Who Should Sell Stocks? Modeling Dividends and Transaction Costs

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 Finance
George 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 useful
information that will enable an investor consistently to outperform a
buy-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 ):
lim
T→∞
1
T
log E (XT + YT )1−γ
1
1−γ
as in Dumas and Luciano (1991), Grossman and Vila (1992), and others.

Dividends as Static Rebalancing
• Budget equation without trading:
dXt = rXt dt + δYt dt
dYt = (µ − δ + r)Yt dt + σYt dWt
• Risky/safe ratio Zt = Yt /Xt equals ratio of portfolio weights Yt
Xt +Yt
/ Xt
Xt +Yt
.
• By Itô’s formula, it satisﬁes
dZt = (µ − δ − δZt )Zt dt + σZt dWt
• No dividends (δ = 0): geometric Brownian motion.
Risky weight converges to one, forcing rebalancing.
• Dividends (δ > 0) make stock weight mean-reverting to 1 − δ
µ .
(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 boundaries
π− < 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
∆
86
88
90
92
94
96
98
100
Π
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 the
exact 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 Condition
Assumption
(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) ,
where
l(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
σ2l(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)
Theorem
Under 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 equivalent
to 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
∆
60
70
80
90
100
Π
Never sell in the blue region. Otherwise classical regime. ε = 1%.

Asymptotics
• Expansion of trading boundaries for small ε:
π± = π∗ ±
3
2γ
π2
∗(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 optimal
than 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.
.
• 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 with
moderate risk aversion, and no intermediate consumption.
• With high consumption and low dividends selling is necessary.
π∗ [πJS
− , πJS
+ ] 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
− , πJS
+ ]), never-sell ([π−, 1]) and static buy-and-hold ([0, 1]) strategies
with πJS
± from Janecek-Shreve.
.
• 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 = ϕ0
t S0
t , Yt = ϕt St ,
• Budget equation:
dXt = rXt dt + δYt dt − (1 + ε)St dϕ↑
t + (1 − ε)St dϕ↓
t ,
dYt = (µ − δ + r)Yt dt + σYt dWt + St dϕ↑
t − St dϕ↓
t .
• Value function V(t, Xt , Yt ) satisﬁes:
dV(t, Xt , Yt ) = Vt dt + Vx dXt + Vy dYt +
1
2
Vyy d Y, Y t
= Vt + rXt Vx + δYt Vx + (µ − δ + r)Yt Vy +
σ2
2
Y2
t Vyy dt
+ St (Vy − (1 + ε)Vx )dϕ↑
t + St ((1 − ε)Vx − Vy )dϕ↓
t + σYt dWt ,

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
1
1 + ε
≤
Vx
Vy
≤
1
1 − ε
.
• In the interior of this region, the drift of V(t, Xt , Yt ) cannot be positive, and
must be zero for the optimal policy,
Vt + rXt Vx + δYt Vx + (µ − δ + r)Yt Vy + σ2
2 Y2
t Vyy = 0, if 1
1+ε < Vx
Vy
< 1
1−ε .
• (i) Value function homogeneous with wealth. (ii) In the long run it should
grow exponentially with the horizon. Guess
V(t, Xt , Yt ) = (Yt )1−γ
v(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) + z2
v (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) + z2
v (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)ex
v (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
σ2ex u
w
− γ +
µ2
− λ2
γσ4
−
2(µ − δ)
σ2
,
w(0, λ) =
u
1 − ε + u
,
w log l(λ)
u(λ) , λ =
l
1 + ε + l
,

Capture Free Boundaries
• Eliminating v (l) and v (l), and setting π− = (1 + ε)/(1 + ε + l),
−
γσ2
2
π2
− + 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 = 2δ
σ2 , N = γ − µ−δ
σ2 − 1 and apply substitution (similar to Jang
(2007))
v(z) =:
B
z
N
exp
B
2z
h
B
z
which leads to the Whittaker equation
0 = h
B
z
+ −
1
4
+
−N
B/z
+
1/4 − m2
(B/z)2
h
B
z
,
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
B
z
= W−N,m
B
z
where W−N,m is a special function deﬁned through the Tricomi function.

Conclusion
• With dividends and proportional transaction costs, never selling is optimal
for 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? Modeling Dividends and Transaction Costs

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