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Illiquidity, Position Limits, and Optimal Investment for Mutual Funds

Min Dai

National University of Singapore

Joint with Hanqing Jin Hong Liu

Oxford University Washington University in St. Louis Workshop on Stochastic Analysis & Finance, 29 Jun-3th July, City University of Hong Kong

Illiquidity, Position Limits, and Optimal Investment for Mutual Funds – p. 1/21

Introduction: Motivation

• Mutual funds are often restricted to allocate certain percentages of fund

assets to certain securities (Almazan, Brown, Carlson, and Chapman 2004,

Clarke, de Silva, and Thorley 2002).

• funds prevented from shorting selling and/or buying-on-margin
• a small cap fund may set a lower bound on its holdings of small cap

stocks.

• Mutual funds can also face significant illiquidity in trading securities

(Chalmers, Edelen, and Kadlec 1999, Delib and Varma 2002)

• The coexistence of position limits and asset illiquidity and the interactions

among them are important for the optimal trading strategy of a mutual fund.

Illiquidity, Position Limits, and Optimal Investment for Mutual Funds – p. 2/21

Motivation (continued)

• The existing literature ignores the coexistence of position limits and asset

illiquidity and the interactions among them.

• Portfolio selection with position limits: Fleming and Zariphopoulou (1991),

Cuoco (1997), Cuoco and Liu (2000).

• Portfolio selection with transaction costs: Constantinides (1986), Davis

and Norman (1990), Shreve and Soner (1994), Liu and Loewenstein (2002), Øksendal and Sulem (2002), Liu (2004), Dai and Yi (2009), Dai et al. (2009).

Illiquidity, Position Limits, and Optimal Investment for Mutual Funds – p. 3/21

The Model

• An investor with a finite horizon T ∈ (0, ∞) maximizes his CRRA utility

from terminal non-liquidated wealth: u(W) = W 1−γ − 1 1 − γ , γ > 0.

• Two assets: 1 liquid stock, and 1 illiquid stock.
• The liquid stock price SLt evolves as

dSLt SLt = µLdt + σLdBLt, where µL and σL > 0 are both constants and BLt is a one-dimensional Brownian motion.

Illiquidity, Position Limits, and Optimal Investment for Mutual Funds – p. 4/21

The Illiquid Stock

• The investor can buy the illiquid stock at the ask price SA

It = (1 + θ)SIt

and sell the stock at the bid price SB

It = (1 − α)SIt, where θ ≥ 0 and

0 ≤ α < 1 represent the proportional transaction cost rates and SIt follows the process dSIt SIt = µIdt + σIdBIt, where µI and σI > 0 are both constants and BIt is another

• ne-dimensional Brownian motion that has a correlation of ρ with BLt

with |ρ| < 1.

Illiquidity, Position Limits, and Optimal Investment for Mutual Funds – p. 5/21

Position Limits

• Let xt and yt be the dollar amount invested in the liquid stock and the

illiquid stock respectively.

• The investor is subject to the following position limits:

b ≤ yt Wt ≤ ¯ b, ∀t ≥ 0,

(1)

where Wt = xt + yt is the non-liquidated wealth process.

Illiquidity, Position Limits, and Optimal Investment for Mutual Funds – p. 6/21

The Dynamic Budget Constraint

When α + θ > 0, we have dxt = µLxtdt + σLxtdBLt − (1 + θ)dIt + (1 − α)dDt,

(2)

dyt = µIytdt + σIytdBIt + dIt − dDt,

(3)

where the processes D and I represent the cumulative dollar amount of sales and purchases of the illiquid stock, respectively. D and I are nondecreasing and right continuous adapted processes with D(0) = I(0) = 0.

Illiquidity, Position Limits, and Optimal Investment for Mutual Funds – p. 7/21

The Solvency Region

• Let Θ(x0, y0) denote the set of admissible trading strategies (D, I) such

that (1), (2), (3), and ˆ Wt ≥ 0, ∀t ≥ 0, where ˆ Wt = xt + (1 − α)y+

t − (1 + θ)y− t is the time t wealth after

liquidation.

y S E L L No-Transaction BUY

x=z y

x

x=z y

Solvency Line

x=-(1- )y x=-(1+ )y

Solvency Line

_ _

α θ

Illiquidity, Position Limits, and Optimal Investment for Mutual Funds – p. 8/21

The Investor’s Problem and HJB Equation

• The investor’s problem is then

sup

(D,I)∈Θ(x0,y0)

E [u(WT )] , which is a singular stochastic control problem with state constraints.

• HJB Equation

max {Vt + LV, (1 − α)Vx − Vy, −(1 + θ)Vx + Vy} = 0, with the boundary conditions (1 − α)Vx − Vy = 0 on y x + y = ¯ b, (1 + θ)Vx − Vy = 0 on y x + y = b, and the terminal condition V (x, y, T) = (x+y)1−γ−1

1−γ

, where LV = 1 2σ2

Iy2Vyy + 1

2σ2

Lx2Vxx + ρσIσLxyVxy + µIyVy + µLxVx

Illiquidity, Position Limits, and Optimal Investment for Mutual Funds – p. 9/21

Analytical Results: The No Transaction Cost Case Theorem 1. Suppose that α = θ = 0. Then the optimal trading policy is

given by π∗

I =

     ¯ b if πM

I

≥ ¯ b πM

I

if b < πM

I

< ¯ b b if πM

I

≤ b , π∗

L = 1 − π∗ I

where πM

I

= µI − µL + γσL (σL − ρσI) γ (σ2

L + σ2 I − 2ρσLσI)

is the optimal fraction of wealth invested in illiquid stock in the unconstraint case.

Illiquidity, Position Limits, and Optimal Investment for Mutual Funds – p. 10/21

The Transaction Cost Case: Change of Variables

V (x, y, t) = (x + y)1−γ V

• x

x + y , y x + y , t

• −

1 1 − γ = (x + y)1−γ V (1 − π, π, t) − 1 1 − γ ≡ (x + y)1−γ ϕ (π, t) − 1 1 − γ , where π = y x + y ∈ (α − 1, ∞).

Illiquidity, Position Limits, and Optimal Investment for Mutual Funds – p. 11/21

A Reduced Equation

It follows max{ϕt + L1ϕ, − (1 − απ) ϕπ − α (1 − γ) ϕ, (1 + θπ) ϕπ − θ (1 − γ) ϕ} = 0, with the boundary conditions − (1 − απ) ϕπ − α (1 − γ) ϕ = 0 on π = ¯ b, (1 + θπ) ϕπ − θ (1 − γ) ϕ = 0 on π = b, and the terminal condition ϕ(π, T) = 1 1 − γ , where L1ϕ = 1 2β1π2 (1 − π)2 ϕππ + (β2 − γβ1π) π (1 − π) ϕπ + (1 − γ)

• β3 + β2π − 1

2γβ1π2

• ϕ.

Illiquidity, Position Limits, and Optimal Investment for Mutual Funds – p. 12/21

A Further Transformation

Let w = 1 1 − γ log [(1 − γ) ϕ] . Then            max

• wt + L2w, −

α 1−απ − wπ, wπ − θ 1+θπ

• = 0

wπ = −

α 1−απ on π = ¯

b, wπ =

θ 1+θπ on π = b,

w(π, T) = 0 in

• − 1

θ , 1 α

• × [0, T), where

L2w = 1 2β1π2 (1 − π)2 wππ + (1 − γ) w2

π

• +(β2−γβ1π)π (1 − π) wπ+β3+β2π−1

2γβ1π2.

Illiquidity, Position Limits, and Optimal Investment for Mutual Funds – p. 13/21

An Equivalent Standard Variational Inequality

Denote v = wπ. Since Lv

∆

= ∂ ∂π (L2w) = 1 2β1π2 (1 − π)2 vππ + [β1 + β2 − (2 + γ) β1π] π (1 − π) vπ + [β2 (1 − 2π) − γβ1π(2 − 3π)] v + (1 − γ) β1π (1 − π) v [(1 − 2π) v + π (1 − π) vπ] + β2 − γβ1π Using the technique developed by Dai and Yi (2009), we can show                    vt + Lv = 0 if −

α 1−απ < v < θ 1+θπ ,

vt + Lv ≤ 0 if v = −

α 1−απ ,

vt + Lv ≥ 0 if v =

θ 1+θπ ,

v = −

α 1−απ on π = ¯

b, v =

θ 1+θπ on π = b,

v(π, T) = 0, in

• − 1

θ , 1 α

• × [0, T).

Illiquidity, Position Limits, and Optimal Investment for Mutual Funds – p. 14/21

The Case with Transaction Costs and Constraints

The following verification theorem shows the existence and the uniqueness of the optimal trading strategy. It also ensures the smoothness of the value function except for a set of measure zero.

Theorem 3.

(i) The HJB equation admits a unique viscosity solution, and the value function is the viscosity solution. (ii) The value function is C2,2,1 in {(x, y, t) : x + (1 − α)y+ − (1 + θ)y− > 0, b < y/(x + y) < ¯ b, 0 ≤ t < T} \ ({y = 0}).

Illiquidity, Position Limits, and Optimal Investment for Mutual Funds – p. 15/21

The Transaction Cost Case Without Constraints Proposition 2. Assume − 1

α + 1 < πM I

< 1

θ + 1. We have ∀t ∈ [0, T],

• 1. for the sell boundary, there exists t < T such that

1 α = πI(s) ≥ πI(t) ≥ πM

I

1 − α (1 − πM

I ), for any t and all s > t;

• 2. for the buy boundary, there exists t < T such that

−1 θ = πI(s) ≤ πI(t) ≤ πM

I

1 + θ (1 − πM

I ), for any t and all s > t.

Illiquidity, Position Limits, and Optimal Investment for Mutual Funds – p. 16/21

The Case with Transaction Costs and Constraints Proposition 4. We have

• 1. for the sell boundary, there exists tb < T such that

b = πc

I(s; b, b) ≥ πc I(t; b, b) ≥ max

• min
• πM

I

1 − α (1 − πM

I ), b

• , b
• , for any t and s > tb;
• 2. for the buy boundary, there exists tb < T such that

min

• max
• πM

I

1 + θ (1 − πM

I ), b

• , b
• ≥ πc

I(t; b, b) ≥ πc I(s; b, b) = b, for any t and s > tb.

• 3. both πc

I(t; b, b) and πc I(t; b, b) are increasing in b and b for all t ∈ [0, T];

Illiquidity, Position Limits, and Optimal Investment for Mutual Funds – p. 17/21

Graphical Illustrations: Optimal Strategy against Time

1 2 3 4 5 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 t πI

Constrained Sell Boundary Unconstrained Sell Boundary Merton Line Constrained Buy Boundary Unconstrained Buy Boundary

Parameters: γ = 2, T = 5, µL = 0.06, σL = 0.20, µI = 0.11, σI = 0.25, ρ = 0.2, α = 0.01,

θ = 0.01, b = 0.60, and ¯ b = 0.80.

• The lower bound is binding for all time, while the sell boundary reaches the upper

bound near maturity.

• The sell strategy is not myopic in the sense that in anticipation of the constraint

becoming binding later, it is optimal to change the early trading strategy.

Illiquidity, Position Limits, and Optimal Investment for Mutual Funds – p. 18/21

Initial Illiquid Stock Holding against Correlation

−0.2 −0.1 0.1 0.2 0.3 0.4 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 ρ πI(0)

α = 5 % α = 5 %

α=2% α=2% α=1% α=1%

Parameters: γ = 2, T = 5, µL = 0.06, σL = 0.20, µI = 0.11, σI = 0.25, θ = α,

b = 0.60, and ¯ b = 0.80.

• The optimal fraction of wealth in the illiquid asset increases with the correlation

coefficient, because of the decrease in the diversification effect of the liquid stock investment.

• The no transaction region widens as the transaction cost rate increases, because

the trading in the illiquid asset becomes more costly.

Illiquidity, Position Limits, and Optimal Investment for Mutual Funds – p. 19/21

Liquidity Premium against Weights Bandwidth

0.2 0.4 0.6 0.8 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 β δ/α σ = 4 % σ =25%

Parameter values changed: b = 0.7292 − 1

2 β, and ¯

b = 0.7292 + 1

2β.

• For very stringent constraints, the liquidity premium (the maximum expected return

an investor is willing to exchange for zero transaction cost) can be much greater than what Constantinides (1986) finds, because imposing stringent constraints can force more frequent transactions and also distort the investment strategy.

• The liquidity premium increases with volatility.
• The liquidity premium against β may not be monotonically decreasing (σ = 0.4).

Illiquidity, Position Limits, and Optimal Investment for Mutual Funds – p. 20/21

Conclusions: Summary

• The behaviors of the optimal buy and sell boundary are characterized.
• Both the sell boundary and the buy boundary can be nonmyopic with

respect to the position limits even for a log utility.

• Position limits can significantly magnify the effect of transaction costs on

liquidity premium and can make it a first-order effect.

• Return correlations significantly affect diversification efficiency and
• ptimal trading strategy.

Illiquidity, Position Limits, and Optimal Investment for Mutual Funds – p. 21/21