Dynamic Portfolio Choice with Frictions

Dynamic Portfolio Choice with Frictions Nicolae Gârleanu UC Berkeley, CEPR, and NBER Lasse H. Pedersen NYU, Copenhagen Business School, AQR, CEPR, and NBER December 2014 Gârleanu and Pedersen Dynamic Portfolio Choice with Frictions, p.1

Dynamic Trading with Predictable Returns and TC Central issue in finance: optimal portfolio choice Dynamic problem: Expected returns vary over time e.g. driven by factors Expected returns mean revert with different alpha decay Transaction costs imply that positions are sticky Investors can continually re-adjust their portfolio Questions: What is the optimal trading strategy? Understand limiting behavior as trading becomes frequent What are the implications for equilibrium excess returns? Gârleanu and Pedersen Dynamic Portfolio Choice with Frictions, p.2

Dynamic Trading with Predictable Returns and TC Central issue in finance: optimal portfolio choice Dynamic problem: Expected returns vary over time e.g. driven by factors Expected returns mean revert with different alpha decay Transaction costs imply that positions are sticky Investors can continually re-adjust their portfolio Questions: What is the optimal trading strategy? Understand limiting behavior as trading becomes frequent What are the implications for equilibrium excess returns? Broadly applicable technology Macro models (e.g., of investment) Monetary economics Micro economics (e.g., firm decisions) Political economy Gârleanu and Pedersen Dynamic Portfolio Choice with Frictions, p.2

Main Results Closed-form solution in general model Gârleanu and Pedersen Dynamic Portfolio Choice with Frictions, p.3

Main Results Closed-form solution in general model Price impact understood as compensation for intermediary risk Gârleanu and Pedersen Dynamic Portfolio Choice with Frictions, p.3

Main Results Closed-form solution in general model Price impact understood as compensation for intermediary risk If intermediary lays off inventory Over one model period: no transaction cost in the limit Over fixed calendar-time period: positive transitory cost in the limit Gradually over (calendar) time: persistent price impact Properties of the optimal portfolio path: With transitory transaction cost: smooth With only persistent price impact: Brownian (with jumps) Gârleanu and Pedersen Dynamic Portfolio Choice with Frictions, p.3

Main Results Closed-form solution in general model Price impact understood as compensation for intermediary risk If intermediary lays off inventory Over one model period: no transaction cost in the limit Over fixed calendar-time period: positive transitory cost in the limit Gradually over (calendar) time: persistent price impact Properties of the optimal portfolio path: With transitory transaction cost: smooth With only persistent price impact: Brownian (with jumps) Stochastic volatility: Risk cost: mean-reverting Trading cost: incurred instantaneously Trade less aggressively in high-volatility environments Gârleanu and Pedersen Dynamic Portfolio Choice with Frictions, p.3

Main Results Closed-form solution in general model Price impact understood as compensation for intermediary risk If intermediary lays off inventory Over one model period: no transaction cost in the limit Over fixed calendar-time period: positive transitory cost in the limit Gradually over (calendar) time: persistent price impact Properties of the optimal portfolio path: With transitory transaction cost: smooth With only persistent price impact: Brownian (with jumps) Stochastic volatility: Risk cost: mean-reverting Trading cost: incurred instantaneously Trade less aggressively in high-volatility environments Equilibrium: Faster decaying alphas are larger Gârleanu and Pedersen Dynamic Portfolio Choice with Frictions, p.3

Aim in Front of the Target Closed-form optimal dynamic trading strategy Gârleanu and Pedersen Dynamic Portfolio Choice with Frictions, p.4

Aim in Front of the Target Closed-form optimal dynamic trading strategy Two portfolio principles 1. Aim in the front of the target 2. Trade partially towards the (current) aim Panel A: Construction of Current Optimal Trade Markowitz t Position in asset 2 old position x t 1 new position x t E t (aim t+1 ) aim t Position in asset 1 Gârleanu and Pedersen Dynamic Portfolio Choice with Frictions, p.4

Aim in Front of the Target Closed-form optimal dynamic trading strategy Two portfolio principles 1. Aim in the front of the target 2. Trade partially towards the (current) aim Aim portfolio : Weighted average of current and future Markowitz portfolios Predictors with Panel slower A: Construction mean reversion of receiver Current higher Optimal weight Trade Markowitz t Position in asset 2 old position x t 1 new position x t E t (aim t+1 ) aim t Position in asset 1 Gârleanu and Pedersen Dynamic Portfolio Choice with Frictions, p.4

Aim in Front of the Target: Illustration Panel A: Construction of Current Optimal Trade Markowitz t Position in asset 2 old position x t 1 new position aim t x t E t (aim t+1 ) Position in asset 1 Panel B: Expected Next Optimal Portfolio Position in asset 2 x t 1 x t Et (x t+1 ) E t (Markowitz t+1 ) Position in asset 1 Gârleanu and Pedersen Dynamic Portfolio Choice with Frictions, p.5

Aim in Front of the Target: Illustration Panel A: Construction of Current Optimal Trade Panel D: Skate to where the puck is going to be Markowitz t Position in asset 2 old position x t 1 new position aim t x t E t (aim t+1 ) Position in asset 1 Panel B: Expected Next Optimal Portfolio Panel E: Shooting: lead the duck Position in asset 2 x t 1 x t Et (x t+1 ) E t (Markowitz t+1 ) Position in asset 1 Gârleanu and Pedersen Dynamic Portfolio Choice with Frictions, p.5

Aim in Front of the Target: Illustration Panel A: Construction of Current Optimal Trade Panel D: Skate to where the puck is going to be Markowitz t Position in asset 2 old position x t 1 new position aim t x t E t (aim t+1 ) Position in asset 1 Panel B: Expected Next Optimal Portfolio Panel E: Shooting: lead the duck Position in asset 2 x t 1 x t Et (x t+1 ) E t (Markowitz t+1 ) Position in asset 1 Panel C: Expected Optimal Future Path Position in asset 2 E t (x t+h ) E t (Markowitz t+h ) Position in asset 1 Gârleanu and Pedersen Dynamic Portfolio Choice with Frictions, p.5

Aim in Front of the Target: Illustration Panel A: Construction of Current Optimal Trade Panel D: Skate to where the puck is going to be Markowitz t Position in asset 2 old position x t 1 new position aim t x t E t (aim t+1 ) Position in asset 1 Panel B: Expected Next Optimal Portfolio Panel E: Shooting: lead the duck Position in asset 2 x t 1 x t Et (x t+1 ) E t (Markowitz t+1 ) Position in asset 1 Panel C: Expected Optimal Future Path Panel F: Missile systems: lead homing guidance Position in asset 2 E t (x t+h ) E t (Markowitz t+h ) Position in asset 1 Gârleanu and Pedersen Dynamic Portfolio Choice with Frictions, p.5

Related Literature Optimal trading with transactions costs, no predictability Constantinides (86), Amihud and Mendelson (86), Vayanos (98), Liu (04) Predictability, no transactions costs Merton (73), Campbell and Viceira (02) Optimal trade execution with exogenous trade: Perold (88), Almgren and Chriss (00) Numerical results with time-varying investment opportunity set Jang, Koo, Liu, and Loewenstein (07), Lynch and Tan (08) Quadratic programming Used in macroeconomics (Ljungqvist and Sargent (04)) and other fields: solve up to Ricatti equations Grinold (06), Collin-Dufresne et al. (2014) Continuous-time TC based on trading intensity Carlin et al. (2008), Oehmke (2009) Gârleanu and Pedersen Dynamic Portfolio Choice with Frictions, p.6

Pure Transitory Transaction Costs: Model Returns: dp t rp t dt = Bf t dt + du t Risk: var t (du t ) = Σ dt Factor decay: df t = µ f (f t )dt + dε t T-cost with dx t = τ t dt : whatever T C(τ t ) = 1 2 τ t Λτ t Maximand: E t [ t e ρ(s t) ( x s Bf s γ 2 x s Σx s 1 2 τ s Λτ s ) ds ] Gârleanu and Pedersen Dynamic Portfolio Choice with Frictions, p.7

Continuous-Time Solution Method Conjecture and verify quadratic value function: V (x, f) = 1 2 x A xx x + x A x (f) + 1 2 A(f) Value function V solves Hamilton-Jacoby-Bellman equation: ρv = sup τ { x Bf γ 2 x Σx 1 2 τ Λτ + V x τ + x DA x (f) + DA(f) Maximizing with respect to the trading intensity results in 1 V τ = Λ. x } Gârleanu and Pedersen Dynamic Portfolio Choice with Frictions, p.8

Optimal Continuous-Time Strategy: General Case Proposition The optimal trading intensity τ = dxt dt where and τ t = Λ 1 A xx [aim t x t ], aim t = (γa 1 xx Σ) A xx = ρ 2 Λ + Λ 1 2 Markowitz s = (γσ) 1 Bf s. t is e (γa 1 xx Σ)(s t) E t [Markowitz s ] ds ( γλ 1 2 ΣΛ 1 ρ ) 1 2 + 4 I 2 Λ 1 2 Gârleanu and Pedersen Dynamic Portfolio Choice with Frictions, p.9

Illustration: Optimal Portfolio with One Asset Markowitz Aim Optimal Position Time Gârleanu and Pedersen Dynamic Portfolio Choice with Frictions, p.10

Optimal Continuous-Time Strategy Proposition If Λ = λσ and µ f (f) = Φf, then the optimal trading intensity is τ t = a λ [aim t x t ], where aim t = (γσ) 1 B (I + aφ/γ) 1 f t a = ρλ + ρ 2 λ 2 + 4γλ. 2 If each factor s decay only depends on itself (i.e., Φ is diagonal), then: ( aim t = (γσ) 1 ft 1 B 1 + aφ 1 /γ,..., ft K ) 1 + aφ K. /γ The rate of trade a λ increases with γ λ. Gârleanu and Pedersen Dynamic Portfolio Choice with Frictions, p.11

Illustration: Expected Optimal Portfolio with Two Assets Markowitz Aim Optimal Markowitz 0 Position in asset 2 aim 0 x 0 Position in asset 1 Gârleanu and Pedersen Dynamic Portfolio Choice with Frictions, p.12

Transitory and Persistent Price Impact: Model Transaction price: p t = p t + D t Price distortion: dd t = RD t dt + Cτ t dt Returns: d p t rp t dt = Bf t dt + dd t rd t dt + du t Risk: var t (du t ) = Σ dt Factor decay: df t = µ f (f t )dt + dε t Temporary t-cost:wh T C(τ t ) = 1 2 τ t Λτ t Maximand: [ E t t e ρ(s t) ( x s Bf s γ 2 x s Σx s 1 2 τ s ) Λτ s ds + ] t e ρ(s t) x s ( (r + R)D s + Cτ s ) ds Gârleanu and Pedersen Dynamic Portfolio Choice with Frictions, p.13

Transitory and Persistent Price Impact: Solution Proposition The value function is quadratic in (x t, D t ). The optimal trading strategy is given by with τ t = M rate ( aim M f (f t ) + M aim f t M aim D D t x t ) (f) = N 2 e N1(s t) N 3 E t [Markowitz s ] dt. t Gârleanu and Pedersen Dynamic Portfolio Choice with Frictions, p.14

Pure Persistent Price Impact: Model Transaction price: p t = p t + D t Price distortion: dd t = RD t dt + Cτ t dt Returns: d p t rp t dt = Bf t dt + dd t rd t dt + du t Risk: var t (du t ) = Σ dt Factor decay: df t = µ f (f t )dt + dε t Temporary t-cost:wh zero (x no longer bounded variation) Maximand: [ E t t e ρ(s t) ( x s (Bf s (r + R)D s ) γ ) 2 x s Σx s ds + t e ρ(s t) x s Cdx s + 1 ] 2 t e ρ(s t) d [x, Cx] s Gârleanu and Pedersen Dynamic Portfolio Choice with Frictions, p.15

Pure Persistent Price Impact: Solution Proposition The value function satisfies V (x, D, f) = V (0, D Cx, f) 1 2 x Cx ˆV (D 0, f) 1 2 x Cx. The function ˆV is quadratic in D 0. The optimal portfolio is given by x t = ˆN 0,f Markowitz t + M 1,f (f t ) M D (D Cx ), with M 1,f (f t ) = ˆN 2 e ˆN 1 (s t) ˆN3 E t [Markowitz s ] ds. 0 Gârleanu and Pedersen Dynamic Portfolio Choice with Frictions, p.16

Illustration: Pure Persistent Impact Markowitz Optimal Position 0 Time Gârleanu and Pedersen Dynamic Portfolio Choice with Frictions, p.17

General Discrete-Time Model Transaction price: p t = p t + D t Returns: p t+1 rp t t = Bf t t + u t Price distortion: D t+1 = R( t )D t + C( t ) x t+1 Risk: var t ( u t+1 ) = Σ t Factor decay: f t+1 = Φf t t + ε t+1 T-cost:whate T C( x t ) = 1 2 x t Λ( t ) x t [ Maximand: E t t e (x ρs t s (Bf s γ 2 x s Σx s ) t 1 2 x s Λ x s + x s ( (r t + R)D s + C x s ) + )] x s C x s+1 + 1 2 x s+1 C x s+1 Gârleanu and Pedersen Dynamic Portfolio Choice with Frictions, p.18

Quadratic TC in Continuous Time: Is It Reasonable? Quadratic transaction cost: T C (1) ( x) = x Λ x Consider splitting the cost over two equal sub-periods: ( ) x ( ) x T C (2) ( x) = 2 Λ 2 2 = 1 2 T C(1) ( x) It follows that T C (n) 0 as n Gârleanu and Pedersen Dynamic Portfolio Choice with Frictions, p.19

Quadratic TC in Continuous Time: Is It Reasonable? Quadratic transaction cost: T C (1) ( x) = x Λ x Consider splitting the cost over two equal sub-periods: ( ) x ( ) x T C (2) ( x) = 2 Λ 2 2 = 1 2 T C(1) ( x) It follows that T C (n) 0 as n But why should the parameter Λ be independent of the frequency of trade? Gârleanu and Pedersen Dynamic Portfolio Choice with Frictions, p.19

Micro Foundation: Transitory Cost Agent trades with specialized, risk-averse intermediaries One unit of intermediaries, risk aversion equal to one Intermediaries require duration of time h( t ) to place inventory with end users, off the market during this period At every trading date: t /h( t ) intermediaries h( t )/ t aggregate risk aversion x units of risky asset Holding period h( t ) Gârleanu and Pedersen Dynamic Portfolio Choice with Frictions, p.20

Micro Foundation: Transitory Cost Agent trades with specialized, risk-averse intermediaries One unit of intermediaries, risk aversion equal to one Intermediaries require duration of time h( t ) to place inventory with end users, off the market during this period At every trading date: t /h( t ) intermediaries h( t )/ t aggregate risk aversion x units of risky asset Holding period h( t ) Implied transaction cost is proportional to x t (Σh( t )) x t 1 t h( t ) t Gârleanu and Pedersen Dynamic Portfolio Choice with Frictions, p.20

Micro Foundation: Transitory Cost Agent trades with specialized, risk-averse intermediaries One unit of intermediaries, risk aversion equal to one Intermediaries require duration of time h( t ) to place inventory with end users, off the market during this period At every trading date: t /h( t ) intermediaries h( t )/ t aggregate risk aversion x units of risky asset Holding period h( t ) Implied transaction cost is proportional to x t (Σh( t )) x t 1 t h( t ) t Implication: Λ( t ) 1 t h( t ) 2 Σ Gârleanu and Pedersen Dynamic Portfolio Choice with Frictions, p.20

Micro Foundation: Transitory Cost Agent trades with specialized, risk-averse intermediaries One unit of intermediaries, risk aversion equal to one Intermediaries require duration of time h( t ) to place inventory with end users, off the market during this period At every trading date: t /h( t ) intermediaries h( t )/ t aggregate risk aversion x units of risky asset Holding period h( t ) Implied transaction cost is proportional to x t (Σh( t )) x t 1 t h( t ) t Implication: Λ( t ) 1 t h( t ) 2 Σ Constant h( t ) : Λ( t ) 1 t If h( t ) t : Λ( t ) t Gârleanu and Pedersen Dynamic Portfolio Choice with Frictions, p.20

Reduced-Form Transaction-Cost Model The number of trades is of order 1 t Transaction cost is of order ( t ) 2 Λ( t ) 1 t Gârleanu and Pedersen Dynamic Portfolio Choice with Frictions, p.21

Reduced-Form Transaction-Cost Model The number of trades is of order 1 t Transaction cost is of order ( t ) 2 Λ( t ) 1 t A transitory cost obtains in continuous time if Λ( t ) 1 t, i.e., h( t ) is constant Corresponds to constant time to liquidate inventory, independent of trading frequency (or modeled, or observed, frequency) If Λ( t ) increases more slowly than 1 t, then TC vanishes in limit Corresponds to time to liquidate that decreases with the trading frequency perhaps due to technological change, other secular trends Gârleanu and Pedersen Dynamic Portfolio Choice with Frictions, p.21

Micro Foundation: Persistent Price Impact If intermediaries sell inventory and return to market gradually persistent impact. Parameter behaviour: R( t ) t C( t ) 0 t Both types of costs obtain if two kinds of intermediaries handle each trade (serially) Gârleanu and Pedersen Dynamic Portfolio Choice with Frictions, p.22

Connection Discrete-Continuous Time Proposition (i) If intermediaries need a fixed amount of calendar time to liquidate inventory, so that Λ( t ) = Λ/ t, then M rate ( t ) lim t 0 t = rate M lim ( t ) t 0 = M aim. Discrete time continuous time with transitory and persistent costs (ii) If intermediaries can liquidate inventory during one trading period, so that Λ( t ) = Λ t, then the optimal portfolio tends to that obtaining in continuous-time model with Λ = 0: Discrete time continuous time with purely persistent costs Gârleanu and Pedersen Dynamic Portfolio Choice with Frictions, p.23

Illustration: Fixed Liquidating Time t=1 Position Time Gârleanu and Pedersen Dynamic Portfolio Choice with Frictions, p.24

Illustration: Fixed Liquidating Time t=1 t=0.25 Position Time Gârleanu and Pedersen Dynamic Portfolio Choice with Frictions, p.24

Illustration: Fixed Liquidating Time t=1 t=0.25 Continuous Position Time Gârleanu and Pedersen Dynamic Portfolio Choice with Frictions, p.24

Stochastic Volatility: Model Suppose that Σt = v t Σ Λt = λσ t dv t = µ v (v t ) dt + σ v (v t ) dw t Gârleanu and Pedersen Dynamic Portfolio Choice with Frictions, p.25

Stochastic Volatility: Model Suppose that Σt = v t Σ Λt = λσ t dv t = µ v (v t ) dt + σ v (v t ) dw t Value function takes the form V (x, f, v) = 1 2 x A xx (v)x + x A xf (v)f + 1 2 f A ff (v)f + A 0 (v) Gârleanu and Pedersen Dynamic Portfolio Choice with Frictions, p.25

Stochastic Volatility: Model Suppose that Σt = v t Σ Λt = λσ t dv t = µ v (v t ) dt + σ v (v t ) dw t Value function takes the form V (x, f, v) = 1 2 x A xx (v)x + x A xf (v)f + 1 2 f A ff (v)f + A 0 (v) Proposition If µ v is decreasing and crosses 0, then trading is more intense for low v t than for high v t : there exists ˆv such that M rate (v t ) > M rate (ˆv) iff v t < ˆv. Gârleanu and Pedersen Dynamic Portfolio Choice with Frictions, p.25

Equilibrium Model Noise trader positions: ( ) dzt l = κ ft l zt l dt df l t = ψ l f l tdt + dw l t. State variables: f (f 1,..., f L, z) where z t = l zl t. It follows E t [df t ] = Φf t dt, with mean-reversion matrix ψ 1 0 0 0 ψ 2 0 Φ =....... κ κ κ Market clearing: x t = z t dx t = dz t Gârleanu and Pedersen Dynamic Portfolio Choice with Frictions, p.26

Equilibrium Expected Returns Proposition The market is in equilibrium iff Bf t = L λσ 2 κ(ψ l + ρ + κ)( ft) l + σ 2 (ρλκ + λκ 2 γ)z t l=1 The coefficients λσ 2 κ(ψ k + ρ + κ) increase in the mean-reversion parameters ψ k and κ and in the trading costs λσ 2. I.e., noise trader selling (ft k < 0) increases the alpha, and especially if its mean reversion is faster and if the trading cost is larger. Gârleanu and Pedersen Dynamic Portfolio Choice with Frictions, p.27

Conclusion Tractable model with frictions Discrete and continuous time General dynamics of investment opportunity Intuitive closed form solution embodying two principles Trade partially towards aim Aim overweights persistent signals Clarifies model dependence on frequency; quadratic costs can be modeled in discrete or continuous time Applications: equilibrium, stochastic volatility Tractable framework has many potential applications, beyond TC: Monetary policy Firm reaction to various consumer-preference signals Political party acting on signals from different constituents Gârleanu and Pedersen Dynamic Portfolio Choice with Frictions, p.28