Optimal Portfolios Under Worst-Case Scenarios
Jit Seng Chen
January 28, 2016
Optimal Portfolios Under Worst-Case Scenarios
• Joint work with Carole Bernard (Grenoble School of
Management) and Steven Vanduffel (Vrije Universiteit
Brussel)
• Optimal portfolios under worst-case scenarios. Quantitative
Finance, 14(4):657-671, 2014
2
Stewardship of Finance
3
Outline
1.
2.
3.
4.
5.
6.
7.
Motivation
Setting
Traditional Diversification Strategies
Optimal Tail Diversification
Examples
Conclusion
Q&A
4
1. Motivation
Standard Portfolio Theories
• Expected Utility Theory (EUT) - dominant theory for making
decisions under risk but inconsistent with observed
behaviour
• Many alternatives have been proposed to capture systematic
behavioural departures from EUT
• Most portfolio theories assume that investors only care about
the terminal distribution of wealth and prefer more to less
6
Law-Invariant and Increasing
1
𝑋 =�
0
1
0
𝑋 =�
1
2
50% 𝑐𝑐𝑐𝑐𝑐𝑐 𝑜𝑜 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎
50% 𝑐𝑐𝑐𝑐𝑐𝑐 𝑜𝑜 𝑛𝑛 − 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎
50% 𝑐𝑐𝑐𝑐𝑐𝑐 𝑜𝑜 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎
50% 𝑐𝑐𝑐𝑐𝑐𝑐 𝑜𝑜 𝑛𝑛 − 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎
Expected utility of contract 1 = Expected utility of contract 2
7
Standard Portfolio Theories
• For an investor with a fixed horizon without intermediate
consumption, optimal portfolios are positively correlated
with a diversified market portfolio
• No downside protection
• Seek security in specific states of the economy
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SP/A Theory
• Security-Potential/Aspiration
• Maximize expected consumption subject to budget constraint and
minimum probability that consumption meets an aspiration level
• Explains the use of capital guaranteed products, CPPI
strategies, and other structured products
• Optimal portfolios still positively correlated with the market
i.e. perform poorly in a crisis
9
Framework
• Consider investors who target some distribution for their final
wealth and seek security in specific states of the economy
• Crisis → market index takes values in the worst 5% of its
distribution
• Allow investors to specify the tail dependence of strategies
with the market under crisis regimes
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Contribution
• Construct strategy that
• Delivers a prescribed wealth distribution
• Preserves a desired dependence with the market during a crisis
• Comes at the cheapest possible price
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2. Setting
Financial Market
• Classical multi-dimensional Black-Scholes with deterministic
risk free rate
• Complete, frictionless and arbitrage free
• Variance-covariance matrix
is positive definite
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Growth Optimal Portfolio
• Constant-mix portfolios with
in the risk-free asset
in
and
invested
• Value of constant-mix portfolio is
where
is normal
• The growth optimal portfolio (GOP)
is a constant-mix
strategy that maximizes the expected growth rate with
proportions
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Growth Optimal Portfolio
• The price of a pay-off equals the average of all its possible
outcomes expressed in units of the GOP i.e. GOP appears
as a numeraire portfolio
• Define a market crisis as the event
where
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3. Traditional Diversification Strategies
Buy-and-Hold
• Initial amount is
the bank account and
, where
is invested in
is invested in the ith stock
• No rebalancing.
• Final wealth
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Constant Mix
• Need dynamic rebalancing to preserve initial allocation
• Initial investment of
, at maturity, grows to
• GOP is a constant-mix portfolio
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Performance During a Crisis
• An optimal strategy must be the cheapest possible one (costefficient) that provides some distribution F for terminal wealth
• A strategy
with distribution F is cost efficient if and only if
• Cost-efficient strategies are monotonic in the GOP → they
provide their worst outcomes in downturns
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Performance During a Crisis
• Consider two payoffs,
and
with the same distribution F,
but only the former is non-decreasing in the GOP
• Can be shown that all investors who prefer more to less will
choose
I over
• Investors who seek value when GOP is low tend to prefer the
latter payoff
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4. Optimal Tail Diversification
Optimal Tail Diversification
• Cheapest strategies that exhibit a fixed distribution and
satisfy some constraints are called “constrained-cost
efficient”
• Provides the cheapest payoff for an investor who wants to
achieve a fixed distribution F for her final wealth at
investment horizon T, as well as a desired interaction with
the market under a crisis regime
• Key result in this paper
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Definitions
• Let F be the investor’s target distribution of terminal wealth
• Investor chooses joint distribution of final wealth with the
market when there is a crisis
• This constraint can be expressed using a copula
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Result
•
•
is any random variable with
distributed.
,
, and
continuously
can be calculated explicitly
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Result
• Construction of constrained optimal strategy requires another
source of uncertainty
•
•
is the cheapest strategy with distribution F that verifies
the constraints on the dependency in the tail
is not unique, depends largely on the choice of
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Applications
• Can find explicit expression for the optimal strategy using
independence, Gaussian, Frank, and Clayton copulas
• Can identify
with another asset in the market at maturity
(path-independent), or the value of the GOP at an earlier
time t<T (path dependent)
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5. Examples
Parameters
• Some numerical results in a two-dimensional Black-Scholes
market
• μ1 = 0.07, σ1 = 0.20; μ1 = 0.08, σ1 = 0.30; ρ12 = 0.25; α = 0.05;
r = 0.05; T = 1
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Standard Diversification Strategies
• Strategy 1: Invest fully in the GOP
• Strategy 2: Equal proportions in risky assets and risk free
account, no rebalancing
• Strategy 3: Equal proportions in risky assets and risk free
account, with rebalancing
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Optimal Diversification Strategies
• Same distribution as GOP at maturity, different tail
dependence during crisis
• Strategy 4: Independent tail
• Strategy 5: Gaussian tail with negative dependence
• Strategy 6: Clayton tail with negative dependence
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Strategies 1-3 vs GOP
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Strategies 4-6 vs GOP
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Measures
• Cost
• Sharpe ratio
• Conditional probabilities related to the following events
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Cost and Sharpe Ratio
Strategy
Cost
Sharpe
1 – Invest fully in GOP
100
0.125
2 – Buy-and-hold
100
0.121
3 – Constant-mix
100
0.123
4 – Independent tail
100.30
0.103
5 – Gaussian tail
100.56
0.083
6 – Clayton tail
100.41
0.095
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Conditional Probabilities
Strategy
P (A|C)
P (B|C)
1 – Invest fully in GOP
1.00
0.19
2 – Buy-and-hold
1.00
0.16
3 – Constant-mix
1.00
0.27
4 – Independent tail
0.48
0.01
5 – Gaussian tail
0.12
0.000
6 – Clayton tail
0.24
0.000
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6. Conclusion
Conclusion
• Optimal strategies that are non-decreasing in the bear
market incur their worst outcomes in bear markets
• Construct the cheapest possible strategy that
• Delivers a given wealth distribution
• Preserves a desired tail dependence with the market during a
crisis
• Trade-off between performance vs protection
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7. Q & A