Optimal Portfolio Under Worst-Case Scenarios Carole Bernard (UW), - - PowerPoint PPT Presentation

Optimal Portfolio Under Worst-Case Scenarios

Carole Bernard (UW), Jit Seng Chen (UW) and Steven Vanduffel (Vrije Universiteit Brussel) SIAM, July 2012.

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Introduction Diversification Strategies Tail Dependence Numerical Example Conclusions

Contributions

1 Understanding issues with traditional diversification strategies

(buy-and-hold, constant mix, Growth Optimal Portfolio) and how lowest outcomes of optimal strategies always happen in the worst states of the economy.

2 Develop innovative strategies to cope with this observation. 3 Implications in terms of assessing risk and return of a

strategy and in terms of reducing systemic risk

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Introduction Diversification Strategies Tail Dependence Numerical Example Conclusions

Part I:

Traditional Diversification Strategies

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Growth Optimal Portfolio (GOP)

• The Growth Optimal Portfolio (GOP) maximizes expected

logarithmic utility from terminal wealth.

• It has the property that it almost surely accumulates more

wealth than any other strictly positive portfolios after a sufficiently long time.

• Under general assumptions on the market, the GOP is a

diversified portfolio.

• Details in Platen & Heath (2006).

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For example, in the 2-dim Black-Scholes model

• Risk-free asset {Bt = B0ert, t 0}

  

dS1

t

S1

t = µ1dt + σ1dW 1

t dS2

t

S2

t = µ2dt + σ2dWt

, where W 1 and W are two correlated Brownian motions under the physical probability measure P.

• Constant-mix strategy: Dynamic rebalancing to preserve the

initial target allocation. The payoff of a constant-mix strategy is Sπ

t = Sπ 0 exp(X π t ) where X π t is normal.

• The Growth Optimal Portfolio (GOP) is a constant-mix

strategy with X π

t =

• µπ − 1

2σ2 π

• t + σπW π

t , that maximizes

the expected growth rate µπ − 1

2σ2 π. It is

π⋆ = Σ−1 · → µ −r

→

1

• .

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Market Crisis The growth optimal portfolio S⋆ can also be interpreted as a major market index. Hence it is intuitive to define a stressed market (or crisis) at time T as an event where the market - materialized through S⋆ - drops below its Value-at-Risk at some high confidence level. The corresponding states of the economy verify Crisis states = {S⋆

T < qα} ,

(1) where qα is such that P(S⋆

T < qα) = 1 − α and α is typically high

(e.g. α = 0.98).

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Strategy 1: GOP We invest fully in the GOP. In a crisis (GOP is low), our portfolio is low!

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60 80 100 120 140 160 180 200 60 80 100 120 140 160 180 200 Growth Optimal Portfolio, S∗(T ) Strategy 1 Strategy 1 vs the Growth Optimal Portfolio

Introduction Diversification Strategies Tail Dependence Numerical Example Conclusions

Strategy 2: Buy-and-Hold The buy-and-hold strategy is the simplest investment strategy. An initial amount V0 is used to purchase w0 units of the bank account and wi units of stock Si (i = 1, 2) such that V0 = w0 + w1 S1

0 + w2 S2 0,

and no further action is undertaken. Example with 1/3 invested in each asset (bank, S1 and S2) on next slide.

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60 80 100 120 140 160 180 200 60 80 100 120 140 160 180 200 220 Growth Optimal Portfolio, S∗(T ) Strategy 2 Strategy 2 vs the Growth Optimal Portfolio

Introduction Diversification Strategies Tail Dependence Numerical Example Conclusions

◮ These traditional diversification strategies do not offer protection during a crisis. ◮ In a more general setting, optimal strategies share the same problem... Optimal Portfolio Selection Problem: Consider an investor with fixed investment horizon: max

XT

U(XT) subject to a given “cost of XT” (equal to initial budget)

• Law-invariant preferences XT ∼ YT ⇒ U(XT) = U(YT)
• Increasing preferences

XT ∼ F, YT ∼ G, ∀x, F(x) G(x) ⇒ U(XT) U(YT)

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Optimal Investment max

{XT / initial budget=x0} U(XT)

Theorem Optimal strategies for U must be cost-efficient. where we recall the definition of cost-efficiency. Definition - Dybvig (1988) A strategy (or a payoff) is cost-efficient if any other strategy that generates the same distribution F under P costs at least as much. A cost-efficient strategy solves the following optimization problem minXT cost(XT) subject to {XT ∼ F} .

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Stochastic Discount Factor and Real-World Pricing: The GOP can be used as numeraire to price under P x0 = Cost of XT at 0

• = EQ[e−rTXT] = EP

XT S⋆

T

• where S⋆

0 = 1.

Cost-efficiency problem: minXT EP

• XT

S⋆

T

• subject to

{XT ∼ F} Theorem A strategy is cost-efficient if and only if its payoff is equal to XT = h(S⋆

T) where h is non-decreasing.

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Stochastic Discount Factor and Real-World Pricing: The GOP can be used as numeraire to price under P x0 = Cost of XT at 0

• = EQ[e−rTXT] = EP

XT S⋆

T

• where S⋆

0 = 1.

Cost-efficiency problem: minXT EP

• XT

S⋆

T

• subject to

{XT ∼ F} Theorem A strategy is cost-efficient if and only if its payoff is equal to XT = h(S⋆

T) where h is non-decreasing.

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Idea of the proof (First method) min

XT

E XT S⋆

T

• subject to
• XT ∼ F

1 S⋆

T ∼ G

Recall that corr

• XT, 1

S⋆

T

• =

E

• XT

S⋆

T

• − E[ 1

S⋆

T ]E[XT]

std( 1

S⋆

T ) std(XT)

. We can prove that when the distributions for both XT and

1 S⋆

T are

fixed, we have

• XT, 1

S⋆

T

• is anti-monotonic ⇔ corr
• XT, 1

S⋆

T

• is minimal.

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Introduction Diversification Strategies Tail Dependence Numerical Example Conclusions

Idea of the proof (Second method) Recall that the joint cdf of a couple (S⋆

T, XT) writes as

P(S⋆

T s, XT x) = C(G(s), F(x))

where G is the marginal cdf of S⋆

T (known: it depends on the

financial market), F is the marginal cdf of XT and C denotes the copula for (S⋆

T, X).

min

XT

E XT S⋆

T

• subject to XT ∼ F

E XT S⋆

T

• =

(G(1/ξ) − C(G(1/ξ), F(x)))dxdξ, (2) The lower bound for E

• XT

S⋆

T

• is derived from the upper bound on C

C(u, v) min(u, v) (where min(u, v) corresponds to the comonotonic copula). then X⋆

T = F −1 (G (S⋆ T)) has the minimum price for the cdf F.

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Part II:

Investment under Worst-Case Scenarios

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Type of Constraints We find optimal strategies with final payoff XT ∼ F ◮ with a set of probability constraints, for example assuming that the final payoff of the strategy is independent of S⋆

T

during a crisis (defined as S⋆

T qα),

∀s qα, x ∈ R, P(S⋆

T s, XT x) = P(S⋆ T s)P(XT x)

Theorem (Optimal Investment with Independence in the Tail) The cheapest path-dependent strategy with cdf F and independent

• f S⋆

T when S⋆ T qα can be constructed as

X ⋆

T =

   F −1

• FS⋆

T (S⋆ T )−α

1−α

• when

S⋆

T > qα,

F −1 (g(S⋆

t , S⋆ T))

when S⋆

T qα,

(3) where g(., .) is explicit and t ∈ (0, T) can be chosen freely.

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60 80 100 120 140 160 180 200 60 80 100 120 140 160 180 200 Growth Optimal Portfolio, S∗(T ) Strategy 4 Strategy 4 vs the Growth Optimal Portfolio

Introduction Diversification Strategies Tail Dependence Numerical Example Conclusions

Proof & Other Types of Dependence Proof: We make use of our results (JAP, 2012) extending Rachev and R¨ uschendorf (1998) and Tankov (JAP, 2011) to derive improved Fr´ echet bounds on copulas when there are constraints on a rectangle. P(S⋆

T s, XT x) = C(G(s), F(x))

where G is the marginal cdf of S⋆

T, F is the marginal cdf of XT

and C is a copula. Optimal strategies can be derived explicitly: ◮ Independence in the tail (C(u, v) = uv): ∀s qα, x ∈ R, P(S⋆

T s, XT x) = P(S⋆ T s)P(XT x)

◮ Gaussian copula in the tail with correlation -0.5. ◮ Similarly for Clayton or Frank dependence.

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Proof & Other Types of Dependence Proof: We make use of our results (JAP, 2012) extending Rachev and R¨ uschendorf (1998) and Tankov (JAP, 2011) to derive improved Fr´ echet bounds on copulas when there are constraints on a rectangle. P(S⋆

T s, XT x) = C(G(s), F(x))

where G is the marginal cdf of S⋆

T, F is the marginal cdf of XT

and C is a copula. Optimal strategies can be derived explicitly: ◮ Independence in the tail (C(u, v) = uv): ∀s qα, x ∈ R, P(S⋆

T s, XT x) = P(S⋆ T s)P(XT x)

◮ Gaussian copula in the tail with correlation -0.5. ◮ Similarly for Clayton or Frank dependence.

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Some numerical results We define two events related to the market, i.e. the market crisis C = {S⋆

T < qα}. Define A =

• XT < x0erT

. T Cost Sharpe P(A|C) GOP 5 100 0.266 1.00 Buy-and-Hold 5 100 0.239 0.9998 Independence 5 101.67 0.214 0.46 Gaussian 5 103.40 0.159 0.12

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Some Implications ◮ Trade-off between losing “utility” and gaining protection during a “crisis”: the new strategies do not incur their biggest losses in the worst states in the economy. ◮ This can be used to reduce systemic risk.

• the idea of assessing risk and performance of a portfolio not
• nly by looking at its final distribution but also by looking at

its interaction with the economic conditions is indeed related to the increasing concern to evaluate systemic risk.

• Acharya (2009) explains that regulators should “be regulating

each bank as a function of both its joint (correlated) risk with

• ther banks as well as its individual (bank-specific) risk”.
• An insight of this work is that if all institutional investors

implement strategies that are resilient against crisis regimes, as we propose, then systemic risk can be diminished.

Do not hesitate to contact me to get updated working papers!

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References

◮ Bernard, C., Boyle P., Vanduffel S., 2011, “Explicit Representation of Cost-efficient Strategies”, available

• n SSRN.

◮ Bernard, C., Jiang, X., Vanduffel, S., 2012. “Note on Improved Fr´ echet bounds and model-free pricing of multi-asset options”, Journal of Applied Probability. ◮ Bernard, C., Maj, M., Vanduffel, S., 2011. “Improving the Design of Financial Products in a Multidimensional Black-Scholes Market,”, North American Actuarial Journal. ◮ Bernard, C., Vanduffel, S., 2011. “Optimal Investment under Probability Constraints,” AfMath Proceedings. ◮ Bernard, C., Vanduffel, S., 2012. “Financial Bounds for Insurance Prices,”Journal of Risk Insurance. ◮ Cox, J.C., Leland, H., 1982. “On Dynamic Investment Strategies,” Proceedings of the seminar on the Analysis of Security Prices,(published in 2000 in JEDC). ◮ Dybvig, P., 1988a. “Distributional Analysis of Portfolio Choice,” Journal of Business. ◮ Dybvig, P., 1988b. “Inefficient Dynamic Portfolio Strategies or How to Throw Away a Million Dollars in the Stock Market,” Review of Financial Studies. ◮ Goldstein, D.G., Johnson, E.J., Sharpe, W.F., 2008. “Choosing Outcomes versus Choosing Products: Consumer-focused Retirement Investment Advice,” Journal of Consumer Research. ◮ Jin, H., Zhou, X.Y., 2008. “Behavioral Portfolio Selection in Continuous Time,” Mathematical Finance. ◮ Nelsen, R., 2006. “An Introduction to Copulas”, Second edition, Springer. ◮ Pelsser, A., Vorst, T., 1996. “Transaction Costs and Efficiency of Portfolio Strategies,” European Journal

• f Operational Research.

◮ Platen, E., 2005. “A benchmark approach to quantitative finance,” Springer finance. ◮ Tankov, P., 2011. “Improved Fr´ echet bounds and model-free pricing of multi-asset options,” Journal of Applied Probability, forthcoming. ◮ Vanduffel, S., Chernih, A., Maj, M., Schoutens, W. 2009. “On the Suboptimality of Path-dependent Pay-offs in L´ evy markets”, Applied Mathematical Finance.

∼∼∼

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