who should sell stocks?

Outline Model Main Result Implications Heuristics

Who Should Sell Stocks?

Paolo Guasoni1,2 Ren Liu3 Johannes Muhle-Karbe3,4

Boston University1

Dublin City University2

ETH Zurich3

University of Michigan4

Mathematical Modeling in Post-Crisis FinanceGeorge Boole 200th Conference, August 26th, 2015


Outline

• Motivation.Buy and Hold vs. Rebalancing. Practice vs. Theory?

• Model:Constant investment opportunities and risk aversion.Dividends and Transaction Costs.

• Result:Buy and Hold vs. Rebalancing regimes. Implications.


Folklore vs. Theory

• Buy and Hold?• Market Efficiency. Malkiel (1999):

The history of stock price movements contains no usefulinformation that will enable an investor consistently to outperform abuy-and-hold strategy in managing a portfolio.

• Portfolio Advice. Stocks for the Long Run (Siegel, 1998)• Warren Buffett (1988):

our favorite holding period is forever.

• Rebalance?• Frictionless theory (Merton, 1969, 1971):

Keep assets’ proportions constants. Rebalance every day.• Transaction costs (Magill, Constantinides, 1976, 1986, Davis, Norman, 1990):

Buy when proportion too low. Sell when too high. Hold in between.• Buy and hold only if optimal frictionless proportion 100%.

Neither robust nor relevant.

• No theoretical result supports buy and hold.


What We Do

• For realistic range of market and preference parameters, it is optimal to:• Buy stocks when their proportion is too low.• Hold them otherwise.• Never sell.

• Assumptions:• Constant investment opportunities and risk aversion (like Merton).• Constant proportional transaction costs (like Davis and Norman).• And constant proportional dividend yield.

• Intuition• When the proportion of stocks is high, dividends are also high.• To rebalance, a better alternative to selling is... waiting.• Qualitative effect. When does it prevail?

• More frictions, less complexity.• Dividends alone irrelevant (Miller and Modigliani, 1961).• Transaction costs alone not enough (Dumas and Luciano, 1991).• With both, qualitatively different solution. Selling can disappear.


Market and Preferences• Safe asset (money market) earns constant interest rate r .• Risky asset traded with constant proportional costs ε. Bid and ask

prices (1− ε)St and (1 + ε)St .• Risky asset pays dividend stream δSt .

Constant dividend yield δ.• Risky asset (stock) mid-price St follows geometric Brownian motion:

dSt

St= (µ− δ + r)dt + σdWt

Constant total excess return µ and volatility σ.• Investor with long horizon and constant relative risk aversion γ > 0.

Maximizes equivalent safe rate of total wealth (cash Xt plus stock YT ):

limT→∞

1T

log E[(XT + YT )

1−γ] 11−γ

as in Dumas and Luciano (1991), Grossman and Vila (1992), and others.


Dividends as Static Rebalancing

• Budget equation without trading:

dXt = rXtdt + δYtdtdYt = (µ− δ + r)Ytdt + σYtdWt

• Risky/safe ratio Zt = Yt/Xt equals ratio of portfolio weights YtXt+Yt

/ XtXt+Yt

.• By Itô’s formula, it satisfies

dZt = (µ− δ − δZt)Ztdt + σZtdWt

• No dividends (δ = 0): geometric Brownian motion.Risky weight converges to one, forcing rebalancing.

• Dividends (δ > 0) make stock weight mean-reverting to 1− δµ .

(Long-run distribution is gamma.)• Selling and waiting are substitutes. Which one is better when?


Main Result (Summary)

• Assumption: frictionless portfolio is long-only.

π∗ :=µ

γσ2 ∈ (0,1)

(Otherwise selling necessary to prevent bankruptcy.)• Classical Regime:

If dividend yield δ small enough, keep portfolio weight within boundariesπ− < π∗ < π+ (buy below π− and sell above π+).

• Never Sell Regime:If dividend yield large, keep portfolio weight withing above π−(buy below π− and never sell).

• Realistic Example:µ = 8%, σ = 16%, γ = 3.45, hence π∗ = 90%. ε = 1%.

• With no dividends, buy below 87.5% and sell above 92.5%.• With 3% dividends, buy when below 90%, otherwise hold. Never sell.


Selling Disappears

1 2 3 4 5 6 7 8

∆ @%D86

88

90

92

94

96

98

100

Π @%D

Buy (bottom) and Sell (top) boundaries (vertical) vs. dividend (horizontal).µ = 8%, σ = 16%, γ = 3.45, ε = 1%.


Main Result (details)

• Define

π−(λ) =µ− εδ/(1 + ε)−

√λ2 − 2µεδ/(1 + ε) + (εδ/(1 + ε))2

γσ2 ,

π+(λ) = min

(µ+ εδ/(1− ε) +

√λ2 + 2µεδ/(1− ε) + (εδ/(1− ε))2

γσ2 ,1

),

• π−(λ), π+(λ) are candidate buy and sell boundaries, identified by theexact value of λ, which is part of the solution.

• π+(λ) = 1 corresponds to never-sell regime.• Expressions for π−(λ), π+(λ) follow from stochastic control derivations.


Classical Regime ConditionAssumption

(CL) There exists λ > 0 such that (i) π+(λ) < 1 and the solution w(x , λ) of

0 =w ′(x) + (1− γ)w(x)2 +(

2γ − 1− 2(µ−δ)σ2 + 2δ

σ2ex u(λ)

)w(x)

−(γ + µ2−λ2

γσ4 − 2(µ−δ))σ2

),

with the boundary condition

w(

log(

l(λ)u(λ)

))= l(λ)

1+ε+l(λ) ,

wherel(λ) = (1 + ε) 1−π−(λ)

π−(λ), u(λ) = (1− ε) 1−π+(λ)

π+(λ),

satisfies the additional boundary condition:

w(0, λ) = u(λ)1−ε+u(λ) .


Never-Sell Regime Condition

Assumption

(NS) There exists λ > 0 such that π+(λ) = 1 and the solution w(x , λ) of

0 =w ′(x) + (1− γ)w(x)2 +(

1− 2γ + 2(µ−δ)σ2 − 2δex

σ2 l(λ)

)w(x)

−(γ + µ2−λ2

γσ4 − 2(µ−δ)σ2

),

with boundary condition

0 = limx→∞ w(x),

satisfies the additional boundary condition:

w(0, λ) =−l(λ)

1 + ε+ l(λ).


Main Result (Statement)

TheoremUnder either condition (CL) or (NS),

• Optimal Strategy:Hold within (π−, π+). At boundaries, trade to keep the risky weight inside[π−, π+]. (π− evaluated at ask price (1 + ε)St , π+ at bid (1− ε)St .)

• Equivalent Safe Rate:Trading the dividend-paying risky asset with transaction costs equivalentto leaving all wealth in a hypothetical safe asset that pays the rate

EsR = r +µ2 − λ2

2γσ2 .

• Reduced value function w(x , λ) has solution in terms of special functions.• λ does not have closed-form expression. Asymptotics.


Who Should Sell Stocks?

0 1 2 3 4 5

∆ @%D

60

70

80

90

100

Π @%D

Never sell in the blue region. Otherwise classical regime. ε = 1%.


Asymptotics

• Expansion of trading boundaries for small ε:

π± = π∗ ±(

32γπ2∗(1− π∗)2

)1/3

ε1/3 +δ

γσ2

(2γπ∗

3(1− π∗)2

)1/3

ε2/3 +O(ε).

• Zeroth order (black): frictionless portfolio.• First order (blue): classical transaction costs.

With (Davis and Norman) or without (Dumas and Luciano) consumption.• Second order (red): effect of dividends, pushing up boundaries.• Small dividends negligible compared to transaction costs.• But 2-3% dividends already large if π∗ is large.• Never-sell regime beyond reach of small ε asymptotics.


Never Sell. No Regrets.

π∗ optimal never sell buy & hold[π−, π+] [π−,1] [0,1]

50% 1.67% 2.00% 4.67%60% 1.76% 1.76% 4.41%70% 1.58% 1.58% 4.21%80% 1.43% 1.43% 3.81%90% 1.52% 1.52% 3.70%

• Even when it is not optimal, the never-sell strategy is closer to optimalthan the static buy-and-hold.

• Relative equivalent safe rate loss (EsR0−EsR)/EsR0 of optimal([π−, π+]), never sell ([π−,1]) and buy-and-hold ([0,1]) strategies.

• Simulation with T = 20, time step dt = 1/250, and sample N = 2× 107.• µ = 8%, σ = 16%, r = 1%, δ = 2%, and ε = 1%.


Never Sell. Never Pay Taxes (on Capital Gains).• Discussion so far neglects effect of taxes on capital gains...• ...which do not affect the never-sell strategy...• ...but reduce the performance of other “optimal ” policies...• ...making never-sell superior after tax.

π∗ [π−, π+] [π−, π+] never sell buy & hold(average) (specific)

50% 2.41% 2.41% 2.07% 4.48%60% 2.13% 2.13% 1.83% 3.96%70% 1.91% 1.91% 1.64% 3.55%80% 1.49% 1.49% 1.49% 3.22%90% 1.36% 1.36% 1.36% 2.94%

• Relative loss (EsR0,τ −EsR)/EsR0,τ with capital gains taxes, for optimal([π−, π+]), never sell ([π−,1]), and buy-and-hold ([0,1]) strategies.

• Simulation with T = 20, time step dt = 1/250, and sample N = 2× 107.• Both taxes on dividends (τ ) and capital gains (α) accounted for.• µ = 8%, σ = 16%, α = 20%, τ = 20%, r = 1%, δ = 2%, and ε = 1%.


Terms and Conditions

• Never Selling superior to rebalancing for long-term investors withmoderate risk aversion, and no intermediate consumption.

• With high consumption and low dividends selling is necessary.

π∗ [πJS− , π

JS+ ] never sell buy & hold

50% 1.00% 1.67% 2.00%60% 0.59% 1.17% 1.47%70% 0.53% 1.05% 1.05%80% 0.48% 0.71% 0.71%90% 0.22% 0.65% 0.65%

• Relative loss (EsR0−EsR)/EsR0 of the asymptotically optimal([πJS− , π

JS+ ]), never-sell ([π−,1]) and static buy-and-hold ([0,1]) strategies

with πJS± from Janecek-Shreve.

• Simulation with T = 20, time step dt = 1/250, and sample N = 2× 107.• µ = 8%, σ = 16%, ρ = 2%, r = 1%, τ = 0%, ε = 1% and δ = 3%.


Wealth and Value Dynamics• Number of safe units ϕ0

t , number of shares ϕt = ϕ↑t − ϕ↓t

• Values of the safe and risky positions (using mid-price St ):

Xt = ϕ0t S0

t , Yt = ϕtSt ,

• Budget equation:

dXt = rXtdt + δYtdt − (1 + ε)Stdϕ↑t + (1− ε)Stdϕ

↓t ,

dYt = (µ− δ + r)Ytdt + σYtdWt + Stdϕ↑t − Stdϕ

↓t .

• Value function V (t ,Xt ,Yt) satisfies:

dV (t ,Xt ,Yt) = Vtdt + VxdXt + Vy dYt +12

Vyy d〈Y ,Y 〉t

=

(Vt + rXtVx + δYtVx + (µ− δ + r)YtVy +

σ2

2Y 2

t Vyy

)dt

+ St(Vy − (1 + ε)Vx)dϕ↑t + St((1− ε)Vx − Vy )dϕ

↓t + σYtdWt ,


HJB Equation

• V (t ,Xt ,Yt) supermartingale for any choice of ϕ↑t , ϕ↓t (increasing

processes). Thus, Vy − (1 + ε)Vx ≤ 0 and (1− ε)Vx − Vy ≤ 0, that is

11 + ε

≤ Vx

Vy≤ 1

1− ε.

• In the interior of this region, the drift of V (t ,Xt ,Yt) cannot be positive, andmust be zero for the optimal policy,

Vt + rXtVx + δYtVx + (µ− δ + r)YtVy + σ2

2 Y 2t Vyy = 0, if 1

1+ε <VxVy< 1

1−ε .

• (i) Value function homogeneous with wealth. (ii) In the long run it shouldgrow exponentially with the horizon. Guess

V (t ,Xt ,Yt) = (Yt)1−γv(Xt/Yt)e−(1−γ)(r+β)t

for some function v and some rate β.


Second Order Linear ODE• Setting z = x/y , the HJB equation reduces to

0 = σ2

2 (−γ(1− γ)v(z) + 2γzv ′(z) + z2v ′′(z)) + (µ− δ)((1− γ)v(z)− zv ′(z)),

+ δv ′(z)− β(1− γ)v(z), if 1− ε+ z ≤ (1− γ)v(z)v ′(z)

≤ 1 + ε+ z.

• Guessing no-trade region {z : 1− ε+ z ≤ (1−γ)v(z)v ′(z) ≤ 1+ ε+ z} of interval

type u ≤ z ≤ l , free boundary problem arises:

0 =σ2

2(−γ(1− γ)v(z) + 2γzv ′(z) + z2v ′′(z)) + (µ− δ)((1− γ)v(z)− zv ′(z))

+ δv ′(z)− β(1− γ)v(z),0 = (1− ε+ u)v ′(u)− (1− γ)v(u),0 = (1 + ε+ l)v ′(l)− (1− γ)v(l).

• Smooth-pasting conditions:

0 = (1− ε+ u)v ′′(u) + γv ′(u),0 = (1 + ε+ l)v ′′(l) + γv ′(l).


First Order Nonlinear ODE

• The substitution

v(z) = e(1−γ)∫ log (z/u(λ))

0 w(y)dy , i.e., w(x) =u(λ)exv ′(u(λ)ex)

(1− γ)v(u(λ)ex),

reduces the boundary value problem to a Riccati equation:

0 = w ′(x) + (1− γ)w(x)2 +

(2γ − 1− 2(µ− δ)

σ2 +2δ

σ2exu

)w(x)

−(γ +

µ2 − λ2

γσ4 − 2(µ− δ)σ2

),

w(0, λ) =u

1− ε+ u,

w(

log(

l(λ)u(λ)

), λ)=

l1 + ε+ l

,


Capture Free Boundaries

• Eliminating v ′′(l) and v ′(l), and setting π− = (1 + ε)/(1 + ε+ l),

−γσ2

2π2− +

(µ− εδ

1 + ε

)π− − β = 0,

whence

π− =µ− εδ/(1 + ε)±

√(µ− εδ/(1 + ε))2 − 2βγσ2

γσ2 ,

and smaller solution is the natural candidate.• Analogously, setting π+ = (1− ε)/(1− ε+ u), leads to the guess

π+ =µ+ εδ/(1− ε) +

√(µ+ εδ/(1− ε))2 − 2βγσ2

γσ2 .


Whittaker ODE

• Set B = 2δσ2 ,N = γ − µ−δ

σ2 − 1 and apply substitution (similar to Jang(2007))

v(z) =:

(Bz

)N

exp(

B2z

)h(

Bz

)which leads to the Whittaker equation

0 = h′′(

Bz

)+

(−1

4+−NB/z

+1/4−m2

(B/z)2

)h(

Bz

),

C = (1− γ)(γ + µ2−λ2

γσ4 − 2(µ−δ)σ2

),m =

√1/4 + N(N + 1) + C.

• Solution is (up to multiplicative constant)

h(

Bz

)= W−N,m

(Bz

)where W−N,m is a special function defined through the Tricomi function.


Conclusion

• With dividends and proportional transaction costs, never selling is optimalfor long-term investors with moderate risk aversion.

• Even when not optimal, close to optimal.• Optimal policy with capital-gain taxes. Regardless of cost basis.• Sensitive to intertemporal consumption. Requires high dividends.• Compounding frictions does not compound their separate effects.

who should sell stocks?

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