Asset pricing under optimal contracts Jakˇ sa Cvitani´ c (Caltech) joint work with Hao Xing (LSE) Thera Stochastics - A Mathematics Conference in Honor of Ioannis Karatzas 1 / 32
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 / 32
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 / 32
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 / 32
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 / 32
Our results ◮ Computing the optimal contract and equilibrium prices ◮ Optimal contract rewards Agent for taking specific risks and not only the systematic risk ◮ Stocks in large supply have high risk premia, while stocks in low supply have low risk premia ◮ Equilibrium asset prices distorted to a lesser extent: Second order sensitivity to agency frictions compared to the first order sensitivity in [BVW 14]. 6 / 32
Outline Introduction Model [BVW 14] Main results Mathematical tools 7 / 32
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 / 32
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 / 32
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 / 32
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 / 32
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 / 32
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 / 32
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 / 32
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 / 32
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 / 32
Outline Introduction Model [BVW 14] Main results Mathematical tools 15 / 32
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 / 32
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 / 32
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 / 32
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 / 32
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 / 32
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. ◮ The term � G − β θ I , G − β θ I � rewards Manager to take the specific risk of individual stocks, and not only the systematic risk of the index. 20 / 32
Optimal strategy Manager’s vector of optimal holdings is given by � η ′ ( µ − r ) Y ∗ = 1 1 R ( µ − r ) + 1 � ρ + ¯ D b ρ Σ − 1 η, (1) r C b r ρ ¯ ρ C b Var η where � 2 ρ � D b = ( ρ + ¯ ρ ) b − + , (2) ρ +¯ ρ ρ ¯ ρ C b = ρ + D b . ρ +¯ 21 / 32
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) 22 / 32
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] 23 / 32
Outline Introduction Model [BVW 14] Main results Mathematical tools 24 / 32
Recommend
More recommend
