Flight to Quality, Flight to Liquidity, and the Pricing of Risk Dimitri Vayanos MIT and NBER February 2003 Introduction 1 INTRODUCTION Liquidity premia seem to vary substantially over time. • Clean experiments: • { T-notes vs. T-bills. (Amihud and Mendelson (1991), Ka- mara (1994), Strebulaev (2002)) { Off- vs. on-the run bonds. (Krishnamurthy (2002)) { Refcorp vs. government bonds. (Longstaff (2002)) { Municipal vs. government bonds. (Chalmers, Kadlec, and Mozumdar (2002)) Effects can be quite strong. • { Variation in spread equals 4-6 times average spread. { Variation in relative prices can be up to 15%. { Flight to liquidity. Effects could be stronger for other asset classes. • { Corporate bonds. { Emerging market bonds. { Stocks. Introduction 2 Why Do Liquidity Premia Vary? Liquidity premia seem to be high when • { Interest-rate volatility is high. (Kamara (1994)) { Consumer confidence is low. (Longstaff (2002)) { Money flows away from equity funds, into money market funds. (Longstaff (2002)) { Stock market goes down. (Chalmers, Kadlec, and Mozum- dar (2002)) Variation in liquidity premia seems to be correlated across mar- • kets. { On-the-run premium is correlated with commercial-paper spread. (Krishnamurthy (2002)) Introduction 3 Questions Questions: • { What is economic mechanism driving variation in liquidity premia? { How is risk associated to changes in liquidity premia priced? Answers could be important for understanding • { An important component of asset price volatility. { An important component of pricing of asset risk. (Espe- cially for illiquid assets.) Introduction 4 This Paper Proposes a theory of time-varying liquidity premia. • Explores its asset-pricing implications. • Dynamic multi-asset equilibrium model. • Introduction 5 Main Assumptions Transaction costs. Exogenous, constant over time, different across assets. • Illiquid assets High TC. • ≡ Time-variation in liquidity premia will not caused by TC. • { Seems realistic: Variation in bid-ask spreads is often small relative to that in liquidity premia. Off-the-run bonds. ∗ Refcorp bonds. (Longstaff (2002)) ∗ Introduction 6 Main Assumptions (cont'd) Stochastic volatility. Asset payoffs have stochastic volatility. • Volatility will drive liquidity premia. • Seems realistic. • { Empirical evidence. { Anecdotal evidence: Traders value liquidity more at times of \uncertainty." Delegated money management. Portfolio decisions are made by fund managers. • When return falls below a threshold, fund is liquidated. • High volatility High liquidation probability Short invest- • ) ) ment horizons High liquidity premia. ) Introduction 7 Results / Empirical Implications Liquidity premia ( price differentials between assets which are • ≡ identical except for TC) increase with volatility. { Flight to liquidity. Market's effective risk aversion ( expected return per unit of • ≡ variance) increases with volatility. { Flight to quality. { Risk aversion varies not because of stochastic utility func- tions, but because of concern of liquidation. Betas depend on volatility. • { Betas of illiquid assets increase with volatility. Correlations depend on volatility. • { Correlations between similar assets increase with volatility. Unconditional two-factor CAPM (factors = market, volatility) • does not price assets correctly. { Understates risk of illiquid assets. Implications for price of volatility factor. • Model 8 MODEL Infinite horizon, discrete time. Time between periods is h. • Assets. Riskless asset, return rh. r is exogenous. • N risky assets. • Volatility v evolves according to • t v = v + γ(v v )h + σ v hη : t+h t − t t t+h p Asset n pays dividend δ h. δ evolves according to • n;t n;t δ = δ +κ(δ δ )h+ v h (φ ζ + η + ξ ) : n;t+h n;t − n;t t n t+h n t+h n;t+h p { ζt+h: systematic shock, independent of ηt+h. { ξn;t+h: idiosyncratic shock. Supply S . • n Transaction costs per share . • n Price (average of bid and ask) p . • n;t Model 9 Fund Management and Liquidation Each investor \manages" a fund of size W . • t Performance-based liquidation: • { In each period, a fund can be \monitored" by its owners with probability µh. { Fund is liquidated if W W < Lph, for L > 0. t+h − t − Random liquidation: • { In each period, a fund can be liquidated with probability λh, regardless of performance. At liquidation: • { Investment in each asset is sold in the market. { Manager can find employment in a new fund, whose size is equal to old fund's liquidation value. Model 10 Fund Managers Infinitely lived, continuum, mass one. • Manager decides how much to allocate in each asset. • Exogenous management fee (aW + b)h. • t Exogenous withdrawal by fund's owners (aW^ + ^b)h. • t Assumptions: a > 0, a + a^ = r=(1 + rh). • Manager maximizes • 1 E exp ( αc βkh) : − − t+kh − Xk=0 Consumption is derived from fee. • Manager can save/borrow in riskless asset. • Equilibrium 11 EQUILIBRIUM Basic Properties Managers buy and hold the market portfolio. • Price of asset n is • 1 κh p = q (v ) + − (δ δ): n;t n t r + κ n;t − Value function of a manager is • exp [ A[W + zw + Z(v )]] ; − − t t t where A αa, z r=[(1 + rh)a], and w are manager's ≡ ≡ t private savings. When h goes to zero, functions q (v) and Z(v) can • f n gn=1;::;N be characterized by a system of N + 1 ODEs. { ODEs are derived from Bellman equation. { They are second order. Equilibrium 12 Preliminaries Notation: • { φ N S φ , M ≡ n=1 n n { N S , M ≡ Pn=1 n n { q (v) N S q (v), M ≡P n=1 n n { N S . M ≡ nP=1 n n Number ofPassets N becomes large, while N S stays fixed. • n=1 n { Sn goes to zero. P { Idiosyncratic shocks do not matter. No Performance-Based Liquidation 13 NO PERFORMANCE-BASED LIQUIDATION ODEs for q (v) and Z(v) have linear solution: • f n gn=1;::;N { q (v) = q q v. n n0 − n1 { q (v) = q q v. M M0 − M1 { Z(v) = Z0 + Z1v. Properties: • { qM1 > 0: Market goes down when volatility increases. { @qn0=@n < 0: Liquidity premia are positive. { @qn1=@n = 0: Liquidity premia are independent of volatil- ity. No Performance-Based Liquidation 14 CAPM ODE for q (v) can be written as • n Et(Rn;t+h) = ACovt(Rn;t+h; RM;t+h) + AZ0(vt)Covt(Rn;t+h; Rv;t+h) + [r + 2λ exp(2AM )] nh; where { Rn;t+h: excess return on asset n. { RM;t+h: excess return on market portfolio. { Rv;t+h: excess return on volatility portfolio. { Excess returns are per share, and between t and t + h. Conditional two-factor CAPM, adjusted for transaction costs. • Linear solution Z0(v ) = Z . • ) t 1 Taking expectations, we find • E(Rn;t+h) = ACov(Rn;t+h; RM;t+h) + AZ1Cov(Rn;t+h; Rv;t+h) + [r + 2λ exp(2AM )] nh; i.e., unconditional two-factor CAPM, adjusted for transaction costs. No Performance-Based Liquidation 15 Volatility Factor Recent research has considered a volatility/liquidity factor. • Pastor and Stambaugh (2001) • { Liquidity factor Price reversals. ≡ Acharya and Pedersen (2002) • { Liquidity factor Aggregate price impact. ≡ Liquidity factor: • { Negative risk premium. { Significant explanatory power. { In our model, aggregate price impact is proportional to volatility Volatility factor corresponds to PS-AP liquidity ) factor. Ang and Hodrick (2002) • { Volatility factor. No Performance-Based Liquidation 16 Volatility Risk Premium Z > 0 Volatility factor carries positive risk premium. • 1 ) { Holding market beta constant, investors prefer assets that pay off when volatility is low. { Opposite to empirical findings. Intuition: • { Holding wealth constant, investors are better off at times of high volatility. Performance-Based Liquidation 17 PERFORMANCE-BASED LIQUIDATION Liquidation condition is W W < Lph. • t+h − t − Probability of this event depends on tails of ζ and η . • t+h t+h Normal distribution is not tractable. • Power laws are, however, very tractable. • Performance-Based Liquidation 18 Power Laws Assume that lower tail of ζ and upper tail of η follow • t+h t+h power laws with exponent b, i.e., c Prob(ζ < y) ζ ; t+h − ∼ yb c Prob(η > y) η : t+h ∼ yb b = 2: Liquidation probability is • 2 2 vt φM M π(v ) = c + c + q0 (v )σ : t L2 ζ (r + κ)2 η r + κ M t " # Linear in vt. b = 4: Liquidation probability is • 2 4 4 vt φM M π(v ) = c + c + q0 (v )σ : t L4 ζ (r + κ)4 η r + κ M t " # Quadratic in vt. Equations are valid for L large relative to v . • t Performance-Based Liquidation 19 Asset Pricing Expected returns are given by • Et(Rn;t+h) = ACovt(Rn;t+h; RM;t+h) + AZ0(vt)Covt(Rn;t+h; Rv;t+h) @π(v ; x) exp(2A ) 1 +µ t M − h @x A n x=S + [r + 2 [λ + µπ(vt)] exp(2AM )] nh: Third term is \liquidation" risk premium. • Linear Liquidation Probability 20 LINEAR LIQUIDATION PROBABILITY Solution is still linear. • { q (v) = q q v n n0 − n1 { q (v) = q q v M M0 − M1 { Z(v) = Z0 + Z1v. New property: • { @qn1=@n > 0: Liquidity premia increase with volatility. Linear Liquidation Probability 21 CAPM Liquidation risk premium depends on covariance between asset • n, and market and volatility portfolios. Can incorporate liquidation risk premium into CAPM risk • ) premium. Conditional two-factor CAPM, with modified risk-aversion • ) coefficients: Et(Rn;t+h) = AM (vt)Covt(Rn;t+h; RM;t+h) +Av(vt)Covt(Rn;t+h; Rv;t+h) + [r + 2 [λ + µπ(vt)] exp(2AM )] nh: In linear case, modified risk-aversion coefficients A (v ) and • M t Av(vt) are constants. Taking expectations, we can derive unconditional two-factor • ) CAPM. Linear Liquidation Probability 22 Risk-Aversion Coefficients For market portfolio: • µ exp(2A ) 1 A A + 2 c M − : M ≡ L2 ζ A { Greater than without performance-based liquidation. For volatility portfolio: • µ exp(2A ) 1 A AZ +2 (c c ) q M M − : v ≡ 1 L2 ζ − η M1 − σ(r + κ) A { Can become negative if cη > cζ (volatility tails fatter than dividend tails).