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 + ψ 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 < L√h, 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 (aWˆ + ˆb)h. • t Assumptions: a > 0, a + aˆ = r/(1 + rh). • Manager maximizes • ∞ 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 • { n }n=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 – ψ N S ψ , 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: • { n }n=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 < L√h. • 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 ψM 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 ψM 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 ψ 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). – Volatility factor can carry negative risk premium. ⇒ – Holding market beta constant, investors prefer assets that pay off when volatility is high. – Intuition: Holding such assets reduces probability of liqui- dation. Linear Liquidation Probability 23

Betas and Correlations

Conditional betas are constant (independent of volatility) and • equal to unconditional betas.

Same for correlations. • Quadratic Liquidation Probability 24

QUADRATIC LIQUIDATION PROBABILITY

Solution is nonlinear. • Highest order new term (for large L) is quadratic in v: • – q (v) = q q v q v2 n n0 − n1 − n2 – q (v) = q q v q v2 M M0 − M1 − M2 2 – Z(v) = Z0 + Z1v + Z2v . ∂q /∂ > 0: Liquidity premia increase with volatility. • n2 n Quadratic Liquidation Probability 25

Conditional CAPM

Conditional two-factor CAPM: •

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 quadratic case, modified risk-aversion coefficients A (v ) • M t and Av(vt) depend on vt. For market portfolio: • µv c φ2 exp(2A ) 1 A (v ) A + 4 t ζ M M − . M t ≡ L4 (r + κ)2 A – Market is more risk averse when volatility is high. – Stochastic risk aversion is not because of stochastic utility functions, but because liquidation probability is convex in volatility. Quadratic Liquidation Probability 26

Betas

Conditional betas are stochastic. • An asset’s market beta increases with volatility if • ψn 1 σqn1 q − r+κ + n2 ψM 2 "σqM1 qM2 # − r+κ φM φn ψM ψn (r+κ)2 + r+κ σqM1 r+κ σqn1 > − − . φh2 i h 2 i M + ψM σq (r+κ)2 r+κ − M1 h i Holds if asset is more price-sensitive to volatility than average. • Holds if • – Asset is more payoff-sensitive to volatility than average.

(ψn small)

– Asset has higher transaction costs than average. (n large) Quadratic Liquidation Probability 27

Correlations

Conditional correlations are stochastic. • For two assets n and n such that φ = φ , ψ = ψ , • 1 2 n1 n2 n1 n2 n1 = n2, correlation increases with volatility. Quadratic Liquidation Probability 28

Unconditional CAPM

Taking expectations in conditional CAPM, we find •

E(Rn,t+h) = E[AM (vt)]Cov(Rn,t+h, RM,t+h)

+E[Av(vt)]Cov(Rn,t+h, Rv,t+h)

+Cov [AM (vt), Covt(Rn,t+h, RM,t+h)]

+Cov [AM (vt), Covt(Rn,t+h, RM,t+h)]

+ [r + 2 [λ + µE(π(vt))] exp(2AM )] nh.

New terms: Covariances between stochastic risk-aversion coef- • ficients and conditional covariances.

– “Risk-aversion/beta” covariances. Quadratic Liquidation Probability 29

CAPM Pricing

Can two assets n and n have same unconditional covariances, • 1 2 but different sum of risk-aversion/beta covariances?

– Yes.

– Sum of risk-aversion/beta covariances is greater for n1 if

qn12 > qn22.

– This is equivalent to n1 > n2. Unconditional CAPM understates risk of illiquid assets. • Intuition: • – Illiquid assets have high betas at times of high volatility. – At these times, market is most risk averse. Quadratic Liquidation Probability 30

Conclusion

This paper:

Proposes a theory of time-varying liquidity premia. • Explores its asset-pricing implications. • Basic idea:

Fund managers face liquidation after poor performance. • High volatility High liquidation probability Short invest- • ⇒ ⇒ ment horizons High liquidity premia. ⇒ Main contributions:

Several phenomena associated to crises / volatile times are • related.

– Large liquidity premia. (Flight to liquidity) – Increased market risk aversion. (Flight to quality) – Increased riskiness of illiquid assets. – Increased correlations.

CAPM understates risk of illiquid assets. • – These assets become very risky in crises, when market is most risk averse.