Asset pricing under optimal contracts Jakˇ sa Cvitani´ c (Caltech) joint work with Hao Xing (LSE) 1 / 31
Motivation and overview ◮ Existing literature: either - Prices are fixed, optimal contract is found or - Contract is fixed, prices are found in equilibrium ◮ An exception: Buffa-Vayanos-Woolley 2014 [BVW 14] ◮ However, [BVW 14] still severely restrict the set of admissible contracts ◮ We allow more general contracts and explore equilibrium implications 2 / 31
Literature ◮ Fixed contracts: Brennan (1993) Cuoco-Kaniel (2011) He-Krishnamurthy (2011) Lioui and Poncet (2013) Basak-Pavlova (2013) —————————————– ◮ Fixed prices: Sung (1995) Ou-Yang (2003) Cadenillas, Cvitani´ c and Zapatero (2007) Leung (2014) Cvitani´ c, Possamai and Touzi, CPT (2016, 2017) 3 / 31
Buffa-Vayanos-Woolley 2014 [BVW 14] ◮ Optimal contract is obtained within the class compensation rate = φ × portfolio return − χ × 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 / 31
As shown in CPT (2016, 2017) ... ◮ 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 / 31
Our results ◮ Computing the optimal contract and equilibrium prices ◮ Equilibrium asset prices distorted to a lesser extent: Second order sensitivity to agency frictions compared to the first order sensitivity in [BVW 14]. 6 / 31
Outline Introduction Model [BVW 14] Main results Technicalities 7 / 31
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 p − p t ) dt + σ p dB p dp t = κ p (¯ t , de it = κ e e i − e it ) dt + σ ei dB e i (¯ 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 = η ′ R t , where η = ( η 1 , . . . , η N ) ′ are the numbers of shares of assets in the market. 8 / 31
Available shares Number of shares available to trade: θ = ( θ 1 , . . . , θ N ) ′ (Some assets may be held by buy-and-hold investors.) We assume that η and θ are not linearly dependent. (Manager provides value to Investor.) 9 / 31
Portfolio manager Portfolio manager’s wealth process follows d ¯ W t = r ¯ W t dt + ( b m t − ¯ c t ) dt + dF t , ◮ ¯ c t is Manager’s consumption rate ◮ F t is the cumulative compensation paid by Investor ◮ b m t is the private benefit from his shirking action m t , b ∈ [0 , 1], [DeMarzo-Sannikov 2006] ◮ No private investment ◮ Chooses portfolio Y for Investor 10 / 31
Investor The reported portfolio value process: � · ( Y ′ G = s dR s − m s ds ) . 0 Investor observes only G and I Her wealth process follows dW t = rW t dt + dG t + y t dI t − c t dt − dF t , ◮ Y t is the vector of the numbers of shares chosen by Manager ◮ y t is the number of shares of index chosen by Investor ◮ c t is Investor’s consumption rate ◮ m t is Manager’s shirking action, assumed to be nonnegative 11 / 31
Manager’s optimization problem Manager maximizes utility over intertemporal consumption: � � ∞ � e − ¯ ¯ δ t u A (¯ V = max c t ) dt c , m , Y E , ¯ 0 ◮ ¯ δ is Manager’s discounting rate c ) = − 1 ρ e − ¯ ρ ¯ ◮ u A (¯ c ¯ 12 / 31
Manager’s optimization problem Manager maximizes utility over intertemporal consumption: � � ∞ � e − ¯ ¯ δ t u A (¯ V = max c t ) dt c , m , Y E , ¯ 0 ◮ ¯ δ is Manager’s discounting rate c ) = − 1 ρ e − ¯ ρ ¯ ◮ u A (¯ c ¯ If Manager is not employed by Investor, he maximizes � � ∞ V u = max � e − ¯ ¯ δ t u A (¯ c u t ) dt c u , Y u E ¯ 0 subject to budget constraint d ¯ W t = r ¯ W t + Y u c u t dR t − ¯ t dt . Manager takes the contact if ¯ V ≥ ¯ V u . 12 / 31
Investor’s maximization problem Investor maximizes utility over intertemporal consumption: � � ∞ � e − δ t u P ( c t ) dt V = max c , F , y E , 0 ◮ δ is Investor’s discounting rate ◮ u P ( c ) = − 1 ρ e − ρ c 13 / 31
Investor’s maximization problem Investor maximizes utility over intertemporal consumption: � � ∞ � e − δ t u P ( c t ) dt V = max c , F , y E , 0 ◮ δ is Investor’s discounting rate ◮ u P ( c ) = − 1 ρ e − ρ c If Investor does not hire Manager, she maximizes � � ∞ V u = max � e − δ t u P ( c u t ) dt c u , y u E 0 subject to budget constraint dW t = rW t + y u t dI t − c u t dt . Investor hires Manager if V ≥ V u . 13 / 31
Equilibrium A price process S , a contract F in a class of contracts F , and an index investment y , form an equilibrium if 1. Given S , ( F , F ), and y , Manager takes the contract, and Y = θ − y η 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 / 31
Outline Introduction Model [BVW 14] Main results Technicalities 15 / 31
Asset prices There exists an equilibrium with asset prices S it = a 0 i + a pi p t + a ei e it (assuming θ and η are not linearly dependent.) Setting a p = ( a p 1 , . . . , a pN ) ′ and a e = diag { a e 1 , . . . , a eN } , we have a i 1 a pi = a ei = , i = 1 , . . . , N , r + κ e r + κ p i (assuming the matrix Σ R = a p σ 2 p a ′ p + a ′ e σ 2 E a e is invertible.) 16 / 31
Asset prices There exists an equilibrium with asset prices S it = a 0 i + a pi p t + a ei e it (assuming θ and η are not linearly dependent.) Setting a p = ( a p 1 , . . . , a pN ) ′ and a e = diag { a e 1 , . . . , a eN } , we have a i 1 a pi = a ei = , i = 1 , . . . , N , r + κ e r + κ p i (assuming the matrix Σ R = a p σ 2 p a ′ p + a ′ e σ 2 E a e is invertible.) Notation: Var η = η ′ Σ R η, Covar θ,η = η ′ Σ R θ, CAPM beta of the fund portfolio: β θ = Covar θ,η . Var η 16 / 31
Asset Returns Asset excess returns are ρ ¯ ρ ρ Σ R θ + r D b Σ R ( θ − β θ η ) , µ − r = r ρ + ¯ where ρ � 2 � D b = ( ρ + ¯ ρ ) b − + . ρ + ¯ ρ ◮ When b ∈ [0 , ρ ρ ], the first best is obtained. ρ +¯ ◮ When θ i η i > β θ , risk premium of asset i increases with b . When θ i η i < β θ , risk premium of asset i decreases with b . 17 / 31
Asset prices/returns In [BVW 14], D b is replaced by ρ � � D BVW = ¯ b − ρ + . b ρ + ¯ ρ Note that D b < D BVW , for any b ∈ (0 , 1) . b 20 15 Expected excess return 10 5 0 -5 -10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Severity of agency friction (b) Figure: Solid lines: our result; Dashed lines: [BVW 14]. 18 / 31
Index and portfolio returns Excess return of the index ρ ¯ ρ η ′ ( µ − r ) = r ρ Covar θ,η . ρ + ¯ Excess return of Manager’s portfolio Var θ − ( Covar θ,η ) 2 ρ ¯ ρ ρ Var θ + r D b � � θ ′ ( µ − r ) = r . ρ + ¯ Var η 35 Agent's portfolio excess return 30 25 20 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Severity of agency friction (b) 19 / 31
Optimal contract 2 ζ d � G − β θ I , G θ − β θ I � t ρ ρ dG t + ξ ( dG t − β θ dI t ) + r dF t = Cdt + ρ +¯ ◮ Optimality in a large class of contracts ◮ Conjecture: It is optimal in general. ρ ◮ ξ = ( b − ρ ) + , ζ = ( ρ + ¯ ρ )( b + ξ )(1 − b − ξ ) ξ ρ +¯ ρ ◮ When b ≤ ρ , ξ = ζ = 0, only the first two terms show up. The ρ +¯ return of the fund is shared between investor and portfolio manager ρ with ratio ρ . ρ +¯ BVW 14 contract corresponds to the two terms in the middle. ◮ The quadratic variation term is new. ◮ � G − β θ I , G − β θ I � can be thought as asttracking 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 / 31
Optimal contract ρ When b ≥ ρ , ρ +¯ ξ is increasing in b , so as to make Manager to not employ the shirking action. Dependence of ζ on b : 8 7 6 5 4 ζ 3 2 1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Severity of agency friction (b) 21 / 31
New contract improves Investor’s value For the asset price in [BVW 14], Investor’s value is improved by using the new contract. 13 12 Principal's certainty equivalence 11 10 9 8 7 6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Severity of agency friction (b) Figure: Solid line: our contract, Dashed line: [BVW 14] 22 / 31
Outline Introduction Model [BVW 14] Main results Technicalities 23 / 31
Admissible contracts: motivation For any Manager’s admissible strategy Ξ = (¯ c , Y , m ), consider Ξ t = { ˆ Ξ admissible | ˆ Ξ s = Ξ s , s ∈ [0 , t ] } . Define Manager’s continuation value process ¯ V (Ξ) as � � ∞ e − ¯ � ¯ δ ( s − t ) u A (¯ V t (Ξ) = ess sup Ξ t E t c s ) ds , t ≥ 0 . t 24 / 31
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