an admissible strategy asa self-financing trading strategy, adapted to the filtration F S and, to preventarbitrage from doubling strategies, such that E (cid:20)(cid:90) T (cid:0) | θ t X t | + θ t (cid:1) dt (cid:21) < ∞ . The set of all admissible strategies is denoted by A S . The wealth processgenerated by an admissible strategy θ t is dW t W t = ( r + θ t σX t ) dt + θ t σdZ St , W = w > . The manager’s compensation is implicitly determined by the value of theAUM at time T , according to f T ( W T , Y T ) W T , where f T is the new fundsflow from investors at time T depending on the relative performance of theportfolio with respect to a benchmark Y . The benchmark Y is the value ofthe constant strategy β and hence it follows dY t Y t = ( r + βσX t ) dt + βσdZ St . The continuously compounded returns on the manager’s portfolio and onthe benchmark over the period [0 , t ] are given by R Wt = ln W t W and R Yt =ln Y t Y , respectively. To compare relative performances, we set Y = W . Thedifference R WT − R YT provides the tracking error of the final wealth relativeto the benchmark. The funds flow to relative performance relationship is At any time t , F St is the right continuous and complete σ -algebra generated by theprocess S up to time t . Specifically, F St := σ { S u , ≤ u ≤ t } ∨ O where O is the collectionof all P -null sets. Notice that F St ⊂ F t , which models the fact the manager has a restrictedinformation on the market. IMPLICIT INCENTIVES AND PARTIAL INFORMATION described by function f T f T ( W T , Y T ) = f L if R WT − R YT < η L f L + ψ · ( R WT − R YT − η L ) if η L ≤ R WT − R YT < η H f H := f L + ψ · ( η H − η L ) if R WT − R YT ≥ η H (1)with f L > , ψ > , and η L ≤ η H and it is illustrated in Figure 1. This sim- -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 R WT - R YT f T ( W T , Y T ) Fund fl ow rate f L f H η L η H Figure 1.
The funds flow f T ( W T , Y T ) as a function of rel-ative performance R WT − R YT , with parameters f L = 0 . , f H = 1 . , η L = − . and η H = 0 . .plified structure of the funds flow to relative performance relationship, calledin the literature collar type , shows that if the manager return is below thebenchmark return of at least η L or above the benchmark return of at least η H , the flow rate received by the fund is flat (with different rates f L < f H ).When the relative performance, measured in terms of tracking error, is be-tween η L and η H , the flow function is a linear segment with a positive slope.The function f T also has two kinks when the difference R WT − R YT reachesthe levels η L and η H . The funds flow to relative performance relationshipin Equation (1) was proposed by Basak et al. [7] to describe an implicitincentive scheme and it is based on the empirical analysis of Chevalier andEllison (1997) [18]. The idea is that, if the fund under-performs with re-spect to the benchmark, investors tend to withdraw their money, the AUMdecreases, and the manager receives a lower compensation. The oppositehappens in the case of over-performance. Citing Basak et al. [7]: “(...) thissimple way of modeling fund flows is able to capture most of the insightspertaining risk-taking incentives of a risk averse manager". MPLICIT INCENTIVES AND PARTIAL INFORMATION 7
The manager maximizes the expected utility of her implicit incentives overthe set of admissible strategies A S , max θ ∈A S E [ u ( W T f T ( W T , Y T ))] (2)with initial budget W = w . We assume that the manager is endowed witha power utility function u ( x ) = 11 − γ x − γ , with nonnegative risk aversion parameter γ (cid:54) = 1 . The case γ = 1 correspondsto the logarithmic utility. Since the market price of risk is not observable,this is an optimization problem under restricted information. To solve it,we first reduce it to a setting with a common information flow by replacingthe unobservable process X t with its conditional expectation. This standardprocedure allows us to consider an equivalent optimization problem underthe available information, see, e.g. Fleming and Pardoux (1982)[23]. Wecharacterize the conditional expectation of X t in the next section via Kalmanfiltering. 3. Optimal wealth and strategies
In this section we solve the problem (2). The first step is to estimate theunobservable market price from stock prices. Applying the Kalman filteringtheory we get that the conditional distribution of market price of risk isGaussian with conditional mean π t = E (cid:2) X t |F St (cid:3) , and conditional variance R t := E (cid:104)(cid:0) X t − E (cid:2) X t |F St (cid:3)(cid:1) |F St (cid:105) . To derive π t and R t we introduce the innovation process I t := Z St + (cid:90) t ( X u − π u ) du. (3)It is well known (see, e.g. Lipster and Shiryaev (2001)[33] or Ceci andColaneri (2012, 2014)[15, 16]) that I t is a Brownian motion with respectto the observable filtration F S . The proposition below, proved for instancein Lipster and Shiryaev (2001)[33], provides the dynamics of π t and R t . Proposition 1.
The conditional mean and variance of the market price ofrisk satisfy the equations dπ t = − λ ( π t − ¯ X ) dt + ( R t + ρσ X ) dI t , π ∈ R , (4) dR t = (cid:2) σ X − λR t − ( R t + ρσ X ) (cid:3) dt, R ∈ R + . (5) Notice that the stock price process S and its log-return generate the same type ofinformation. This is a key feature since the drift of the log-return is a linear function of X , and hence the Kalman filter applies. The same setting has been considered for instancein Colaneri et al. (2020)[19]. IMPLICIT INCENTIVES AND PARTIAL INFORMATION
From (5) we see that the conditional variance of the market price of riskis deterministic and satisfies a Riccati ordinary differential equation. UsingEquation (3), we also get the equivalent dynamics of the stock, the wealthand the benchmark under partial information dS t S t = ( r + σπ t ) dt + σdI t ,dW t W t = ( r + θ t σπ t ) dt + θ t σdI t , (6) dY t Y t = ( r + βσπ t ) dt + βσdI t . All processes are driven by the innovation process and hence the sub-marketrestricted to those claims that can be replicated by strategies in A S is com-plete. We solve the optimization problem (2) using the martingale method(see, for instance Cox and Huang (1989)[20]), transforming the dynamicoptimization problem (2) where the control variable is a strategy into anequivalent static problem where the control variable is the terminal wealth.To identify the terminal wealths reachable from the initial budget w withfeasible strategies, we introduce the unique state price density process dξ t ξ t = − rdt − π t dI t , ξ = 1 . The static optimization problem, equivalent to (2) is max W T E [ u ( W T f T ( W T , Y T ))] , (7)with budget constraint w = E [ ξ T W T ] (8)The objective function in problem (7)–(8) is not concave in W T . To overcomethis issue we apply the concavification procedure proposed by Carpenter(2000)[14]. Following the approach in Proposition 2 of Basak et al. (2007)[7],we define the optimal final wealth relative to the benchmark, which is givenby V T = W (cid:63)T Y T , where W (cid:63)T is the optimal final wealth in problem (7)-(8). Oneof the advantages of working with this quantity is that it has an explicitrepresentation (e.g. equation ( A in Basak et al. (2007)[7] or equation (6) in Nicolosi et al. (2018)[36]) given, for completeness, by equation (11)in Appendix A. One key characteristic is that V T is a function of ζ T := ξ T Y γT only. Computing V T , enables us to characterize the optimal terminalwealth. However this is not sufficient to obtain the trading optimal portfoliostrategy, for which, we need to know the value of optimal wealth at any time t ∈ [0 , T ] , and consequently, we must determine the relative wealth V t = W (cid:63)t Y t .To compute V t we consider the benchmarked market , where we discount allprocesses with the numéraire Y t . Due to market completeness there exists an Notice that the benchmark is a positive self-financing portfolio, and hence it can betaken as numéraire.
MPLICIT INCENTIVES AND PARTIAL INFORMATION 9 equivalent risk neutral probability measure Q for the benchmarked market.Put in other words there exists a probability measure Q which is equivalentto P and such that the price process of any benchmarked traded asset (i.e.any traded asset discounted with Y t ), is a martingale under Q .We define the process ζ t = ξ t Y γt and derive its distribution under Q . Letthe conditional moment generating function of ln( ζ t ) under the measure Q be given by H ( t, ζ, π ; z ) = E Q (cid:2) ζ zT |F St (cid:3) = E Q [ ζ zT | ζ t = ζ, π t = π ] for some complex number z . The function H ( t, ζ, π ; z ) plays a key role insolving the optimization problem (see Proposition 2 below). It is character-ized in the following technical lemma: Lemma 1.
Under the usual regularity conditions, the conditional momentgenerating function of ln( ζ T ) under the measure Q is given by: H ( t, ζ, π ; z ) = ζ z e A ( t ; z )+ B ( t ; z ) π + C ( t ; z ) π where A ( t ; z ) , B ( t ; z ) and C ( t ; z ) are deterministic functions satisfying asystem of Riccati Equations. The proof of Lemma 1 and the Riccati equations for the functions A ( t ; z ) , B ( t ; z ) and C ( t ; z ) are given in Appendix B and goes along the same lines asin Nicolosi et al. (2018)[36]. We remark that in this particular case, becauseboth the drift and volatility in the dynamics of the filter π t are not constant(4), the coefficients of the Riccati equations that characterize the functions A, B and C are time-dependent. The solutions of non-homogeneous Riccatiequations are discussed in Appendix C.In the next step we use Fourier Transform to compute the optimal relativewealth and the optimal strategy at any time t ≤ T . Proposition 2.
Let R j , for j = 1 , , , be real numbers such that R < − /γ , R > − /γ and H ( t, ζ, π ; R j ) = E Q (cid:104) e R j ln( ζ T ) | ζ t = ζ, π t = π (cid:105) < ∞ . Then, (i) the relative wealth V t is given by V t = 12 π (cid:88) j =1 (cid:90) + ∞−∞ ˆ ϕ j ( u + iR j ) H ( t, ζ, π ; R j − iu ) du (9) where the functions ˆ ϕ j ( z ) of the complex variable z are given in Ap-pendix A; Introducing the measure Q allows us to circumvent technical difficulties: for instance,to get the optimal wealth W (cid:63)t under the physical measure P , one should know the jointdistribution of Y γ and ξ . This is unnecessary if we perform the change of measure, whereone can use the martingale property and get the distribution of W (cid:63)t more directly. See,e.g. Proposition 2 in Basak et al. (2007)[7] (ii) the optimal strategy is θ t = β + 1 σV t (cid:18) ∂V∂ζ ζ t ( γβσ − π t ) + ∂V∂π ( R t + ρσ X ) (cid:19) . (10)A sketch of the proof of Proposition 2 is given in Appendix B. It is impor-tant to notice that Proposition 2 provides a useful semi-closed solution for theoptimal relative wealth and the optimal strategy. Examples of applicationsof those formulas are given in the next Section 4.4. A numerical illustration
In this part of the paper we study the implications of considering pa-rameter uncertainty on optimal strategies of a portfolio manager subject toimplicit incentives. The goal is to show that the impact depends on a com-bined effect of risk aversion and of market conditions. In a relatively stablemarket (i.e. low volatility) with lower expected returns, portfolio managerstend to increase their exposure to the risky asset when underperformingthe benchmark and decrease it when overperforming. The opposite happenswhen the market is more volatile and expected returns are higher. Risk-aversion has a direct influence on the view of the manager regarding theuncertain estimates. Managers with a risk-aversion parameter larger than ,fear that the true value may be below the current estimate and hence reducetheir leverage. On the contrary, managers with risk aversion lower than hope that the true value may be higher than the current estimate and hencetilt their strategy in the opposite way.To illustrate such behavior with an example, we consider a simplifiedversion of our model, where the market price of risk is constant but unknown,and is represented by a random variable X . The manager who is uncertainabout the value of the market price of risk assumes that the exact value of X is drawn from a normal random variable with mean π and variance R .This setting is analogous to that of Brennan (1998), where the manager doesnot know the value of the drift of the price process and can only estimate itsexpected value m , which is related to the market price of risk π by π = m − rσ . Setting λ = 0 and σ X = 0 , we get a stochastic conditional mean π t = R t dI t , π ∈ R , and conditional variance R t = R R t + 1 . To highlight the effects of uncertainty of parameter estimates, as a com-parison we use the strategy of a manager who believes that she knows exactlythe value X of the market price of risk. We call this manager myopic be-cause she does not adjust her strategy to hedge for future changes on theestimates. We denote by θ t the fraction of wealth invested by the myopic MPLICIT INCENTIVES AND PARTIAL INFORMATION 11 manager in the risky asset and by θ t the optimal strategy of the partiallyinformed manager given by equation (10). For comparison reason, we alsoconsider the Merton level θ N = γ µ − rσ that is the optimal investment of themyopic manager who optimizes only the utility of terminal wealth, withoutother incentives.Figure 2 represents the myopic strategy θ t (dotted line) and the optimalstrategy under partial information θ t (continuous line) as functions of therelative return of the portfolio with respect to the benchmark, that is R Wt − R Yt , at time t = 0 . , either for γ = 0 . (left panels) or for γ = 2 (rightpanels). The parameters of the implicit incentives structure at time T = 1 in Equation (1) are the same as in Figure 1.The top panels show the strategies when the Merton level is higher thanthe investment in the risky asset of the benchmark portfolio, that is when θ N > β . This setting corresponds to a market situation with relatively smallvolatility and returns, that is called Economy (a) by Basak et al. (2007),obtained by taking σ = 0 . , r = 0 and m = 0 . in our model.The bottom panels show the strategies when the Merton level is lower thaninvestment in the risky asset of the benchmark, that is when θ N < β . Thisis Economy (b) in Basak et al. (2007) which describes a more volatile andremunerative market and it is obtained by setting σ = 1 , r = 0 and m = 0 . .The level of uncertainty on the initial estimate is given by R = 0 . for allthe panels.The panels on the left represent a less risk averse manager γ = 0 . , thoseon the right a more risk-averse one γ = 2 . By comparing the left to theright panels we see that the investment in risky asset decreases with riskaversion for both Economies (a) and (b). The strategies for a myopic anda non-myopic manager are qualitatively similar to each other but the non-myopic manager invests always more or less than the myopic one, dependingon the risk-aversion parameter. When the risk aversion parameter γ is equalto (that is the case of logarithmic utility), the strategies of the myopicand of the non-myopic manager coincide. A non-myopic manager with arisk aversion smaller than (left panels) tends to be more exposed to therisky asset than a myopic manager with the same risk aversion. In this case,the non-myopic manager acts optimistically, as she believes that, increasingthe precision of the estimates of the market price, the correct value will behigher than the current one. Instead, the more risk-averse manager (rightpanels) is pessimistic and reduces the exposition to the risky asset fearingthat the future estimates will be lower than the current one. By comparingtop to bottom panels in Figure 2 we see the effects of the overall economiccondition on the optimal strategy, depending on the current results of theportfolio management strategy. When the relative performance is either toolow or too high for the incentives to have an effect on the final reward, theoptimal strategy approaches a constant level that corresponds to the optimalrisky exposure without incentives and hence the myopic investment convergesto the Merton level. If the manager is underperforming but still hopes to -4 -2 0 2 412345678 -4 -2 0 2 412345678 tt0 -4 -2 0 2 400.10.20.30.40.50.60.7 -4 -2 0 2 400.10.20.30.40.50.60.7 tt0 Figure 2.
Optimal strategies for different economies and dif-ferent managers. The optimal strategy θ t (continuous line)and the myopic one θ t (dotted line) at time t = 0 . are re-ported as functions of the relative return of the portfolio withrespect to the benchmark R Wt − R Yt . Left panels representthe strategies of managers with risk aversion γ = 0 . , rightpanels those of more risk-averse managers ( γ = 2 ). Top pan-els represent economy (a), that is a less volatile market withhigher returns ( σ = 0 . , π t = 0 . ), the bottom panels arereferred to economy (b), a more volatile market with lowerreturns ( σ = 1 , π t = 0 . ). The parameters of the payoff func-tion are the same as in Figure 1. The others parameters are T = 1 , r = 0 , and R = 0 . .recover, or if she is slightly ahead, but still fearing to end behind, she adjuststhe portfolio strategy in a way that depends on the economic conditions.In the case of economy (a) (top panels), she increases the exposure when MPLICIT INCENTIVES AND PARTIAL INFORMATION 13 trailing and decreases it when leading. The economy (b), representing amore volatile market and higher expected returns (bottom panels), inducesthe same manager to take opposite choices.5.
Conclusions
We studied a portfolio optimization problem for a manager who is com-pensated depending on the performance of her portfolio relative to a bench-mark. The manager can invest in a risk-free asset and in a risky asset whosereturn depends on a latent variable representing the market price of risk.Hence she solves an optimization problem under partial information. Dueto the implicit incentives given by the funds flow to relative performance,the utility function of the manager is not concave and hence existence of theoptimum does not trivially hold. We solve the optimization problem usingthe martingale approach and a concavification procedure. This approachcan be successfully applied due to completeness of the market under par-tial information. Optimal wealth and consequently the optimal strategy arecharacterized in a semi-explicit form via Fourier transform. We illustratedour results with an example, where we assume that the market price of riskis constant but unknown. We observed that the level of risk aversion hasan influence on the manager’s estimate of the market conditions, and con-sequently on her investment choices. Managers with a small risk aversionparameter are optimistic: they tend to increase their investment in the riskyassets compared to myopic managers, believing that the true value of themarket price of risk (and hence of the asset return) is more favourable thanher estimate. It is also seen that if the market is not subject to large fluctu-ations, managers invest more in risky assets when they are underperformingthe benchmark, in anticipation to retrieve benchmark revenues, and investless in the risky asset when overperforming, to avoid possible downwardmovements of the market.
Appendix A. Optimal final wealth
In this section we characterize the final wealth relative to the benchmark V t , for every t ∈ [0 , T ] . We first consider the optimal final wealth relative tothe benchmark, given by the random variable V T = W ∗ T Y T . Its expression hasbeen computed in Basak et al. (2007) and also reported in Nicolosi et al.(2018), and in our framework, is given by V T = ϕ ( ζ T ; ζ ) + ϕ ( ζ T ; ζ , ζ ) + ϕ ( ζ T ; ζ , ζ ) + ϕ ( ζ T ; ζ ) (11) where ζ T = ξ T Y γT and functions ϕ j , for j = 1 , . . . , are ϕ ( ζ ; ζ ) = f /γ − H y − /γ ζ − /γ ζ<ζ (12) ϕ ( ζ ; ζ , ζ ) = e η H ζ ≤ ζ<ζ (13) ϕ ( ζ ; ζ , ζ ) = h ( ζ ) ζ ≤ ζ<ζ (14) ϕ ( ζ ; ζ ) = f /γ − L y − /γ ζ − /γ ζ ≥ ζ . (15)The value y ∈ R is the Lagrange multiplier that ensures that the budget con-straint of the optimization problem w = E [ ξ T W ∗ T ] is satisfied; the function h ( ζ ) is the solution of the equation ddV u ( V f L + V ψ (ln V − η L )) = yζ. Parameters ζ , ζ and ζ , and hence the value of V T , depend on the followingrelation, called Condition A : γ − γ (cid:18) f H + ψf L (cid:19) − /γ + (cid:18) f H + ψf H (cid:19) − − γ ≥ . This condition is related to concavification and defines the region in whichthe optimum of the maximization problem with the utility function and theoptimum of the optimization problem where the utility function is replacedby the smallest concave function above it, is reached in a point where thetwo functions coincide.Proposition 2 of Basak et al. (2007) shows that, if
Condition A holds, then ζ , ζ and ζ in (11) satisfy ζ = f − γH e − γη H /y , and ζ = ζ > ζ satisfying g ( ζ ) = 0 , with g ( ζ ) = (cid:32) γ (cid:18) yf L ζ (cid:19) − /γ − ( f H e η H ) − γ (cid:33) / (1 − γ ) + e η H yζ. Hence, in this case, ϕ ( ζ ; ζ , ζ ) is the indicator function of the empty setand therefore it is zero.When Condition A is not met, Basak et al. (2007) show in Appendix Cthat ζ = f − γH e − γη H /y , ζ = ( e η H f H ) − γ ( f H + ψ ) /y and ζ = ( f L V ) − γ f L /y with V being the left boundary of the region where the objective function isnot concave. Denoting with V the right boundary of the non concave region, V and V can be computed as the points where the straight line betweenthese two points is tangent to the objective function.Next, we provide a representation for the function ˆ ϕ j ( z ) , j = 1 , . . . , ,which are used to compute the optimal relative wealth V t given in Proposition2. The functions ˆ ϕ j ( z ) , j = 1 , . . . , , are the Fourier transforms of the MPLICIT INCENTIVES AND PARTIAL INFORMATION 15 functions in (12)-(13)-(14)-(15) and they are given by ˆ ϕ ( z ) = f /γ − H y − /γ ( ζ ) − /γ + iz − /γ + iz ˆ ϕ ( z ) = e η H ( ζ ) iz − ( ζ ) iz iz ˆ ϕ ( z ) = f /γ − L y − /γ ( ζ ) − /γ + iz − /γ + iz . Numerical computations of the Fourier transform ˆ ϕ ( z ) , which are neededonly when Condition A is not satisfied, are given in Section 4.1 of Nicolosiet al. (2018)[36].
Appendix B. Proofs
This section contains the proofs of Lemma 1 and Proposition 2.
Proof of Lemma 1.
We define the process I Qt = I t − (cid:90) t ( σβ − π s ) ds. By Girsanov Theorem this is a Q -brownian motion (see, e.g. Chap. 26 ofBjörk (2009)[9]). Then the Q -dynamics of the filter π t is given by dπ t = (cid:0) λ ( ¯ X − π t ) + ( R t + ρσ X )( βσ − π t ) (cid:1) dt + ( R t + ρσ X ) dI Qt . Using Ito’s Lemma, we get that ζ t = ξ t Y γ under Q has the following dy-namics dζ t ζ t = (cid:18) r ( γ −
1) + 12 γ ( γ + 1) β σ − ( γ + 1) βσπ t + π t (cid:19) dt + ( γβσ − π t ) dI Qt . Since the process H ( t, ζ t , π t ; z ) is a ( F S , Q ) -martingale, equating the dt -termto zero leads to the partial differential equation (for simplicity we drop thearguments of the functions) ∂H∂t + ∂H∂ζ ζ (cid:18) r ( γ −
1) + 12 γ ( γ + 1) β σ − ( γ + 1) βσπ + π (cid:19) + ∂H∂π (cid:0) λ X ( ¯ X − π ) + ( R t + ρσ X )( βσ − π ) (cid:1) + 12 ∂ H∂π ( R t + ρσ X ) + ∂ H∂ζ∂π ζ ( R t + ρσ X )( γβσ − π ) + 12 ∂ H∂ζ ζ ( γβσ − π ) with the boundary condition at time TH ( T, ζ, π ; z ) = ζ z , ζ ∈ R + , π ∈ R , z ∈ C . (19) We use a similar approach as in the optimization problem under full informa-tion (see Nicolosi et al. (2018)[36]), and consider an exponential-polynomialansatz of the type H ( t, ζ, π ; z ) = ζ z e A ( t ; z )+ B ( t ; z ) π + C ( t ; z ) π where A ( t ; z ) , B ( t ; z ) and C ( t ; z ) are deterministic functions. From theboundary condition (19) we get that A ( T ; z ) = 0 , B ( T ; z ) = 0 , C ( T ; z ) = 0 . Moreover, substituting the partial derivatives of the function H into (18) andimposing that coefficients of π , π and the constant terms are equal to zero,we obtain the system of ordinary differential equations for A ( t ; z ) , B ( t ; z ) and C ( t ; z ) ∂C∂t = − ( R t + ρσ X ) C + 2( λ X + (1 + z )( R t + ρσ X )) C − z ( z + 1) (20) ∂B∂t = (cid:0) λ X + ( z + 1)( R t + ρσ X ) − ( R t + ρσ X ) C (cid:1) B − (cid:0) λ X ¯ X + (1 + zγ ) βσ ( R t + ρσ X ) (cid:1) C + z (1 + zγ ) βσ (21) ∂A∂t = zr (1 − γ ) − zγ (1 + zγ ) β σ −
12 ( R t + ρσ X ) B − ( λ X ¯ X + (1 + zγ ) βσ ( R t + ρσ X )) B −
12 ( R t + ρσ X ) C. (22)Notice that this is a coupled system of equations of Riccati type, with non-homogeneous coefficients. (cid:3) Proof of Proposition 2.
The proof of part ( i ) follows the same lines of Ni-colosi et al. (2018)[36, Proposition 2.1]. Here we summarize the idea. Sincethe market model under partial information is complete, after applying con-cavification we get that the relative final wealth V T is given by the formula(11) in Appendix A. Plugging the expression of V T into V t = E Q (cid:2) V T |F St (cid:3) and then using Fourier transform we can calculate the value at time t ofthe optimal relative value. For part ( ii ) , we first determine the dynamicsof W (cid:63)t = Y t V t = Y t V ( t, ζ t , π t ) via Ito’s product rule. Then comparing thisequation with equation (6) provides the expression for θ t in (10). Notice thatthe integrals in (9) are principal value integrals and the partial derivativesof the function V in (10) can be computed from (9) by taking the derivativeunder the integral sign. (cid:3) Appendix C. Solutions to non-Homogeneous Riccati ODEs
We discuss the solution of the system of non-homogeneous system of Ric-cati equations arising in the expression of the conditional moment generatingfunction of ln( ζ T ) . Precisely, we show how to solve the system of equations(20) – (21). Equation (22) can be computed by direct integration, and wedo this numerically. Following, for instance, Brendle (2006) and Colaneri etal. (2020), it can be proved that the functions B and C satisfy MPLICIT INCENTIVES AND PARTIAL INFORMATION 17 C ( t ; z ) = C o ( t ; z )1 + z C o ( t ; z ) R t B ( t ; z ) = B o ( t ; z )1 + z C o ( t ; z ) R t for some functions C o ( t ; z ) and B o ( t ; z ) which solve the homogeneous systemof Riccati equations below ∂C o ∂t = (cid:18) z (1 − ρ ) − ρ (cid:19) σ X C o + 2( λ X + (1 + z ) ρσ X ) C o − z ( z + 1) ∂B o ∂t = (cid:18) λ X + ( z + 1) ρσ X + (cid:18) z (1 − ρ ) − ρ (cid:19) σ X C o (cid:19) B o − (cid:0) λ X ¯ X + (1 + zγ ) βσρσ X (cid:1) C o + z (1 + zγ ) βσ with boundary conditions B o ( T, z ) = 0 , C o ( T, z ) = 0 . (25)Equations (23)–(24) have a solution in closed form see for instance Filipović(2009)[22, Lemma 10.12]. References [1] Altay, S., Colaneri, K., and Eksi, Z. (2018). Pairs trading under driftuncertainty and risk penalization. International Journal of Theoretical andApplied Finance, 21(7).[2] Altay, S., Colaneri, K., and Eksi, Z. (2020). Optimal Converge Trad-ing with Unobservable Pricing Errors. The Annals of Operations Research.https://doi.org/10.1007/s10479-020-03647-z.[3] Barucci, E., and Marazzina, D. (2015). Risk seeking, nonconvex remuner-ation and regime switching.
International Journal of Theoretical and Ap-plied Finance Equations (23)–(24) are related to the conditional moment generating function of theprocess ln( ζ T ) , under full information. In fact, by Markovianity it holds that (cid:101) H ( t, ζ, π ; z ) := E Q [ ζ zT |F t ] . Here using the ansatz (cid:101) H ( t, ζ, π ; z ) = ζ z e A o ( t ; z )+ B o ( t ; z ) π + C o ( t ; z ) π we get that B o ( t ; z ) and C o ( t ; z ) solve (23)–(24) with the boundary condition (25) and A o ( t ; z ) satisfies ∂A o ∂t = zr (1 − γ ) − zγ (1 + zγ ) β σ − ( λ X ¯ X + B o (1 + zγ ) βσσ X ) − σ X ( C o + B o ) , with the boundary condition A o ( T ; z ) = 0 . [6] Basak, S., and Pavlova, A. (2013). Asset Prices and Institutional In-vestors. American Economic Review. 103 (5): 1728–1758[7] Basak, S., Pavlova, A., and Shapiro, A. (2007). Optimal asset allocationand risk shifting in money management. The Review of Financial Studies
Journal of EconomicTheory , 49, 33-83.[21] Cuoco, D., and Kaniel, R. (2011). Equilibrium prices in the presence ofdelegated portfolio management. Journal of Financial Economics, 101(2),264-296.[22] Filipovic, D. (2009). Term-Structure Models. A Graduate Course.Springer.[23] Fleming, W. H. and Pardoux, É. (1982). Optimal control for partiallyobserved diffusions, SIAM Journal on Control and Optimization, 20 (2),261–285.