# Tail Risk Constraints and Maximum Entropy

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Left Tail Risk as the Central Portfolio Constraint

#### 1.1. The Barbell as seen by E.T. Jaynes

“It may be that a macroeconomic system does not move in response to (or at least not solely in response to) the forces that are supposed to exist in current theories; it may simply move in the direction of increasing entropy as constrained by the conservation laws imposed by Nature and Government.”

## 2. Revisiting the Mean Variance Setting

**Normal World:**The joint distribution $g(\overrightarrow{x})$ of asset returns is multivariate Gaussian $N(\overrightarrow{\mu},\sum )$. Assuming normality is equivalent to assuming $g(\overrightarrow{x})$ has maximum (Shannon) entropy among all multivariate distributions with the given first- and second-order statistics $\overrightarrow{\mu}$ and Σ. Moreover, for a fixed mean $\mathbb{E}(X)$, minimizing the variance V (X) is equivalent to minimizing the entropy (uncertainty) of X. (This is true since joint normality implies that X is univariate normal for any choice of weights and the entropy of a N(μ, σ^{2}) variable is $H=\frac{1}{2}(1+\mathrm{log}(2\pi {\sigma}^{2}))$.) This is natural in a world with complete information. (The idea of entropy as mean uncertainty is in Philippatos and Wilson (1972) [18]; see Zhou et al. (2013) [19] for a review of entropy in financial economics and Georgescu-Roegen (1971) [20] for economics in general.)**Unknown Multivariate Distribution:**Since we assume we can estimate the second-order structure, we can still carry out the Markowitz program, i.e., choose the portfolio weights to find an optimal mean-variance performance, which determines $\mathbb{E}(X)=\mu $ and V (X) = σ^{2}. However, we do not know the distribution of the return X. Observe that assuming X is normal N(μ, σ^{2}) is equivalent to assuming the entropy of X is maximized since, again, the normal maximizes entropy at a given mean and variance, see [18].

^{2}by two left-tail value-at-risk constraints and to model the portfolio return as the maximum entropy extension of these constraints together with a constraint on the overall performance or on the growth of the portfolio in the non-danger zone.

#### 2.1. Analyzing the Constraints

_{−}< K, the value-at-risk constraints are:

**Tail probability:**$$\mathbb{P}(X\le K)={\displaystyle {\int}_{-\infty}^{K}f(x)\mathrm{d}x=\u03f5.}$$**Expected shortfall (CVaR):**$$\mathbb{E}(X|X\le K)={v}_{-}.$$

_{−}), let Ω

_{var}(θ) denote the set of probability densities f satisfying the two constraints. Notice that Ω

_{var}(θ) is convex: f

_{1}, f

_{2}∈ Ω

_{var}(θ) implies αf

_{1}+(1−α)f

_{2}∈ Ω

_{var}(θ). Later we will add another constraint involving the overall mean.

## 3. Revisiting the Gaussian Case

^{2}. In principle it should be possible to satisfy the VaR constraints since we have two free parameters. Indeed, as shown below, the left-tail constraints determine the mean and variance; see Figure 1. However, satisfying the VaR constraints imposes interesting restrictions on μ and σ and leads to a natural inequality of a “no free lunch” style.

^{−}

^{1}(ϵ), where Φ is the c.d.f. of the standard normal density ϕ (x). In addition, set

**Proposition 1.**If X ∼ N(μ, σ

^{2}) and satisfies the two VaR constraints, then the mean and variance are given by:

_{ϵ↓}

_{0}B(ϵ) = −1.

^{−1}is the inverse of the regularized incomplete beta function I, and s the solution of $\u03f5=\frac{1}{2}{I}_{\frac{\alpha {s}^{2}}{{(k-m)}^{2}+\alpha {s}^{2}}}\left(\frac{\alpha}{2},\frac{1}{2}\right)$.)

#### 3.1. A Mixture of Two Normals

_{1}= ν

_{−}, in which case μ

_{2}is constrained to be $\frac{\mu -\u03f5{v}_{-}}{1-\u03f5}$. It then follows that the left-tail constraints are approximately satisfied for σ

_{1}, σ

_{2}sufficiently small. Indeed, when σ

_{1}= σ

_{2}≈ 0, the density is effectively composed of two spikes (small variance normals) with the left one centered at ν

_{−}and the right one centered at at $\frac{\mu -\u03f5{v}_{-}}{1-\u03f5}$. The extreme case is a Dirac function on the left, as we see next.

#### 3.1.1. Dynamic Stop Loss, A Brief Comment

## 4. Maximum Entropy

_{j}(X) = c

_{j}, j = 1, …, J. Assuming Ω is non-empty, it is well-known that f

_{MEE}is unique and (away from the boundary of feasibility) is an exponential distribution in the constraint functions, i.e., is of the form

_{1}, …, λ

_{M}) is the normalizing constant. (This form comes from differentiating an appropriate functional J(f) based on entropy, and forcing the integral to be unity and imposing the constraints with Lagrange mult1ipliers.) In the special cases below we use this characterization to find the MEE for our constraints.

_{−}) together with those parameters associated with the additional constraints.

#### 4.1. Case A: Constraining the Global Mean

_{+}satisfies ϵν

_{−}+ (1 − ϵ)ν

_{+}= μ.

_{−}and f

_{+}integrate to one. Then

- ${\int}_{-\infty}^{K}{f}_{MEE}(x)\mathrm{d}x=\u03f5};$
- ${\int}_{-\infty}^{K}x\phantom{\rule{0.2em}{0ex}}{f}_{MEE}(x)\mathrm{d}x}=\in {v}_{-$
- $\int}_{K}^{\infty}x\phantom{\rule{0.2em}{0ex}}{f}_{MEE}(x)\mathrm{d}x=(1-\u03f5){v}_{+$.

_{MEE}has an exponential form in our constraint functions:

#### 4.2. Case B: Constraining the Absolute Mean

_{−}(x) as above, and let

_{1}can be chosen such that

#### 4.3. Case C: Power Laws for the Right Tail

_{+}(x) will have a power law, namely

#### 4.4. Extension to a Multi-Period Setting: A Comment

^{A}(t)

^{n}converges to that of an n-summed Gaussian. Further, the characteristic function of the limit of the average of strategies, namely

_{+}+ ϵ (ν

_{−}− ν

_{+}).

## 5. Comments and Conclusion

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

**Proof of Proposition 1:**Since X ∼ N(μ, σ

^{2}), the tail probability constraint is

^{−}

^{1}(ϵ) yields ϵ = P (X < η(ϵ)) ≤ − ϵB(ϵ) or 1 + B(ϵ) ≤ 0. Since the upper tail inequality is asymptotically exact as x → ∞ we have B(0) = −1, which concludes the proof.

## References

- Chicheportiche, R.; Bouchaud, J.-P. The joint distribution of stock returns is not elliptical. Int. J. Theor. Appl. Financ.
**2012**, 15. [Google Scholar] [CrossRef] - Markowitz, H. Portfolio selection. J. Financ.
**1952**, 7, 77–91. [Google Scholar] - Kelly, J.L. A new interpretation of information rate. IRE Trans. Inf. Theory
**1956**, 2, 185–189. [Google Scholar] - Bell, R.M.; Cover, T.M. Competitive optimality of logarithmic investment. Math. Oper. Res.
**1980**, 5, 161–166. [Google Scholar] - Thorp, E.O. Optimal gambling systems for favorable games. Revue de l’Institut International de Statistique
**1969**, 37, 273–293. [Google Scholar] - Haigh, J. The kelly criterion and bet comparisons in spread betting. J. R. Stat. Soc. D
**2000**, 49, 531–539. [Google Scholar] - MacLean, L.; Ziemba, W.T.; Blazenko, G. Growth versus security in dynamic investment analysis. Manag. Sci.
**1992**, 38, 1562–1585. [Google Scholar] - Thorp, E.O. Understanding the kelly criterion. In The Kelly Capital Growth Investment Criterion: Theory and Practice; MacLean, L.C., Thorp, E.O., Ziemba, W.T., Eds.; World Scientific Press: Singapore, Singapore, 2010. [Google Scholar]
- Xu, Y.; Wu, Z.; Jiang, L.; Song, X. A maximum entropy method for a robust portfolio problem. Entropy
**2014**, 16, 3401–3415. [Google Scholar] - Tsallis, C.; Anteneodo, C.; Borland, L.; Osorio, R. Nonextensive statistical mechanics and economics. Physica A
**2003**, 324, 89–100. [Google Scholar] - Jizba, P.; Kleinert, H.; Shefaat, M. Rényi’s information transfer between financial time series. Physica A
**2012**, 391, 2971–2989. [Google Scholar] - Hicks, J. R. Value and CapitalClarendon Press: Oxford, U.K, 1939; Volume 1946, 2nd ed. [Google Scholar]
- Tobin, J. Liquidity preference as behavior towards risk. Rev. Econ. Stud.
**1958**, 25, 65–86. [Google Scholar] - Markowitz, H.M. Portfolio Selection: Efficient Diversification of Investments; Wiley: New York, NY, USA, 1959. [Google Scholar]
- Merton, R.C. An analytic derivation of the efficient portfolio frontier. J. Financ. Quant. Anal.
**1972**, 7, 1851–1872. [Google Scholar] - Ross, S.A. Mutual fund separation in financial theory—the separating distributions. J. Econ. Theory
**1978**, 17, 254–286. [Google Scholar] - Jaynes, E.T. How Should We Use Entropy in Economics; University of Cambridge: Cambridge, UK, 1991. [Google Scholar]
- Philippatos, G.C.; Wilson, C.J. Entropy, market risk, and the selection of efficient portfolios. Appl. Econ.
**1972**, 4, 209–220. [Google Scholar] - Zhou, R.; Cai, R.; Tong, G. Applications of entropy in finance: A review. Entropy
**2013**, 15, 4909–4931. [Google Scholar] - Georgescu-Roegen, N. The Entropy Law and the Economic Process; Harvard University Press: Cambridge, MA, USA, 1971. [Google Scholar]
- Richardson, M.; Smith, T. A direct test of the mixture of distributions hypothesis: Measuring the daily flow of information. J. Financ. Quant. Anal.
**1994**, 29, 101–116. [Google Scholar] - Ane, T.; Geman, H. Order flow, transaction clock, and normality of asset returns. J. Financ.
**2000**, 55, 2259–2284. [Google Scholar] - Taleb, N.N. Dynamic Hedging: Managing Vanilla and Exotic Options; Wiley: New York, NY, USA, 1997. [Google Scholar]
- Brigo, D.; Mercurio, F. Lognormal-mixture dynamics and calibration to market volatility smiles. Int. J. Theor. Appl. Financ.
**2002**, 5, 427–446. [Google Scholar] - Frittelli, M. The minimal entropy martingale measure and the valuation problem in incomplete markets. Math. Financ.
**2000**, 10, 39–52. [Google Scholar]

**Figure 1.**By setting K (the value at risk), the probability ϵ of exceeding it, and the shortfall when doing so, there is no wiggle room left under a Gaussian distribution: σ and μ are determined, which makes construction according to portfolio theory less relevant.

**Figure 2.**Dynamic stop loss acts as an absorbing barrier, with a Dirac function at the executed stop.

**Figure 5.**Case C: Effect of different values of on the shape of the fat-tailed maximum entropy distribution.

**Figure 6.**Case C: Effect of different values of on the shape of the fat-tailed maximum entropy distribution (closer K).

**Figure 7.**Average return for multiperiod naive strategy for Case A, that is, assuming independence of “sizing”, as position size does not depend on past performance. They aggregate nicely to a standard Gaussian, and (as shown in Equation (1)), shrink to a Dirac at the mean value.

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Geman, D.; Geman, H.; Taleb, N.N.
Tail Risk Constraints and Maximum Entropy. *Entropy* **2015**, *17*, 3724-3737.
https://doi.org/10.3390/e17063724

**AMA Style**

Geman D, Geman H, Taleb NN.
Tail Risk Constraints and Maximum Entropy. *Entropy*. 2015; 17(6):3724-3737.
https://doi.org/10.3390/e17063724

**Chicago/Turabian Style**

Geman, Donald, Hélyette Geman, and Nassim Nicholas Taleb.
2015. "Tail Risk Constraints and Maximum Entropy" *Entropy* 17, no. 6: 3724-3737.
https://doi.org/10.3390/e17063724