Dynamic Contracts and the Sharpe Ratio: Theory and Evidence

Transcription

1 Dynamic Contracts and the Sharpe Ratio: Theory and Evidence Raymond C. W. Leung June 29, 2017 Abstract We show theoretical and empirical asset pricing implications of long-term dynamic contracts between households and financial intermediaries. In our continuous-time joint contracting and asset pricing model, dynamic contracts induce households to have endogenous absolute- and downside-risk aversions over the intermediary s wealth. In equilibrium, these two risk aversions are priced into the Sharpe ratio of the intermediated economy. Financial sector volatilities and credit spreads, which we use as empirical proxies for the two respective risk aversions, affect the broad equity market Sharpe ratio in the time series as predicted by our model. In the cross-section of 135 equity and non-equity portfolios, short-run shocks to the two proxies is a priced factor ( intermediary insurance factor) with a large negative risk price suggesting households willingness to pay a significant insurance premium to hedge away contracting breakdowns with the intermediaries. Our results stress the role of intermediaries through dynamic contracts and the consequent endogenous risk aversions, even without capital constraints. Keywords: dynamic contracting, intermediated asset pricing, market price of risk, time series regression, cointegration, cross-sectional regression, intermediary insurance factor Department of Finance, Cheung Kong Graduate School of Business. raymondleung@ckgsb.edu.cn. I thank Rui Albuquerque, Lorenzo Garlappi, Jennifer Huang, Ou-Yang Hui, Nengjiu Ju, and Ron Kaniel for very helpful discussions and suggestions. I thank the seminar participants at the Cheung Kong Graduate School of Business, the 2016 Five-Star Workshop in Finance, 2017 Asian Meeting of the Econometric Society and 2017 China Meeting of the Econometric Society for their helpful comments. I thank Zeyu Li for very helpful research assistance. This paper is based on a chapter from my doctoral dissertation completed at UC Berkeley, Haas School of Business. I am deeply indebted to and grateful for the countless hours, encouragement, patience and comments of both of my advisers, Robert M. Anderson and Gustavo Manso, on this project and other research endeavors. This paper was previously circulated as Financial Intermediation and the Market Price of Risk: Theory and Evidence. All errors are mine and mine alone.

2 It is well known that the majority of US household wealth is delegated to and managed by financial intermediaries. In effect, these financial intermediaries and not households as classical asset pricing theories would assume are the true marginal investors in the modern financial markets. By nature of delegation, the question is how contracts to these financial intermediaries influence their asset allocation decisions, and ultimately the aggregate demand of risky assets and equilibrium prices. The existing literature predominantly addresses this contract-induced asset pricing problem by assuming short-term or exogenous contracts. However, these assumptions are at odds with the fact that the financial sector is highly concentrated towards a few large broker-dealers and household investors who want to transact complex financial products must contract with these same large players for the long-run. Through the stochastic discount factor, the core driver of almost all classical asset pricing theories is the market price of risk (or Sharpe ratio). In this paper we ask: how do long-term contracts for a financial intermediary impact the equilibrium market Sharpe ratio? We approach this question both theoretically and empirically. Our model is built upon long-term dynamic contracts between client investors ( households ) and financial intermediaries ( specialists ). Long-term contracts endogenously generate two key definitions of our paper: (A) household s absolute risk aversion over the financial intermediary s wealth; and (B) household s downside risk aversion over the financial intermediary s wealth. The core asset pricing prediction of our model is, Market Sharpe ratio t (Household s absolute risk aversion over the financial intermediary s wealth t ) 1 + Household s downside risk aversion over the financial intermediary s wealth t, (M) That is, household s absolute risk aversion is negatively related to the market price of risk, while household s downside risk aversion is positively related. In our model, a risk neutral household cannot access the financial markets and must contract with and incentivize a risk averse financial intermediary for access. Moreover, the financial intermediary must exert costly private effort to ensure high dividend growth rates of a representative risky asset. Given an arbitrary but committed long-term contract, the specialist decides how much effort to exert, how much portfolio wealth to allocate to the risky asset, and how much portfolio wealth to withdraw for his own consumption. In return, the 2

3 household receives dividends from the representative risky asset and decides how much to consume and how much to compensate the specialist. In equilibrium, both the consumption good and security markets clear. Existing papers in the literature use short-term contracts and exogenous capital constraints on the financial intermediary to drive equilibrium asset pricing results. In contrast, our paper considers endogenous long-term dynamic contracts for a financial intermediary who does not face capital constraints. We show that long-term dynamic contracts induce the household to have endogenous risk aversions (A) and (B) over the financial intermediary s wealth. Absolute risk aversion (A) captures the household s concern for uncertain bi-directional swings of the specialist s wealth. Downside risk aversion (B) captures the household s concern for uncertain one-sided directional swings of the specialist s wealth. Generically, such endogenous risk aversions arise only when long-term dynamic contracts are considered. The optimal long-term contract can be implemented as a linear-quadratic form over the specialist s wealth. Long-term contracts, unlike short-term contracts, have long-lasting effects on the specialist s portfolio choice trajectory. The specialist takes the mean and variability of his contract pay into account when making portfolio choices into the risky asset. However, the contract pay mean and variability depend on the household s endogenous absolute- and downside-risk aversions over the financial intermediary s wealth. As the specialist is the marginal investor in the economy, upon market clearing, long-term contract implies the household s enogenous risk aversions are priced into equilibrium asset prices. In all, this culminates to our key asset pricing prediction on the market Sharpe ratio (M). Next we turn to the empirical tests of our theory prediction (M). We use monthly data from January 1985 to December We empirically proxy: (i) the household s absolute risk aversion over the financial intermediary s wealth (A) by financial sector volatility; and (ii) the household s downside risk aversion (B) by financial sector credit spreads. The empirical option pricing literature provides evidence that volatility and absolute risk aversion are related to each other. The empirical corporate bond literature provides evidence that credit spreads may be largely driven by investor s downside risk aversions. We present both static and dynamic time series results. For static time series results, we run monthly time series regressions of the market Sharpe ratio on the two proxies. For various definitions of the market, including the Fama-French market factor and all major 3

4 US equity indices (e.g. S&P 500, Russell 2000, NASDAQ 100, Wilshire 5000 and the Dow Jones Industrial Average), the estimated coefficients on proxies (i) and (ii) are statistically significant, economically large, and most importantly having the same signs as predicted by the model (M). As (M) is a dynamic prediction, the relationship between the market Sharpe ratio, household s absolute risk aversion (A), and downside risk aversion (B) should also hold dynamically over time empirically. Via cointegration tests, we show the aforementioned relationship indeed hold dynamically in the data. Our results are robust to alternative specifications on proxies (i) and (ii), econometric inference technicalities, and controls for the financial intermediary capital ratio variables of Adrian, Etula, and Muir (2014) and He, Kelly, and Manela (2016). These static and dynamic time series results offer strong empirical validity for our theory that the effects of long-term contracts are priced into the market Sharpe ratio. Given the results of Adrian, Etula, and Muir (2014) and He, Kelly, and Manela (2016) in using financial intermediary variables for asset pricing of cross-sectional excess returns, we naturally ask whether our two proxies for the household s risk aversions can enjoy a similar success. We propose the intermediary insurance factor. The intermediary insurance factor is defined as the residuals of a time series regression of proxy (i) onto proxy (ii). The econometric motivation for the intermediary insurance factor comes from the fact that financial sector volatility and financial sector credit spreads are cointegrated, and thereby their residuals yield a single stationary process suitable for cross-sectional regressions. We discuss the economic motivation of our factor after we highlight the empirical results. We find a surprisingly large negative risk price for the intermediary insurance factor. Using a total of 41 test assets 1 we find a risk price of 34.60% (annualized) for the intermediary insurance factor in the monthly cross-section. If we consider instead the quarterly cross-sectional regressions of He, Kelly, and Manela (2016) and include 135 test assets 2 we find a risk price of 11.09% (annualized) for our intermediary insurance factor. Unlike positive risk prices, the large negative risk price suggests households are willing to pay a large insurance premium to hedge away this priced risk factor. Guided by our model, the intermediary insurance factor represents short-run shocks to 1 e.g. 25 size and book-to-market + 10 momentum + 6 US Treasury portfolios. 2 e.g. 25 size and book-to-market + 10 momentum + 20 US Treasury and corporate bonds + 20 credit default swaps + 18 options + 6 sovereign bonds + 24 commodities + 24 foreign exchange portfolios. 4

5 the long-run relationship between the household s absolute- and downside-risk aversions over the financial intermediary s wealth. When there are short-run shocks to these two risk aversions, the long-run contract from the household to the specialist becomes less effective in incentivizing the specialist to make correct portfolio choices. From the data, these short-run shocks are highly correlated with macroeconomic downturns and times of financial distress. It is conceivable that during these extreme times, other constraints binding on the specialist force him to at least temporarily deviate from any long-term contracts with the household. From the household s perspective, such unilateral deviations from long-term contracts are a systematic risk, which is why the household is willing to pay a large insurance premium to hedge away this risk. This large insurance premium is precisely the large negative risk price of our factor. Indeed, this systematic risk does not earn a positive risk price because the household has no other choice but to access the financial markets through contracting with the intermediary. Related literature This paper contributes to both the theory and empirical literature of intermediated asset pricing. A few studies examine the interaction between asset pricing and optimal contracting. The joint papers of He and Krishnamurthy (2012, 2013) are interesting steps toward this direction, but the contract forms they consider are not long-term dynamic contracts, and the financial intermediary faces exogenous capital constraints. Buffa, Vayanos, and Woolley (2014) considers the impact of static benchmarked contracts on the asset prices that are affine in the state variables. Cvitanić and Xing (2017) extends the framework considered in Buffa, Vayanos, and Woolley (2014) and show that a benchmarked contract can be endogenously generated and study the resulting equilibrium asset prices. However, both Buffa, Vayanos, and Woolley (2014) and Cvitanić and Xing (2017) primarily focus on the effects of benchmarked contracts. Moreover, unlike Buffa, Vayanos, and Woolley (2014), we do not make exogenous assumptions on the asset price form. Cuoco and Kaniel (2011) is one of the first papers to explicitly consider how contracts of managers can affect equilibrium asset prices in a delegated portfolio management setting, but the contracts considered are exogenously specified. Sung and Wan (2013) is an interesting contribution to this line of thinking by considering a discrete-time two-period general equilibrium model with many 5

6 firms; however, our contribution is to cast the problem in continuous-time, and practically only in the dynamic framework do we get interesting dynamics in relating the specialist s wealth to the market price of risk. Ou-Yang (2005) is another attempt to integrate asset pricing with moral hazard. However, Ou-Yang (2005) assumes a priori that the equilibrium asset prices are affine in the dividends, while we do not assume such a restricted form in this paper. From the optimal contracting perspective, the principal of Ou-Yang (2005) can only compensate the agent at the end of the contracting period, whereas we allow for full dynamic optimal contracting over all periods of time. Kaniel and Kondor (2013) considers a delegated Lucas tree with portfolio choice dynamics, but they consider an exogenous contract form for the manager and does not discuss dynamic optimal contracts. There is a small but rapidly growing empirical literature on the impact of the financial sector on the broader market and macroeconomy. Some recent examples include Adrian, Etula, and Muir (2014), He, Kelly, and Manela (2016), Kelly, Lustig, and Nieuwerburgh (2016) and Gilchrist and Zakrajšek (2012). Our results are among one of the first to explicitly link the Sharpe ratios of various reference market portfolios to both the financial sector volatility and the financial sector credit spread. Indeed, guided by the predictions of our model, time series variations of the Sharpe ratios can be attributed to uncertainty risks and downside risks of the financial sector. Finally, we show our two empirical proxies for the households risk aversions over the financial intermediarys wealth is a priced factor with a large negative risk price in a large panel of equity and non-equity portfolios. This large negative risk price can be seen as an insurance premium that the household is willing to pay to hedge away contracting failure of the financial intermediaries. All theory proofs are in the online Theory Appendix A, while additional empirical results are in the online Empirical Appendix B. 1 Model setup We fix a probability space (Ω, F, P). There are two representative individuals in this economy, a principal (household) and an agent (specialist). There exists a representative risky asset producing a stream of dividends D = {D t } of the consumption good. The financial markets consist of a single risky asset that is a claimant on the stream of dividends. The household has no access to the financial markets nor any labor income, but is initially en- 6

7 dowed with the entire supply (which we normalize to unity) of this risky asset. We assume there does not exist a risk-free asset in the market, and hence the only asset in the economy is the risky asset itself 3. The risky asset may be thought of as a portfolio of equities, or a portfolio of complex financial instruments (i.e. CDS, CDOs, etc) for which only the specialist has the ability to access and monitor. 4 The delegation time is infinite, that is the time span is [0, ]. 1.1 Risky asset and dividends The representative risky asset outputs a dividend stream D = {D t } with dynamics, dd t D t = h(a t )dt + ΣdZ A t, Σ > 0 (1.1) where A = {A t } is the private costly effort exerted by the specialist. We interpret the P A - Brownian motion Z A in the sense as discussed in Section A.1 of the Theory Appendix. In particular, we take the following conditions on the dividend drift: Assumption 1.1 (Dividend drift and effort). The set of feasible effort choices is the closed interval [a L, a H ], where 0 < a L < a H <. The reward function h : [a L, a H ] R ++ is C 1 ([a L, a H ]), strictly positive and strictly increasing, and h(a L ) Σ 2 /2 > 0. The effort here can be interpreted as the specialist s private cost in monitoring a basket of complex risky assets. Next, we make the following assumption on the form of the equilibrium gains process. 3 Clearly there is some loss of generality by assuming the nonexistence of the risk free asset. However, in general equilibrium, we want to focus on the effects of delegation on the risky asset drift without effects from the risk free asset, arising from the consumption smoothing motive of the specialist. We will have more to say on this issue in Remark We should note that in contrast, He and Krishnamurthy (2012, 2013) does indeed allow the household to directly invest into the risk free asset, but not the risky asset. But due to differences in how they model the moral hazard problem and also the contracting environment, allowing the household here to have (partial) access to the financial markets substantially complicates our problem. It is conjectured that one could attempt to modify the approach considered by Basak and Cuoco (1998) to this paper, but nonetheless, the presence of the moral hazard and contracting problem substantially complicates the analysis. We leave this problem for future research. 7

8 Assumption 1.2 (Gains process). The price of the risky asset is S and the gains process is such that, ds t + D t dt S t = µ t dt + σ t dz A t, (1.2) where the drift µ and volatility σ are {F A t }-adapted processes that are to be determined in equilibrium. 1.2 Specialist The specialist has time separable logarithmic utility over his consumption with subjective time discount rate ρ M. 5 The specialist has three choice variables: the amount of wealth θ to allocate to the risky asset, his consumption level C M, and the amount of effort A to exert in order to increase the expected dividend growth rate h(a t ) of the risky asset. The specialist restricts his choices to self-financing portfolios P. objective function, 6 sup C M,θ,A subject to the self-financing condition, [ ρ M E 0 e ρ M t log C M (t)dt 0 0 Thus, the specialist has the ] e ρ M t g(a t )dt, (1.3) ( ) dst + D t dt dp t = θ(t) C M (t)dt (1.4) S t = θ(t) ( µ t dt + σ t dz A t where the last equality follows from (1.2). ) CM (t)dt, (1.5) Assumption 1.3 (Manager s private cost function). The specialist s private cost function g : [a L, a H ] R ++ is strictly positive, C 2 ([a L, a H ]), and strictly convex. The specialist has an individual rationality (IR) constraint. The specialist s objective (1.3) for contracting with the household must be greater than or equal to his constant 5 We use M in our notations to denote various parameters and variables for the specialist, where M stands for manager. We avoid using S as it conflicts with the notation for the risky asset price. 6 The time discount factor term ρ M in front of the expectation is just for convenient scaling in the future discussion. 8

9 outside option of w 0 0. That is, sup C M,(α,θ),A 1.3 Household [ ρ M E 0 e ρ M t log C M (t)dt 0 0 ] e ρ M t g(a t )dt w 0. (1.6) As discussed, since the household has no access to the financial markets nor any labor income, the household s consumption C H must come from the dividend distribution D. The household offers a contract Y = {Y (t)} to incentivize and reward the specialist. The household faces a budget constraint C H D Y for all t. That is, the household can only consume C H up to the after-fees dividend distribution D Y. Finally, we assume the household is risk neutral and has the subjective discount rate ρ > 0. 7 Thus, the household has the objective of choosing the amount of consumption C H and the amount of compensation Y to the specialist: sup C H,Y E 0 [ 0 ] e ρt C H (t)dt = sup Y E 0 [ 0 ] e ρt (D t Y (t))dt. (1.7) The equality in (1.7) holds because of the risk neutrality of the household, where he will optimally consume all the after-fees dividend distributions; that is, C H = D Y. This implies the household s only remaining choice variable is the optimal contract Y for the specialist. In addition, we assume that the household is willing to contract with the specialist as long as the net benefits of contracting is greater than not contracting at all. Assumption 1.4 (Limited liability). The household contracts with the specialist as long as the household s limited liability condition is met: from (1.7), sup Y E 0 [ 0 ] e ρt (D t Y t )dt 0. (1.8) 7 DeMarzo and Sannikov (2006) and He (2009) considers the difference in patience level between the specialist and household. Earlier versions of the paper considered allowing the household to have a generic utility function u. However, using a generic utility function u necessitates the need for two state variables (i.e. the dividend process and the agent s continuation utility), while with risk neutral preferences, only one state variable is needed. In Leung (2015), we consider a similar model where the specialist has a generic utility u. 9

10 where D a L is the dividend process (1.1) of perpetual low effort, where h(a t ) h(a L ). 1.4 Financial markets We give the standard definition of market clearing. Definition 1.1 (Market clearing). The consumption good market clears if, D = C M + C H. (1.9) As the supply of risky assets is assumed to be unity, then the securities market clears if, θ(t) = S t. (1.10) Next, we state an assumption that facilitates tractability of our model. Assumption 1.5 (Dynamically complete markets). For any effort process A = {A t } chosen by the specialist, the financial markets are dynamically complete with no arbitrage. As a consequence, for each effort process A = {A t }, a state price density process ξ A =: ξ = {ξ t } exists and is unique, and has dynamics, dξ t = ξ t κ t dz A t, (1.11) where the market price of risk κ is defined as, κ t := µ t σ t. (1.12) This assumption is a stronger claim than what is usually considered in the asset pricing theory literature. For instance, for a fixed single dividend process D, Anderson and Raimondo (2008) proves the conditions needed for the existence of dynamically complete markets. In contrast, here we assume that for any arbitrarily chosen effort process A (that is, for all dividend processes D A ), we still have dynamically complete financial markets. 1.5 Equilibrium concept We define our concept of equilibrium in this economy. 10

11 Definition 1.2 (Complete market competitive subgame perfect Nash equilibrium). A complete market subgame perfect Nash equilibrium (or equilibrium) (µ, σ, S, θ, A, X, C M, C H ) is one such that: 1. The drift and volatility pair (µ, θ) admits a solution such that the risky asset gains process (1.2) holds and admits the existence and uniqueness of the state price density ξ A of Assumption 1.5; 2. The consumptions (C M, C H ), portfolio θ, and risky asset price S satisfy the conditions for market clearing of Definition 1.1; 3. The specialist s choice variables (C M, θ, A) satisfy the specialist s individual rationality (IR) constraint (1.6); 4. The specialist s choice variables (C M, θ, A) are incentive compatible (IC), meaning it is a solution to the specialist s problem (1.3) and (1.5); and 5. The contract Y is a solution to the household s problem (1.7). The solution strategy is to separate the problem into two distinct steps first we solve the portfolio-consumption choice problem of the specialist, then second we solve the dynamic contracting problem of the household. 2 Financial markets equilibrium We begin by solving the portfolio-consumption choice problem of the specialist. Proposition 2.1 (Portfolio choice and equilibrium asset price). Fix any given effort A and contract Y. Then, (i) The specialist s dollar amount portfolio choice into the risky asset is, κ t θ(t) = P t σ(t), (2.1) 11

12 where we recall P t is the time t portfolio wealth value of the specialist. (ii) Under the specialist s optimal portfolio choice of (2.1), the specialist s portfolio is, P t = C M(t) ρ M (2.2) and its dynamics are, dp t = P t (κ 2 t ρ M )dt + P t κ t dz A t. (2.3) After consumption good and security markets clear: (iii) The equilibrium consumption at time t for the specialist is, C M (t) = Y (t) (2.4) (iv) The equilibrium risky asset price at time t is, S t = Y (t) ρ M κ t σ(t). (2.5) The key observation is that the specialist s risky asset allocation amount θ of (2.1) depends, in equilibrium, on the contract value Y. After the consumption market clears, we find that the specialist s optimal consumption C M must equal to the household s contract payment Y. As a result, the specialist s portfolio wealth P depends directly on the contract payment Y. After the security market clears, and as the specialist is the marginal investor in this economy, this implies the equilibrium price S of the risky asset also depends on the household s contract value Y. In classical general equilibrium models where a representative agent has log preferences and has access to a single risky asset, the agent s risky asset allocation is identical to (2.1) and his portfolio wealth is also equal to (2.2). However, because the representative agent is the only individual in the market, after the consumption goods market clears, his equilibrium consumption must equal to the total dividends of the risky asset. And after security market clears, the equilibrium price of the risky asset will depend just on the dividends. Simply put, the equilibrium in the classical representative agent framework would have dividends D in 12

13 place of the contract compensation Y in (2.4) and (2.5). This is the largest wedge between our model and classical representative agent models. Using the results of Proposition 2.1, we can rewrite the specialist s objective function from (1.3) as follows: for any given contract Y, the specialist s objective function is equivalent to, sup A [ E 0 ρ M 0 ] e ρ M t [log Y (t) g(a t )]dt. (2.6) From here, we can discuss the optimal contracting problem between the household and the specialist. 3 Contracting problem Once the portfolio θ and consumption C M choices by the specialist have been made, we consider the contracting step of the household s problem. 3.1 Incentive compatibility We consider the set of incentive compatible contracts. resembles the discussion of Sannikov (2008). The development of this section Proposition 3.1 (Specialist s continuation utility). For any contract Y and any effort A, define the specialist s continuation value, W A,Y t := E A t [ρ M t ] e ρ M s [log Y (s) g(a s )]ds, (3.1) where E A t [ ] := E A [ Ft ZA ] is the conditional expectation generated by the probability measure induced by action A, conditioning on the filtration generated by dividends D. (i) There exists η A such that the specialist s continuation value has the dynamics, [ dw A,Y t = ρ M W A,Y t [ = ρ M W A,Y t ] + g(a t ) log Y (s) ] + g(a t ) log Y (s) dt + ηt A dzt A ( dt + ηa t D t ddt Σ D t (3.2a) ) h(a t )dt. (3.2b) 13

14 (ii) An effort process A is incentive compatible for the specialist if and only if η A is such that, A t = arg max a [a L,a H ] η A t Σ h(a) ρ Mg(a). (3.3) (iii) Suppose we seek an equilibrium where the household wishes to implement high effort A t a H at all times. Then η A must be such that, η A t ρ M Σ g (a H ) h (a H ). (3.4) Once we have characterized the incentive compatibility conditions as per Proposition 3.1, we then consider the household s optimization problem. We first make an assumption on the household s desired effort level by the specialist. Assumption 3.2 (Constantly High Effort Equilibrium). high effort A t a H ; 8 and (i) The specialist always exerts (ii) The sensitivity η A in (3.4) of Proposition 3.1 to implement the always high effort equilibrium can be held at equality, so η A = η a H =: η, η t ρ M Σ g (a H ) h (a H ). Condition (ii) says the performance sensitivity imposed on the specialist can be implemented at the cheapest and minimal fees. What is ruled out by Assumption 3.2 is job shirking by the specialist; that is to say, the specialist is not allowed to choose A t < a H. We henceforth assume and enforce Assumption Household s optimization problem Putting together the household s objective function (1.7), the specialist s continuation utility dynamics (3.2b), the specialist s portfolio and consumption choices, and applying Assump- 8 The assumption of the principal desiring a constantly high level of effort exerted by the agent is common in several applied continuous-time principal-agent problems. See DeMarzo and Sannikov (2006) and He (2009), for instance. Zhu (2013) is an example of continuous-time principal-agent problems in which the principal might optimally incentivize the agent to shirk. 14

15 tion 3.2, the household s optimization problem is the following. subject to constraints: Ṽ (w, δ) = sup Y E 0 [ 0 ] e ρt (D t Y (t))dt, (4.1) (a) State variable dynamics (i.e. dynamic incentive compatibility condition for the specialist, and dividend dynamics) dw t = ρ M [W t + g(a H ) log Y (t)] dt + ρ M Σ g (a H ) h (a H ) dza H t, (4.2a) dd t = D t h(a H )dt + D t ΣdZ a H t ; (4.2b) (b) Household s limited liability condition (1.8); and (c) Specialist s individual rationality condition (1.6). We solve for the household s optimal contract Y and the subsequent household s (indirect) utility function Ṽ (w, δ). Proposition 4.1 (Optimal contract). (i) Provided that ρ > h(a H ), the household s (indirect) utility function Ṽ (w, δ) is, where, Ṽ (w, δ) = V (w) = inf Y δ V (w), (4.3) ρ h(a H ) E 0 [ is subject to constraints (a), (b) and (c) from the above. 0 ] e ρt Y (t)dt, (4.4) (ii) The function V = V (w) of (4.4) is the solution to the ordinary differential equation, 0 = ρv +ρ M (1+w+g(a H ) log V )V + (ρ MΣ) 2 2 ( ) g 2 (a H ) V, on (w h 0, w 1 ), (4.5) (a H ) 15

16 subject to the boundary conditions: V (w 0 ) = 0, V (w 1 ) = δ ρ h(a H ), (4.6a) (4.6b) provided w 1 > w 0, and where and w 0 is from the (1.6). (iii) In particular, the optimal contract Y is, w 1 := w 1 (δ) = log δ g(a H ) + h(a H) Σ 2 /2 ρ M. (4.7) Y (t) = ρ M V (W t ) > 0. (4.8) Here, Ṽ (w, δ) is the household s (indirect) utility function for contracting with the specialist, when the specialist has an initial continuation utility of W 0 = w, and when the risky asset has an initial dividend level of D 0 = δ. The household s indirect utility Ṽ has a straightforward interpretation. The value Ṽ is equal to the present value of the growing perpetuity of dividends δ/(ρ h(a H )), less the present value of compensations V to the specialist. Thus, V is precisely the expenditure function of the household. With some abuse of terminologies in the subsequent discussions, we name V as the value function, even though the true utility of the household is Ṽ. Next, we link Assumption 1.4 with the boundary conditions (4.6). When the specialist s continuation utility equates to his own reservation utility, w = w 0, the specialist will be indifferent between contracting or not contracting with the household. In this case, the household need not pay the specialist for his indifference, and thus the expenditure function is nil at V (w 0 ) = 0. In the other case, the maximal utility the specialist can attain is w = w 1 of (4.7). This occurs when the specialist becomes the perpetual sole owner of the risky asset. As a result, the household has no residual claimant to the dividends. From Assumption 1.4 the household will reach the limited liability outside value of zero, and thus his value function equals to (4.6). We note that (4.5) is a second order nonlinear ordinary differential equation with Dirichlet 16

17 boundary conditions on the interval [w 0, w 1 ]. Unfortunately, this ODE does not have a closed form solution, but it is straightforward to obtain numerical solutions. Furthermore, several analytical properties are known (details are in the Section A.5 of the Theory Appendix), and the most important for us are that V > 0, V > 0 and V > 0 (under mild parameter conditions). Indeed, these are sufficient to derive many of the key qualitative asset pricing results of our paper. Note that although the household truly has risk neutral time separable preferences, the household becomes endogenously risk averse (over the specialist s continuation utility) due to dynamic long-term contracting. This is a standard result in the continuous-time principalagent literature (e.g. Sannikov (2008)). Our contribution here is to understand the household s endogenous risk aversion to equilibrium asset pricing. 5 Links between household risk aversion, continuation utility, and specialist s wealth While the concept of continuation utility of an agent is a useful theoretical tool to solve optimal contracts in dynamic principal-agent problems, it is often difficult to directly map this concept to empirically observable variables with clean economic interpretations. 9 Thanks to our competitive equilibrium setup, we can directly identify the specialist s continuation utility to his portfolio wealth value. This identification is particularly important for subsequent empirical testing of our theory. Putting together the specialist s portfolio value (2.2), his equilibrium consumption value (2.4) and the optimal contract (4.8), we have that P t = Y (t)/ρ M = ρ M V (W t )/ρ M = V (W t ). That is to say, in equilibrium and under the optimal contract, the specialist s portfolio value P t is equal to the household s marginal expenditure V (W t ) over the specialist s continuation value W t. We then have a simple rewriting, 10 W t = (V ) 1 (P t ). (5.1) 9 Some papers, namely DeMarzo and Sannikov (2006) can rewrite an agent s continuation utility economically motivated financial securities and contracts. 10 In the Theory Appendix A, we show that that V > 0, and thus V is monotonically increasing. Hence, (V ) 1 uniquely exists. 17

18 Thus (5.1) establishes a one-to-one relationship between the specialist s continuation value W t and his portfolio wealth P t. Hence, this suggests the economic identification: Specialist s continuation utility Specialist s portfolio wealth value. (5.2) The specialist s portfolio value is economically more transparent to interpret and potentially empirically observable. This simple yet critical identification facilitates the subsequent discussion of our model and the empirical testing of our predictions. 5.1 Household s two risk aversions due to financial intermediary s wealth The key point of this section and of this paper is Definition 5.1, where we introduce the household s absolute- and downside-risk aversions over the financial intermediary s wealth. We motivate this definition as follows. Fixing a dividend level δ, we consider a third order Taylor expansion of the household s value function Ṽ (, δ) around a specialist s continuation utility w, Ṽ (w, δ) = Ṽ ( w, δ) + Ṽw( w, δ)(w w) + 1 2Ṽww( w, δ)(w w) Ṽwww( w, δ)(w w) 3. Recalling the optimal contract form and the relationship between the household s indirect utility function Ṽ and the household s expenditure function V from Proposition 4.1, and by rearranging, we see that, Ṽ ( w, δ) Ṽ (w, δ) = 1 Y ( w)/ρ M ρ M [ w w + 1 V ( w) 2 V ( w) (w w)2 + 1 ] V ( w) 6 V ( w) (w w)3. Motivated by (5.3), we make the following key definition. (5.3) Definition 5.1 (Household s risk aversions due to financial intermediary s wealth). We define the household s absolute- and downside-risk aversions over the financial intermediary s 18

19 wealth as V /V and V /V, respectively It is worth emphasizing that the household s risk aversions are defined over the financial intermediary s wealth, and not over the household s own consumption. This is because the state variable that drives the household s per-period utility is not his own consumption, but rather the compensation to the specialist. The optimal contract depends on the specialist s continuation utility, or equivalently, thanks to the identification (5.2), the optimal contract depends on the specialist s portfolio wealth. In all, the household utility depends on the specialist s wealth, and hence, the household s risk attitudes also depend on the specialist s wealth. We justify why the terms V /V and V /V in Definition 5.1 are indeed risk aversions of the household, and remark on their signs. In the classical von Neumann-Morgenstern expected utility framework, a risk averse agent maximizes his consumption c over a strictly concave utility function u. As a result, u > 0 and u < 0 imply the Arrow-Pratt coefficient of absolute-risk aversion u /u is positively signed. However in our case, we have V > 0, V > 0, and hence we choose to define a positively signed coefficient +V /V instead of a negatively signed one. 13 The literature has agreed upon that risk aversion is captured by a negative second derivative u < 0, and that the intensity of risk aversion can be captured by the Arrow-Pratt measure u /u. Likewise, the literature for instance, Menezes, Geiss, and Tressler (1980), Kimball (1990), and others has agreed that downside-risk aversion is captured by the third derivative u. However, there is no unanimous measure of intensity for downsiderisk aversion. For instance, Kimball (1990) characterizes such intensity through prudence u /u, while Modica and Scarsini (2005) considers u /u as the coefficient of downside risk aversion. See also Jindapon and Neilson (2007) and Denuit and Eeckhoudt (2010) for 11 It is equivalent to the define the household s risk aversions using his indirect utility function Ṽ of (4.1), or using his expenditure function V of (4.4). By the additive form in (4.1), we have that Ṽww/Ṽw = V /V and Ṽwww/Ṽw = V /V. 12 Sometimes for the sake of brevity, we will just call V /V and V /V as household s absolute- and downside-risk aversions, respectively, with implicit reference that it is defined over the specialist s wealth. 13 There remains a slight subtlety in our discussion. In terms of our expenditure function V, we have that V > 0 and V > 0 (again for details, see Theory Appendix A). This actually means V resembles the utility function of an individual with risk loving preferences. Usually, the ideas of prudence or downside risk aversion is defined for a risk averse agent. Recent results by Crainich, Eeckhoudt, and Trannoy (2013) show that the even risk lovers can be prudent. For completeness, see also Ebert (2013) for a comment to Crainich et al. (2013). Applying these results, the interpretation of V /V as downside risk aversion remains valid. 19

20 further studies on higher-order risk attitudes. Crainich and Eeckhoudt (2008) also discusses the differences and similarities between the measures u /u and u /u. In this paper, we focus on V /V as downside-risk aversion. Returning back to (5.3), we show the optimal contract from the household to the specialist is heavily dependent on the household s two risk aversions. For the sake of discussion, consider a fixed point equilibrium of the agent s continuation utility and regard ŵ as the expected value of that fixed point equilibrium. The numerator on the left hand side of (5.3) represents the household s indirect utility change from choosing a contract away from the equilibrium, along which the specialist s continuation utility consequently moves from w to w. The denominator is the fees the household pays to the specialist for this contract change. Thus, the left hand side of (5.3) can be thought of as the ratio of marginal utility change of the household relative to the marginal utility change of the specialist for a contract change. By taking a formal expectation operator to (5.3) and invoking the identification (5.2) which links the specialist s continuation utility to his portfolio wealth, we see the following relationship, Ratio of household s MU change absolute Household s risk to specialist s MU change aversion Volatility of specialist s wealth + downside Household s risk aversion Skewness of specialist s wealth. (5.4) As in most general principal-agent problems, the household s objective for optimal contracting is to influence the ratio of the household s (principal s) marginal utility change to that of the specialist (agent). In our model, this culminates to the left hand side of (5.4). The right hand side of (5.4) shows the optimal contract is determined by the household s absolute- and downside-risk aversions, modified by the specialist s wealth volatility and skewness. Simply put, household s absolute risk aversion captures the household s concern for uncertain bi-directional changes in the specialist s wealth; downside risk aversion captures the household s concern for one-sided directional changes in the specialist s wealth. 5.2 Optimal contract and specialist s portfolio position Having understood the importance of the household s risk aversions over the specialist s wealth in Definition 5.1, we study dynamics of the optimal contract. By a direct application 20

21 of Ito s lemma to the optimal contract in Proposition 4.1, we have the dynamics of the optimal contract as follows. Proposition 5.1 (Optimal contract dynamics). The optimal contract has the dynamics, dy (t) Y (t) = V V dw t + 1 V 2 V (dw t) 2 (5.5) [ V = ρ M (W V t + g(a H ) log Y (t)) + 1 ( ) ] 2 2 ρ M Σ g (a H ) V dt + ρ h (a H ) V M Σ g (a H ) V h (a H ) V dza H t (5.6) We show the optimal contract can be implemented as a linear-quadratic contract. Paired with the identification (5.2) where we identify the specialist s continuation utility W t to his portfolio wealth P t, (5.5) shows we can interpret the percentage change dy t /Y t of the optimal contract as, dy t V Y t V dp t + 1 V 2 V (dp t) 2. (5.7) Equation (5.7) suggests the optimal contract to the specialist can be implemented in a linearquadratic form over the stochastic performance of the specialist s wealth. The form of the optimal contract has two components: a reward component linear in dp t scaled by the household s absolute-risk aversion V /V, and a risk sharing component quadratic in (dp t ) 2 scaled by the household s downside-risk aversion V /V. We note in a partial equilibrium delegated portfolio management setting, Stoughton (1993) discusses the optimality of quadratic contracts. We show below how the optimal contract influences the specialist s optimal portfolio position. The proof is immediate from Proposition 2.1 by taking the log of θ(t) and applying Ito s lemma. Proposition 5.2 (Portfolio choice and optimal contracts). Under the optimal contract, the log difference d log θ(t) of the specialist s allocation into the risky asset is, d log θ(t) = d log κ t + d log Y (t) (5.8) σ t ( ( ) ) 2 dy (t) = Y (t) + d log µ 1 dyt t + d log σt 2 (5.9) 2 21 Y t

22 From (5.8), we see the log difference of the specialist s portfolio position can be decomposed into two components: (i) effects from changes in the investment opportunity set d log(κ t /σ t ); and (ii) effects coming from changes in the optimal contract d log Y (t). Component (i) represents the changes in the mean and volatility of the risky asset returns, and is present in practically all classical portfolio choice models. Component (ii) shows the specialist is effectively offloading the risk-return trade-offs of the optimal contract into the risky asset. This is a similar channel to Admati and Pfleiderer (1997) whereby a delegated portfolio manager takes any compensation contract into account and accordingly modifies his portfolio positions. Unlike our setting where these contract-adjusted portfolio positions influence the aggregate demand for the risky asset, Admati and Pfleiderer (1997) has no general equilibrium implications. Equation (5.9) shows that in determining the portfolio position, the specialist will group together the mean return d log µ t of the risky asset along with the mean fee pay rate dy (t)/y (t), and group together the variance d log σt 2 of the risky asset along with the variability of the fees (dy (t)/y (t)) 2. We recall from Proposition 5.1 that the dynamics of the optimal contract dy (t)/y (t) depend explicitly on the household s absolute- and downside-risk aversions. Chaining this argument together, household s risk aversions impact the specialist s portfolio positions. In all, a long-term optimal contract endogenously modifies the mean and variance of the specialist s portfolio investment opportunity set through the household s absolute- and downside-risk aversions. Remark 5.3. The contemporary paper to our theory model is He and Krishnamurthy (2012), but Proposition 5.2 marks a significant departure of our model from theirs. The main message of their paper is the importance of financial constraints of a financial intermediary equilibrium on asset pricing. Their paper motivates a similar contracting relationship between the household and the specialist and a moral hazard problem. However, to keep tractability of their model, they only use short-term intermediation contracts between the household and the intermediary in (t, t + dt], and afterwards the intermediation relationship is broken and then refreshed. Moreover, although their specialist budget constraint has history dependence on the past budget levels, but because intermediation is short-term, the budget constraint is not directly based on the past actions of the specialist or past compensations by the household. Thus the percentage change in the specialist s allocation into the risky asset of their paper is mostly driven by capital constraints and the changes in the investment opportunity set. 22

23 In contrast, we lift the capital constraints of the specialist and shift our attention to long-term dynamic contracts. Proposition 5.2 clearly shows the important influence of longterm dynamic contracts on portfolio allocations, in addition to changes in the investment opportunity set of the risky asset. Of course, the trade-off of our complex dynamic contracting structure is that we have fewer closed form asset pricing expressions than He and Krishnamurthy (2012). 6 Asset Pricing Dynamics Having explored the effects of the household s risk aversions on the specialist s portfolio choice, we are now ready to explore the equilibrium asset pricing implications. Proposition 6.1 (Cum-dividend log returns). The equilibrium cum-dividend risky asset log return is equal to d log θ(t). Proof of Proposition 6.1. This is from applying the security market clearing condition (1.10) that θ(t) = S t. By market clearing, Proposition 6.1 shows the expression of d log θ(t) is precisely equal to the cum-dividend risky asset log return. While closely related, this cum-dividend risky asset log return is still different from the equilibrium drift µ t of the gains process in Assumption 1.2, which includes the returns from dividend distributions. However, the key asset pricing parameter that we want to understand in the model is the effects of contracting (or more precisely, the household s risk aversions) on the market price of risk κ t. We arrive at the core theory result of this paper the market price of risk due to dynamic contracting with a financial intermediary. Proposition 6.2 (Market price of risk with long-term contracts). The equilibrium market price of risk κ at time t is, κ t = ρ M Σg (a H )/h (a H ) [ W t + g(a H ) log V + 1 V /V + ρ MΣ 2 2 ( g (a H ) h (a H ) ) ] 2 V. (6.1) V 23

24 From Proposition 6.2, the market price of risk in our intermediated economy can be decomposed as: (i) Precision of contract volatility {}}{ [ ρ M κ t = W Σg (a H )/h t + g(a H ) log V + (a H ) } {{ } (ii) Contracting net cost (iii) Absolute risk tolerance {}}{ 1 + ρ MΣ 2 V /V 2 ( ) ] g 2 (a H ) V /V h (a H ) V /V }{{} (iv) Ratio of downside- to absolute-risk aversions (6.2) Term (i) is a scaling factor that incorporates the volatility effects of the contract into the equilibrium volatility of the risky asset. Term (ii) represents a direct fee effect on the market price of risk. The specialist s per-period benefit for contracting with the household is log V (W t ) = log Y (t) log ρ M, and g(a H ) is per-period private costs for exerting high effort. When the household increases the payment Y (t) to the specialist in order to maintain correct incentivization, the specialist s wealth W t inherently increases. With more available wealth, the specialist will increase his future allocations to the risky asset, which then affects the market price of risk in equilibrium. The key contributions of our theory culminate to terms (iii) and (iv) in the market price of risk. Indeed, abstracting away the inherently empirically unobservable terms (i) and (ii), (6.2) is the core result (M) alluded the introduction of our paper, where we highlighted that the market price of risk is priced by the household s absolute- and downside-risk aversions over the financial intermediary s wealth. We reiterate the critical mechanisms that drive our results. From the motivation of Definition 5.1, the long-term optimal contract induces the household to have endogenous absolute- and downside-risk aversions over the financial intermediary s wealth. In turn, Proposition 5.1 shows the two risk aversions affect the rate of pay dy (t)/y (t) from the household to the specialist. When the specialist decides how much wealth d log θ(t) to allocate into the risky asset, Proposition 5.2 shows the household s two risk aversions, through the optimal long-term contract, modifies the specialist s investment opportunity set. As the specialist is the marginal investor in the financial markets, this means in equilibrium, the specialist s portfolio position for the risky assets equates to the aggregate demand for the risky asset. Thus upon market clearing for the risky asset, the two risk aversions affect the 24