Who Should Sell Stocks?
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Transcript of Who Should Sell Stocks?
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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
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Outline Model Main Result Implications Heuristics
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.
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Outline Model Main Result Implications Heuristics
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.
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Outline Model Main Result Implications Heuristics
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.
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Outline Model Main Result Implications Heuristics
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.
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Outline Model Main Result Implications Heuristics
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?
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Outline Model Main Result Implications Heuristics
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.
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Outline Model Main Result Implications Heuristics
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%.
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Outline Model Main Result Implications Heuristics
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.
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Outline Model Main Result Implications Heuristics
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) .
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Outline Model Main Result Implications Heuristics
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).
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Outline Model Main Result Implications Heuristics
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.
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Outline Model Main Result Implications Heuristics
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%.
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Outline Model Main Result Implications Heuristics
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.
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Outline Model Main Result Implications Heuristics
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%.
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Outline Model Main Result Implications Heuristics
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%.
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Outline Model Main Result Implications Heuristics
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%.
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Outline Model Main Result Implications Heuristics
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 ,
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Outline Model Main Result Implications Heuristics
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.
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Outline Model Main Result Implications Heuristics
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).
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Outline Model Main Result Implications Heuristics
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
,
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Outline Model Main Result Implications Heuristics
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 .
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Outline Model Main Result Implications Heuristics
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.
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Outline Model Main Result Implications Heuristics
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.