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Who Should Sell Stocks?

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Never selling stocks is optimal for investors with a long horizon and a realistic range of preference and market parameters, if relative risk aversion, investment opportunities, proportional transaction costs, and dividend yields are constant. Such investors should buy stocks when their portfolio weight is too low, and otherwise hold them, letting dividends rebalance to cash over time rather than selling. With capital gain taxes, this policy outperforms both static buy-and-hold and dynamic rebalancing strategies that account for transaction costs. Selling stocks becomes optimal if either their target weight is low, or intermediate consumption is substantial.

Published in: Economy & Finance

Who Should Sell Stocks?

  1. 1. 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
  2. 2. 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.
  3. 3. Outline Model Main Result Implications Heuristics 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.
  4. 4. 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.
  5. 5. 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 ): 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.
  6. 6. Outline Model Main Result Implications Heuristics 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?
  7. 7. 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 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.
  8. 8. Outline Model Main Result Implications Heuristics 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%.
  9. 9. 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 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.
  10. 10. Outline Model Main Result Implications Heuristics 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) .
  11. 11. 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 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) .
  12. 12. Outline Model Main Result Implications Heuristics 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.
  13. 13. Outline Model Main Result Implications Heuristics 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%.
  14. 14. Outline Model Main Result Implications Heuristics 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.
  15. 15. 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 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%.
  16. 16. 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%.
  17. 17. Outline Model Main Result Implications Heuristics 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. • 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%.
  18. 18. 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 = 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 ,
  19. 19. 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 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.
  20. 20. 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) + 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).
  21. 21. 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)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 ,
  22. 22. Outline Model Main Result Implications Heuristics 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 .
  23. 23. Outline Model Main Result Implications Heuristics 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.
  24. 24. Outline Model Main Result Implications Heuristics 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.

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