Presentation by Michael-Paul James on Optimal Portfolio Choice and the Valuation of Illiquid Securities by Francis A. Longstaff

1. Optimal Portfolio Choice
and the Valuation of
Illiquid Securities
Paper by Francis A. Longstaff
Presentation by Michael-Paul James
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2. Table of contents
Introduction Illiquidity
Definition, Contrast
Story, Questions, Context, Issues
Results
Numerical Testing
Portfolio Constraint
Illiquidity and Portfolio Choice
01 02
04 05
Continuous Time
Solving the investor’s portfolio choice
problem in the stochastic volatility
framework
Conclusion
Key Points and Takeaways
03
06
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3. Introduction
01
Story, Questions, Context, Issues
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4. Misleading Assumptions
● Fundamental assumption in portfolio choice
○ Investors can continuously trade securities at any quantity.
○ Stochastic process of unbounded variation
● Reality
○ Investors face liquidity constraints.
○ Illiquidity facilitates different portfolio choices than unconstrained
optimality.
○ Illiquid versus liquid assets should reflect welfare loss.
■ Large discounts to illiquid assets.
● Evidence
○ Equivalent Illiquid treasury notes & liquid treasury bills: 35 pt spread
○ Liquid versus illiquid Japanese bonds: 50 pt spread
○ Rule 144 Letter Stock versus equivalent liquid stock: 35 pt spread
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5. Illiquidity Model
● Illiquidity
○ Traditionally measured in bid ask spread for securities.
○ Risk a trader may not be able to exit a position quickly & costlessly.
● Method
○ Compare optimal portfolio strategy of an investor with and without
liquidity constraints.
● Shadow price of liquidity
○ Compare constrained to unconstrained utilities of wealth to
determine the price discount of an illiquid asset.
● Differences from literature
○ Past focuses on exogenous transaction costs or borrow constraints.
○ Paper focuses on endogenous effects of illiquidity on trading
strategies and security pricing.
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6. Illiquidity
02
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7. Illiquidity Definitions
● Liquidity
○ Defined as bid ask spread or transaction costs of trading securities.
● Illiquidity
○ Higher costs
■ Periods of increased trading and execution costs.
■ Geography and other markets can also increase costs.
○ Thin Market
■ Ability to buy or sell securities at any price is limited or
restricted.
■ A thin market is a period characterized by a few buyers and
sellers.
● Model attempts to capture real world events by limiting trading
frequency.
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8. Continuous Time
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Solving the investor’s portfolio choice problem
in the stochastic volatility framework
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9. Deriving Wealth Utility & Optimal Portfolio Weight
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10. Deriving Wealth Utility & Optimal Portfolio Weight
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11. Deriving Wealth Utility & Optimal Portfolio Weight
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12. Deriving Wealth Utility & Optimal Portfolio Weight
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The derived utility of wealth (Eq.17) and optimal portfolio weight (Eq.13) provide a complete
solution to the investor's portfolio choice problem in this stochastic volatility framework.

13. Portfolio Constraint
04
Illiquidity and Portfolio Choice
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14. Deriving Wealth Utility & Optimal Portfolio Weight
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15. Deriving Wealth Utility & Optimal Portfolio Weight
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The derived utility of wealth depends linearly on w(t) on the first term in the integral, and
quadratically on w(t) on the second term.

16. Results
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Numerical testing
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17. Table
1 Table 1 Horizon denotes the
investor's horizon in years; α is
the maximum number of
shares that can be traded per
year where the total number
of shares that could be
initially purchased is
normalized to one; volatility is
the current volatility of
returns on the risky asset;
unconstrained is the initial
optimal portfolio weight for
the risky asset in the absence
of liquidity constraints; and σ
is the volatility of volatility
parameter. The expected
return parameter μ is set
equal to .10 and the market
price of volatility risk λ equals
zero.
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Table 1: Optimal initial portfolio weight for the risky asset in the presence of liquidity restrictions
Horizon α Volatility Unconstrained σ = .00 σ = .20 σ = .40 σ = .60
1 0.00 0.7071 0.200 0.182 0.176 0.163 0.144
1 0.00 0.4472 0.500 0.499 0.487 0.458 0.417
1 0.00 0.3536 0.800 0.800 0.782 0.734 0.663
1 0.00 0.3162 1.000 0.994 0.968 0.903 0.813
1 0.00 0.2236 2.000 1.000 1.000 1.000 1.000
1 0.00 0.1414 5.000 1.000 1.000 1.000 1.000
1 0.10 0.7071 0.200 0.199 0.191 0.182 0.168
1 0.10 0.4472 0.500 0.495 0.489 0.470 0.427
1 0.10 0.3536 0.800 0.789 0.773 0.741 0.678
1 0.10 0.3162 1.000 1.000 0.934 0.879 0.817
1 0.10 0.2236 2.000 1.000 1.000 1.000 1.000
1 0.10 0.1414 5.000 1.000 1.000 1.000 1.000
2 0.00 0.7071 0.200 0.162 0.155 0.135 0.114
2 0.00 0.4472 0.500 0.497 0.482 0.437 0.380
2 0.00 0.3536 0.800 0.809 0.780 0.699 0.604
2 0.00 0.3162 1.000 0.998 0.956 0.848 0.723
2 0.00 0.2236 2.000 1.000 1.000 1.000 0.985
2 0.00 0.1414 5.000 1.000 1.000 1.000 0.999
2 0.10 0.7071 0.200 0.169 0.177 0.182 0.170
2 0.10 0.4472 0.500 0.466 0.492 0.466 0.411
2 0.10 0.3536 0.800 0.837 0.782 0.720 0.634
2 0.10 0.3162 1.000 1.000 0.923 0.855 0.756
2 0.10 0.2236 2.000 1.000 1.000 1.000 1.000
2 0.10 0.1414 5.000 1.000 1.000 1.000 1.000
Michael-Paul James

18. Table
2 Table 2 Horizon denotes the
investor's horizon in years; α is
the maximum number of
shares that can be traded per
year where the total
number of shares that could
be initially purchased is
normalized to one; volatility is
the current volatility of
returns on the risky
asset; unconstrained is the
initial optimal portfolio
weight for the risky asset in
the absence of liquidity
constraints; and σ is
the volatility of volatility
parameter. The expected
return parameter μ is set
equal to .10 and the market
price of volatility risk
λ equals zero.
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Table 2: Percentage liquidity discounts for the risky asset
Horizon α Volatility Unconstrained σ = .00 σ = .20 σ = .40 σ = .60
1 0.00 0.7071 0.200 0.105 0.207 0.509 1.132
1 0.00 0.4472 0.500 0.033 0.265 0.977 2.480
1 0.00 0.3536 0.800 0.000 0.364 1.488 3.853
1 0.00 0.3162 1.000 0.001 0.455 1.858 4.795
1 0.00 0.2236 2.000 2.473 3.158 5.452 10.509
1 0.00 0.1414 5.000 14.798 16.150 20.721 30.428
1 0.10 0.7071 0.200 0.067 0.115 0.338 0.877
1 0.10 0.4472 0.500 0.026 0.191 0.819 2.240
1 0.10 0.3536 0.800 0.000 0.303 1.340 3.628
1 0.10 0.3162 1.000 0.003 0.426 1.744 4.576
1 0.10 0.2236 2.000 2.473 3.158 5.451 10.497
1 0.10 0.1414 5.000 14.798 16.150 20.721 30.428
2 0.00 0.7071 0.200 0.493 0.838 2.152 5.984
2 0.00 0.4472 0.500 0.229 1.041 4.198 13.348
2 0.00 0.3536 0.800 0.056 1.355 6.327 20.210
2 0.00 0.3162 1.000 0.023 1.662 7.845 24.583
2 0.00 0.2236 2.000 4.894 7.502 17.658 44.244
2 0.00 0.1414 5.000 27.409 32.027 48.463 79.911
2 0.10 0.7071 0.200 0.269 0.517 1.407 4.904
2 0.10 0.4472 0.500 0.178 0.677 3.412 12.275
2 0.10 0.3536 0.800 0.043 1.048 5.613 19.275
2 0.10 0.3162 1.000 0.021 1.435 7.163 23.717
2 0.10 0.2236 2.000 4.894 7.502 17.608 43.915
2 0.10 0.1414 5.000 27.409 32.027 48.463 79.898
Michael-Paul James

19. Conclusion
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Key Points and Takeaways
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20. Intertemporal Portfolio Choice
● Paper proposes a solution to trading strategies under constraint
(bounded variation)
● Closely approximates thin markets in the real world.
● Optimal trading strategy endogenously imposes borrowing and
short-selling constraints on the investor.
● Trading constraints exposes levered investors to more bankruptcy risks
and lower welfare gains.
● To avoid risks, investors receive discounts for illiquidity (pay a premium
for liquidity).
● The model provides results similar to empirical observations under
specific parameters.
● Further analysis might include a full equilibrium framework with
multiple agents and securities.
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21. You are Amazing
Ask me all the questions you desire. I will do my best to answer honestly
and strive to grasp your intent and creativity.
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