Asset pricing under optimal contracts

Transcription

1 Asset pricing under optimal contracts Jakša Cvitanić (Caltech) joint work with Hao Xing (LSE) 1/44

2 Motivation and overview I Existing literature: either - Prices are fixed, optimal contract is found or - Contract is fixed, prices are found in equilibrium I An exception: Bu a-vayanos-woolley 2014 [BVW 14] I However, [BVW 14] still severely restrict the set of admissible contracts I We allow more general contracts and explore equilibrium implications 2/44

3 Literature I Fixed contracts: Brennan (1993) Cuoco-Kaniel (2011) He-Krishnamurthy (2011) Lioui and Poncet (2013) Basak-Pavlova (2013) I Fixed prices: Sung (1995) Ou-Yang (2003) Cadenillas, Cvitanić and Zapatero (2007) Leung (2014) Cvitanić, Possamai and Touzi, CPT (2016, 2017) 3/44

4 Bu a-vayanos-woolley 2014 [BVW 14] I Optimal contract is obtained within the class compensation rate = portfolio return index return. Our questions: 1. What is the optimal contract when investors are allowed to optimize in a larger class of contracts? (Linear contract is optimal in [Holmstrom-Milgrom 1987]) 2. What are the equilibrium properties? 4/44

5 As shown in CPT (2016, 2017)... I The optimal contract depends on the output, its quadratic variation, the contractible sources of risk (if any), and the cross-variations between the output and the risk sources. 5/44

6 Our results I Computing the optimal contract and equilibrium prices I Optimal contract rewards Agent for taking specific risks and not only the systematic risk I Stocks in large supply have high risk premia, while stocks in low supply have low risk premia I Equilibrium asset prices distorted to a lesser extent: Second order sensitivity to agency frictions compared to the first order sensitivity in [BVW 14]. 6/44

7 Outline Introduction Model [BVW 14] Main results Technicalities Other examples for optimal contracting 7/44

8 Assets Riskless asset has an exogenous constant risk-free rate r. Prices of N risky assets will be determined in equilibrium. Dividend of asset i is given by D it = a i p t + e it, where p and e i follow Ornstein-Uhlenbeck processes dp t = apple p ( p p t )dt + p db p t, de it = apple e i (ē i e it )dt + ei db e it. Vector of asset excess returns per share dr t = D t dt + ds t rs t dt. The excess return of index I t = 0 R t, where =( 1,..., N ) 0 are the numbers of shares of assets in the market. 8/44

9 Available shares Number of shares available to trade: =( 1,..., N ) 0 (Some assets may be held by buy-and-hold investors.) We assume that and are not linearly dependent. (Manager provides value to Investor.) 9/44

10 Portfolio manager Portfolio manager s wealth process follows d W t = r W t dt +(bm t c t )dt + df t, I c t is Manager s consumption rate I F t is the cumulative compensation paid by Investor I bm t is the private benefit from his shirking action m t, b 2 [0, 1], [DeMarzo-Sannikov 2006] I No private investment I Chooses portfolio Y for Investor 10 / 44

11 Investor The reported portfolio value process: G = Investor observes only G and I Her wealth process follows Z 0 (Y 0 s dr s m s ds). dw t = rw t dt + dg t + y t di t c t dt df t, I Y t is the vector of the numbers of shares chosen by Manager I y t is the number of shares of index chosen by Investor I c t is Investor s consumption rate I m t is Manager s shirking action, assumed to be nonnegative 11 / 44

12 Manager s optimization problem Manager maximizes utility over intertemporal consumption: h Z 1 V =max E c,m,y 0 i e t u A ( c t )dt, I is Manager s discounting rate I u A ( c) = 1 e c 12 / 44

13 Manager s optimization problem Manager maximizes utility over intertemporal consumption: h Z 1 V =max E c,m,y 0 i e t u A ( c t )dt, I is Manager s discounting rate I u A ( c) = 1 e c If Manager is not employed by Investor, he maximizes subject to budget constraint h Z 1 V u =max E c u,y u 0 e tu i A ( c t u )dt d W t = r W t + Y u t dr t c u t dt. Manager takes the contact if V V u. 12 / 44

14 Investor s maximization problem Investor maximizes utility over intertemporal consumption: h Z 1 V =max E e c,f,y 0 i t u P (c t )dt, I is Investor s discounting rate I u P (c) = 1 e c 13 / 44

15 Investor s maximization problem Investor maximizes utility over intertemporal consumption: h Z 1 V =max E e c,f,y 0 i t u P (c t )dt, I is Investor s discounting rate I u P (c) = 1 e c If Investor does not hire Manager, she maximizes subject to budget constraint h Z 1 V u =max E e c u,y u 0 i t u P (ct u )dt dw t = rw t + y u t di t c u t dt. Investor hires Manager if V V u. 13 / 44

16 Equilibrium ApriceprocessS, a contract F in a class of contracts F, andanindex investment y, formanequilibrium if 1. Given S, (F, F), and y, Manager takes the contract, and Y = y solves Manager s optimization problem. 2. Given S, Investor hires Manager, and (F, y) solves Investor s optimization problem, and F is the optimal contract in F. 14 / 44

17 Outline Introduction Model [BVW 14] Main results Technicalities Other examples for optimal contracting 15 / 44

18 Asset prices There exists an equilibrium with asset prices S it = a 0i + a pi p t + a ei e it (assuming and are not linearly dependent.) Setting a p =(a p1,...,a pn ) 0 and a e = diag{a e1,...,a en },wehave a pi = a i r + apple p a ei = 1 r + apple e, i =1,...,N, i (assuming the matrix R = a p 2 p a 0 p + a 0 e 2 E a e is invertible.) 16 / 44

19 Asset prices There exists an equilibrium with asset prices S it = a 0i + a pi p t + a ei e it (assuming and are not linearly dependent.) Setting a p =(a p1,...,a pn ) 0 and a e = diag{a e1,...,a en },wehave a pi = a i r + apple p a ei = 1 r + apple e, i =1,...,N, i (assuming the matrix R = a p Notation: 2 p a 0 p + a 0 e 2 E a e is invertible.) Var = 0 R, Covar, = 0 R, CAPM beta of the fund portfolio: = Covar, Var. 16 / 44

20 Asset Returns Asset excess returns are µ r = r + R + rd b R ( ), where D b =( + ) b I When b 2 [0, + ], the first best is obtained. I When i i >, risk premium of asset i increases with b. When i i <, risk premium of asset i decreases with b. 17 / 44

21 Asset prices/returns In [BVW 14], D b is replaced by Note that D BVW b = b. + + D b < D BVW b, for any b 2 (0, 1) Expected excess return Severity of agency friction (b) Figure: Solid lines: our result; Dashed lines: [BVW 14]. 18 / 44

22 Index and portfolio returns Excess return of the index 0 (µ r) =r + Covar,. Excess return of Manager s portfolio 0 (µ r) =r + Var + rd b Var (Covar, ) 2 Var. 35 Agent's portfolio excess return Severity of agency friction (b) 19 / 44

23 Optimal contract df t = Cdt + + dg t + (dg t di t )+ r 2 dhg I, G I i t I Optimality in a large class of contracts I Conjecture: It is optimal in general. I =(b + ) +, =( + )(b + )(1 b ) I When b apple +, = = 0, only the first two terms show up. The return of the fund is shared between investor and portfolio manager with ratio +. BVW 14 contract corresponds to the two terms in the middle. I The quadratic variation term is new. I hg I, G I i can be thought as a tracking gap. Tracking gap is rewarded to motivate Manager to take the specific risk of individual stocks, and not only the systematic risk of the index. 20 / 44

24 Optimal contract When b +, is increasing in b, so as to make Manager to not employ the shirking action. Dependence of on b: ζ Severity of agency friction (b) 21 / 44

25 New contract improves Investor s value For the asset price in [BVW 14], Investor s value is improved by using the new contract Principal's certainty equivalence Severity of agency friction (b) Figure: Solid line: our contract, Dashed line: [BVW 14] 22 / 44

26 Outline Introduction Model [BVW 14] Main results Technicalities Other examples for optimal contracting 23 / 44

27 Admissible contracts: motivation For any Manager s admissible strategy = ( c, Y, m), consider t = {ˆ admissible ˆ s = s, s 2 [0, t]}. Define Manager s continuation value process V( ) as V t ( ) = ess sup t E t h Z 1 t e (s i t) u A ( c s )ds, t / 44

28 Admissible contracts: motivation For any Manager s admissible strategy = ( c, Y, m), consider t = {ˆ admissible ˆ s = s, s 2 [0, t]}. Define Manager s continuation value process V( ) as V t ( ) = ess sup t E t h Z 1 t e (s i t) u A ( c s )ds, t 0. W t V t ( ) = r V t ( ); (ii) Transversality condition: lim t!1 E e t V t ( ) = 0; (iii) Martingale principle: Z t Ṽ t ( ) = e t V t ( ) + e s u A ( c s )ds, 0 is a supermartingale for arbitrary admissible strategy, and is a martingale for the optimal strategy. 24 / 44

29 Admissible contracts: definition (Motivated by CPT (2016), (2017)) A contract F is admissible if 1. there exists a constant V 0, 2. for any Agent s strategy there exist F G,I -adapted processes Z, U, G, I, GI such that the process V ( ), defined via where X t = H = d V t ( ) =X t h(bm t c t )dt + Z t dg t + U t di t sup c,m 0,Y G t dhg, Gi t I tdhi, I i t + + V t ( )dt H t dt, V0 ( ) = V 0, r V t ( ) and H is the Hamiltonian n u A ( c)+x i GI t dhg, I i t h bm c Zm + ZY 0 (µ r)+u 0 (µ r) satisfies lim t!1 E e t Vt ( ) = 0. G Y 0 R Y I 0 R + io GI Y 0 R, 25 / 44

30 Manager s optimal strategy Lemma Given an admissible contract with X > 0, Z b, and G < 0, the Manager s optimal strategy is the one maximizing the Hamiltonian, c =(u 0 A) 1 (X ), m =0, and we have Y + y = Z G 1 R (µ r) GI G, V ( ) = ˆV( ). 26 / 44

31 Do we lose on generality? [CPT 2016, 2016] considered the finite horizon case, d V t =X t hbm t dt + Z t dg t + U t di t G t dhg, Gi t I tdhi, I i t + i GI t hg, I i t H t dt. V T = C T is the lump-sum compensation paid. They showed the set of C that can be represented as V T is dense in the set of all (reasonable) contracts. Hence, there is no loss of generality in their framework. Their proof is based on the 2BSDE theory, e.g., [Soner-Touzi-Zhang 2011,12,13]. Conjecture: A similar result holds for the infinite horizon case. (Work in progress by Lin, Ren, and Touzi.) 27 / 44

32 Representation of admissible contracts Lemma An admissible contract F can be represented as df t =Z t dg t + U t di t G t dhg, Gi t r dhz G + U I, Z G + U I i t where Z G = R 0 Z sdg s and I t dhi, I i t + GI t r + H t dt, H t = 1 log( r V 0 ) 1 +(Z ty t + U t ) 0 (µ t r) G t (Y t ) 0 R Y t I t 0 R + In particular, F is adapted to F G,I (as it should be). GI t (Y t ) 0 R. dhg, I i t 28 / 44

33 Investor s problem 1. Guess Investor s value function V (w) =Ke r w, 2. Treat Z, U, G, GI as Investor s control variables. 3. Work the with HJB equation satisfied by V. 29 / 44

34 Conclusion I We find an asset pricing equilibrium with the contract optimal in a large class. (Maybe the largest.) I Price/return distortion less sensitive to agency frictions. I The contract based on the tracking gap and its quadratic variation. Future work: I Square root, CIR dividend processes 30 / 44

35 Outline Introduction Model [BVW 14] Main results Technicalities Other examples for optimal contracting 31 / 44

36 Example: delegated portfolio management The portfolio value process X t follows the dynamics where dx = v 1 S 1 ds 1 + v 2 S 2 ds 2 ds i,t /S i,t = b i dt + db i t, where B i are independent Brownian motions and b i are constants. We have then dx t =[v 1,t b 1 + v 2,t b 2 ] dt + v 1,t db 1 t + v 2,t db 2 t. The principal hires an agent to manage the values of v t =(v 1,t, v 2,t ). 32 / 44

37 Expected utility The agent is paid at the final time T in the amount C T,anddraws expected utility E he R R T A(C c(v1(s),v2(s))ds)i T 0 where the agent s running cost is of the form c(v 1, v 2 )= (v 1 1 ) (v 2 2 ) 2 The principal s expected utility is E he i R P (X T C T ) 33 / 44

38 First best Given a bargaining-power parameter >0, the first-best (risk-sharing) problem is " Z # T max max E U P (X T C T )+ U A (C T c(v 1 (s), v 2 (s))ds) v C T The first order condition for C T is then UP 0 (X T C T ) UA 0 (C T KT v ) = With CARA utilities, we obtain 1 C T = R P X T + R A K v RA T + log R A + R P R P 0 34 / 44

39 Second best We consider linear contracts based on the path of the observable portfolio value X, the observable quadratic variation of X, and, possibly, on S 1 via B 1, and the co-variation of X and B 1.Indicator1 O indicates whether S 1 is observed. Z T C T = C 0 + Z X s dx s + Ys X dhx i s + 1 O Zs 1 dbs 1 + Ys 1 dhx, B 1 i s + H s ds, 0 (1) for some constant C 0, and some adapted processes Z X, Z 1, Y X, Y 1 and H. Transformation of variables: Y X = 1 2 X + R A (Z X ) 2, Y 1 = 1 + R A Z X Z 1, H = G R A(Z 1 ) / 44

40 SIMPLE CRUCIAL OBSERVATION: C T = agent s value function at T. We will argue then that the natural choice for G t is G t := G(Z X t, Z 1 t, X t, 1 t ), where G(Z X, Z 1, X, 1 ):=supg(v 1, v 2, Z X, Z 1, X, 1 ) v 1,v 2 1 := sup X (v1 2 + v2 2 )+Z X b v c(v 1, v 2 )+1 1 O v 1. v 1,v 2 2 The agent is maximizing E P t h e R R T A 0 [gs Gs ]dsi apple 1, Any pair (v 1 (s), v 2 (s)) that maximizes g s := g(z X s, Z 1 s, X s, 1 s )is optimal. 36 / 44

41 Contractible S 1 :firstbestisattained Optimal (v1, v 2 ) is obtained by maximizing g = (v 1 1 ) (v 2 2 ) 2 +Z X b v + 1 v X kvk 2 + Z 1 b 1 +(Z 1 ) 2 +2Z 1 Z X v 1. Assume, for example, that b 2 6= 0, 2 apple 1. Suppose the principal sets Then, X t 2, Z X t 2 2 /b 2, 1 t = 1 1 Zt X b 1 +( 1 2 )v1 FB, Z 1 0. apple 1 g =( 2 1 ) 2 v 1 2 v 1 v1 FB + const. Agent is indi erent with respect to v 2, and he chooses v 1 = v FB / 44

42 Non-contractible S 1 Optimal (v1, v 2 ) is obtained by maximizing g(v 1, v 2 )= (v 1 1 ) (v 2 2 ) 2 + Z X b v X kvk 2. Assume, for example, 2 apple 1.If X < 2 apple 1, the optimal positions are v i = Z X b i + i i The principal maximizes, over Z and, i X. b v (Z, ) 1 2 [R AZ 2 + R P (1 Z) 2 ]kv k 2 c(v (Z, )). 38 / 44

43 Main messages from numerics - 1. The percentage loss in the principal s second best utility certainty equivalent relative to the first best, when varying initial risk expoosure 2, can be significant for extreme values of 2.Thatis,whentheinitial risk exposure is far from desirable, the moral hazard cost of providing incentives to the agent to modify the exposure is high The loss in the principal s second best certainty equivalent relative to the one she would obtain if o ering the contract that is optimal among those that are linear in the output, but do not depend on its quadratic variation, can also be large The principal uses quadratic variation as an incentive tool: for low values of the initial risk exposure she wants to increase the risk exposure by rewarding higher variation, and for its high values she wants to decrease it by penalizing high variation. 39 / 44

44 Example: Quadratic cost, drift e ort; C., Wan and Zhang (2009) dx t = t dt + db t c 2 R T 0 2 t dt], while Principal is Agent is maximizing E[U A (C T ) maximizing E[U P (X T C T )] Proposition. Assuming that Principal s value function V P (t, x, y) isin class C 2,3,3, we have, for some constant L, V P y (t, X t, Y t )= 1 c V P (t, X t, Y t ) L In particular, the optimal contract C T satisfies Ũ 0 P (X T C T ) U 0 A (C T ) = 1 c U P(X T C T )+L (2) 40 / 44

45 Proof: The HJB equation for Principal s value function v(t, x, y) =V P (t, x, y) is 8 < t v +sup 2 zv x z 2 v y + 1 z c 2c 2 : v(t, x, y) =U P (x U 1 (y)). Optimizing over z gives A 2 v xx + z 2 v yy + 2 zv xy =0, z = v x + cv xy v y + cv yy We have that v(t, X t, Y t ) is a martingale under the optimal measure P, satisfying dv = (v x + z v y )dw Then, compare to dv y, with boundary condition v y (T, x, y) = U0 P (x U 1 A (y)) U 0 A (U 1 A (y)) 41 / 44

46 Risk-neutral principal and logarithmic agent Suppose c =1, U P (C T )=X T C T, U A (C T ) = log C T. We also assume dx t = t X t dt + X t dbt. The optimal contract payo C T satisfies C T = 1 2 X T + const. 42 / 44

47 Example: Risk-Sharing in complete markets Cadenillas, Cvitanić and Zapatero (2007) Assume a complete market with no cost on choosing the portfolio strategy. Using these methods we recover the result from the above paper that the optimal payo F (X T ) is given by solving the ODE UP 0 (x F (x)) = applef 0 (x) (F (x)) U 0 A This gives a linear contract for CARA utilities. Also for CRRA utilities, but only with the same risk aversions. 43 / 44

48 Thank you for your attention! 44 / 44

49 Figure 1: Percentage loss in principal's certainty equivalent relative to first best, as function of α. Parameter values: =10, =0.58, α =0.5, β =0.4, β =1, 1 =0.4, =1, =0.

50 Figure 2: Percentage loss in principal's certainty equivalent when not using quadratic variation, as function of α. Parameter values: =10, =0.58, α =0.5, β =0.4, β =1, =0.4, =1, =0.

51 Figure 3: Optimal contract's sensitivity to quadratic variation, as function of α. Parameter values: =10, =0.58, α =0.5, β =0.4, β =1, =0.4, =1, =0.