Risk, Coherency and Cooperative Game

Risk, Coherency and Cooperative Game Haijun Li lih@math.wsu.edu Department of Mathematics Washington State University Tokyo, June 2015 Haijun Li Risk, Coherency and Cooperative Game Tokyo, June 2015 1 / 30

Outline 1 Risk Assessment Risk = Volatility? Modern Portfolio Theory (MPT) Risk = Volatility + Loss 2 Coherent Risks: An Axiomatic Approach Coherent Risk and Its Dual Representation Expected Shortfalls Relation Between CVaR and VaR 3 Relation with Cooperative Game Convex Game Distortion 4 Concluding Remarks Haijun Li Risk, Coherency and Cooperative Game Tokyo, June 2015 2 / 30

Risk Factors and Univariate Risk measures Fix a probability space (Ω, F, P) generated by a stochastic system (e.g., financial portfolio). Let L denote a set of random risk factors defined on (Ω, F, P); e.g., profit/loss variables or returns X in a financial portfolio. Assume that L is a convex cone. A univariate risk measure ϱ : L R is a functional satisfying some operational properties. Multivariate (or set-valued) risk measures have also been studied in the literature. Haijun Li Risk, Coherency and Cooperative Game Tokyo, June 2015 3 / 30

Although risk measures are closely related to survivability measures in reliability modeling in engineering and survival analysis used in medical fields, we will stick with the context of financial applications (due to widely available public data in finance). Why should we care about good risk measures? 1 Risk assessment ϱ(x) is used to set up capital reserves at banks (Basel II International Banking Accords, 2004). 2 Risk assessment is widely used in capital allocations in diversification of risky investment. 3 Risk measure ϱ can be used to price financial derivatives and calculate insurance premiums. Haijun Li Risk, Coherency and Cooperative Game Tokyo, June 2015 4 / 30

Risk = Volatility? One method for risk assessment is to use some volatility measures, such as the variance (and co-variance). Definition The variance of a risk factor X on (Ω, F, P) is defined as Var(X) := (X(ω) µ) 2 P(dω) = X µ L2 = E(X µ) 2, Ω where µ = X(ω)P(dω) denotes the mean value of X. The risk measure Var(X) is appropriate for risk factors that have Gaussian (normal) distributions, because a Gaussian distribution is completely determined by its mean and variance. The standard deviation σ = Var(X) is also used as a risk measure. Haijun Li Risk, Coherency and Cooperative Game Tokyo, June 2015 5 / 30

Modern Portfolio Theory (MPT) Harry Markowitz s MPT (developed in 1950-1970, received the Nobel Memorial Prize in 1990) is a mathematical formulation of diversification in investing. The basic ingredients are as follows. Investors are assumed to be rational and financial markets are assumed to be efficient. Asset s returns are assumed to be jointly normally distributed, and so the portfolio return has a Gaussian distribution. Risk is measured by the standard deviation σ of returns. MPT aims to minimize the variance of the portfolio for a given level of expected return, by carefully choosing the proportions (weights) of its assets. Investing is a tradeoff between risk and expected return. MPT explains how to find the best possible diversification strategy for specific definitions of risk and return. Haijun Li Risk, Coherency and Cooperative Game Tokyo, June 2015 6 / 30

Modern Portfolio Theory (MPT) Mean-Variance portfolio_fig1_wl.jpg (JPEG Image, Efficient 750x643 pixels) Frontier http://www.mathworks.com/company/newsletters/news_notes/oct06/imag... (mathworks.com) Haijun Li Risk, Coherency and Cooperative Game Tokyo, June 2015 7 / 30

Modern Portfolio Theory (MPT) Does MPT really model financial markets? The main criticisms: Market participants are not rational (e.g., herding), and markets are not efficient (such as information asymmetry and statistical arbitrage). The assumptions of MPT are strongly challenged by behavioral economists (e.g., Robert Shiller). Asset returns are not (jointly) normally distributed. The variance (or standard deviation) is a scale parameter and it cannot be used to measure extreme risk at the tails of loss distributions. The L 2 -norm based risk measures, such as the variance, overlook clustering of extreme values that frequently appears in financial data. Haijun Li Risk, Coherency and Cooperative Game Tokyo, June 2015 8 / 30

Modern Portfolio Theory (MPT) Financial Log-Return Series (X n, n 1) Stylized Facts Extreme returns appear in clusters. Returns are heavy-tailed (i.e., P(X n > x) x β as x ). Return series are not iid although they show little temporal correlation. Series of absolute or squared returns show profound temporal correlation (higher order long-range dependence). These facts seem to apply to the majority of risk-factor changes, such as log-returns on equities, indexes, exchange rates, and commodity prices. These facts hold in multiple time scales: intra-daily, daily, weekly and monthly returns. Haijun Li Risk, Coherency and Cooperative Game Tokyo, June 2015 9 / 30

Modern Portfolio Theory (MPT) Returns are Heavy-Tailed Histogram of Monthly U.S. SP500 Financia Exploratory QQ Plot Density 0.00 0.02 0.04 0.06 0.08 Exponential Quantiles 0 1 2 3 4 5 6 xi = 0 30 20 10 0 10 20 mr[, 30] 10 0 10 20 Ordered Data Haijun Li Risk, Coherency and Cooperative Game Tokyo, June 2015 10 / 30 16 4 Threshold Figure: S&P 500 Financials 1990-2010 (Erin Lu, WSU Department of Mean Excess Plot Finance) 6.6100 3.7500 1.7800 0.0118

Modern Portfolio Theory (MPT) Autocorrelations of U.K. FTSE 100: 1998-2007 showimage.php (PNG Image, 500x500 pixels) http://fedc.wiwi.hu-berlin.de/quantnet/graphics/showimage.php?i=ed360 Figure: Left = returns, Middle = returns 2, Right = returns Haijun Li Risk, Coherency and Cooperative Game Tokyo, June 2015 11 / 30

Modern Portfolio Theory (MPT) (JPEG Image, 480x480 pixels) https://mail.google.com/mail/?ui=2&ik=2a21981ea2&view=att&th Government National Mortgage Association Data Figure: Ginnie Mae MBS, 1990-2010 (Erin Lu, WSU Dept of Finance) Haijun Li Risk, Coherency and Cooperative Game Tokyo, June 2015 12 / 30

Risk = Volatility + Loss Value-at-Risk (VaR) of a Loss Variable X The VaR with confidence level 1 α (0 α 1) is defined as VaR α (X) := sup{x : P(X > x) α}. For example, VaR α (X) = σφ 1 (1 α) + µ for a normally distributed loss X with mean µ and standard deviation σ, where var.jpg (JPEG Image, 468x394 pixels) Φ is the CDF of the standard Gaussian. http://www.ima.umn.edu/2009-2010/mm8.2-11.10/var.jpg Haijun Li Risk, Coherency and Cooperative Game Tokyo, June 2015 13 / 30

Risk = Volatility + Loss VaR is a widely used risk measure of the risk of loss. VaR was developed by a division at J. P. Morgan in 1994, which was later spun off into an independent for-profit business, called RiskMetrics Group. The Basel II recommends VaR as the preferred measure of market risk. The Actuarial exams test regularly VaRs of various loss distributions. VaR enjoys some nice operational properties: 1 X 1, X 2 L such that X 1 X 2 almost surely, VaR α (X 1 ) VaR α (X 2 ). 2 X L, λ > 0, VaR α (λx) = λvar α (X). 3 X L, l R, VaR α (X + l) = VaR α (X) + l. Haijun Li Risk, Coherency and Cooperative Game Tokyo, June 2015 14 / 30

Risk = Volatility + Loss Drawbacks of VaR VaR focuses on the manageable risks near the center of profit/loss distributions and underestimate tail risk, leading to excessive risk-taking and over-leverage at financial institutions. VaR is not subadditive; that is, VaR α (X 1 + X 2 ) > VaR α (X 1 ) + VaR α (X 2 ), X 1, X 2 L. That is, the diversification can lead to increase of risk, if risk is measured by VaR. VaR is subadditve within the class of loss variables that have elliptical distributions (including Gaussian, generalized hyperbolic distributions,...). Haijun Li Risk, Coherency and Cooperative Game Tokyo, June 2015 15 / 30

Coherent Risk and Its Dual Representation Univariate Coherent Risk Measures Let L be the convex cone consisting of all the variables X which may represent losses of portfolios at the end of a given period. Artzner, Delbaen, Eber and Heath (1999) A functional ϱ : L R is called a coherent risk measure if ϱ satisfies the four coherent axioms: (monotonicity) For X 1, X 2 L with X 1 X 2 almost surely, ϱ(x 1 ) ϱ(x 2 ). (subadditivity) For all X 1, X 2 L, ϱ(x 1 + X 2 ) ϱ(x 1 ) + ϱ(x 2 ). (positive homogeneity) For all X L and every λ > 0, ϱ(λx) = λϱ(x). (translation invariance) For all X L and every l R, ϱ(x + l) = ϱ(x) + l. Haijun Li Risk, Coherency and Cooperative Game Tokyo, June 2015 16 / 30

Coherent Risk and Its Dual Representation Dual Representation (Delbaen, 2000) Under some regularity conditions (Fatou property,...), a coherent risk measure ϱ(x) arises as the supremum of expected values of loss X under various scenarios: ϱ(x) = sup Q S E Q (X) where S is a convex set of probability measures on physical states, that are absolutely continuous with respect to the underlying measure P. Q P (absolute continuity): P(A) = 0 implies that Q(A) = 0, A F. Q P implies that Q(A) = A f (ω)p(dω), where f = dq/dp is known as the Radon-Nikodym derivative. Haijun Li Risk, Coherency and Cooperative Game Tokyo, June 2015 17 / 30

Coherent Risk and Its Dual Representation Interpretation in Finance The measure P describes what random events might occur. Any measure Q S describes likelihoods of a generalized scenario in the future that takes some uncertainty (e.g., interest rate hike,...) into account. A scenario measure Q, which can be just finitely additive, is also called a distortion measure. Any coherent risk measure is the worst expected loss under a collection of generalized scenarios that could happen in the real world. The SPAN margin system (Chicago Mercantile Exchange, 1995) considered 16 scenarios (14 regular scenarios + 2 extreme scenarios ) to measure risk for margin requirements. Haijun Li Risk, Coherency and Cooperative Game Tokyo, June 2015 18 / 30

Coherent Risk and Its Dual Representation Another Interpretation... Let C := {X : ϱ(x) 0} is clearly a convex cone. C is a set of positions with acceptable risk.... From Regulator/Supervisor s Point of View For any L -closed convex cone C L (Ω, F, P) such that L (Ω, F, P) C, is a coherent risk measure. ϱ(x) = inf{l : l X C} The risk ϱ(x) for loss X corresponds to the amount of extra capital requirement that has to be invested in some secure instrument so that the resulting position ϱ(x) X is acceptable to regulator/supervisor. Haijun Li Risk, Coherency and Cooperative Game Tokyo, June 2015 19 / 30

Coherent Risk and Its Dual Representation A general theory of coherent risk measures for arbitrary, univariate loss variables was developed in Delbaen (2000). Set-valued coherent risk was studied in Jouini, Meddeb and Touzi (2004). The asymptotic properties of set-valued coherent risk were studied in Joe and Li (2010) using the intensity measures of multivariate extremes. One can also consider a wider class of the convex risk measures that combine sub-additivity and positive homogeneity into the convexity property. The convex risk measures was extended to stochastic processes in Cheridito, Delbaen and Klüppelberg (2004), and to a general space that may include deterministic, stochastic, single or multi-period cash-stream structures (Föllmer and Schield, 2002). Haijun Li Risk, Coherency and Cooperative Game Tokyo, June 2015 20 / 30

Expected Shortfalls Worst Conditional Expectation Consider the following class of scenario probability measures: S α := {P( A) : P(A) > α, A F} = {Q : dq/dp L 1/α}, 0 < α < 1. The corresponding coherent risk is called the worst conditional expectation and given by, WCE α (X) = sup{e(x A) : P(A) > α, A F}. Since α is usually small, S α contains all the scenario probability measures conditioning on events with occurring probability at least α, including rare events. WCE α (X) is the worst expected loss which could be incurred from various random events with occurring probability at least α. Haijun Li Risk, Coherency and Cooperative Game Tokyo, June 2015 21 / 30

Expected Shortfalls Conditional VaR = Tail Conditional Expectation If the profit/loss variable X is continuous, then WCE α (X) = E(X X > VaR α (X)) =: CVaR α (X) which is called the tail conditional expectation (or conditional VaR or expected shortfall) with confidence level 1 α. CVaR has been adopted by the finance and insurance regulators both in Canada and US. Let t = VaR α (X), then 0 E(X t X > t) = CVaR α (X) VaR α (X) is known as the mean residual life widely used in reliability modeling and in survival analysis long before coherent risk was introduced in 1999. Haijun Li Risk, Coherency and Cooperative Game Tokyo, June 2015 22 / 30

Relation Between CVaR and VaR CVaR α (X) = 1 α α 0 VaR ξ(x)dξ. For any light-tailed loss variable X (i.e., its distribution tails decay exponentially), CVaR α (X) VaR α (X), as α is small. For a heavy-tailed loss variable X (i.e., P(X > x) x β as x ), if tail index β > 1, then CVaR α (X) β β 1 VaR α(x), when α is small. Haijun Li Risk, Coherency and Cooperative Game Tokyo, June 2015 23 / 30

Relation Between CVaR and VaR CVaR α (X) = 1 α α 0 VaR ξ(x)dξ. For any light-tailed loss variable X (i.e., its distribution tails decay exponentially), CVaR α (X) VaR α (X), as α is small. For a heavy-tailed loss variable X (i.e., P(X > x) x β as x ), if tail index β > 1, then CVaR α (X) β β 1 VaR α(x), when α is small. var.jpg (JPEG Image, 468x394 pixels) http://www.ima.umn.edu/2009-2010/mm8.2-11.10/var.jpg Haijun Li Risk, Coherency and Cooperative Game Tokyo, June 2015 23 / 30

Relation Between CVaR and VaR CVaR is Better, But Is CVaR the Best One? Question: VaR is not coherent, but can we find the smallest coherent risk measure, say ϱ, that dominates VaR (i.e., ϱ(x) VaR α (X), X L)? Theorem (Delbaen, 2000) Assume that some regularity conditions (such as the Fatou property) hold, and 0 < α < 1 and X L are fixed. 1 VaR α (X) = inf{ϱ(x) : ϱ is coherent with ϱ VaR α }. 2 If X is continuous and ϱ is coherent, then ϱ VaR α implies that ϱ CVaR α. Answer: Yes, CVaR is the best one in the sense that (1) it is coherent and (2) it dominates VaR. Haijun Li Risk, Coherency and Cooperative Game Tokyo, June 2015 24 / 30

Convex Game Consider a set function ν : F R + with ν( ) = 0. Intuitively, ν(a) = payoff gained by forming the coalition A F. Convex Game (Shapley, 1971) A set function ν is called a convex game defined on (Ω, F, P) if 1 (supermodular) ν(a B)+ν(A B) ν(a)+ν(b), A, B F. 2 (absolutely continuous) for coalitions A, B F with P(A = B) = 1, ν(a) = ν(b). The supermodularity is equivalent to ν(a {ω}) ν(a) ν(b {ω}) ν(b), A B, ω Ω. That is, one should join a larger coalition because the payoff is larger. The absolute continuity is equivalent to that P(A) = 0 implies that ν(a) = 0. Haijun Li Risk, Coherency and Cooperative Game Tokyo, June 2015 25 / 30

Convex Game The Core Define the core of a convex game ν: C(ν) := {µ : µ is a finitely additive, non-negative measure, µ(ω) = ν(ω), µ(a) ν(a), A F}. Intuitively, the core contains all the profit distribution rules that make every member in the game happy (no incentive to leave the grand coalition). The core of a convex game is non-empty. The Shapley value (the unique fair profit distribution rule) of a convex game is the center of gravity of its core. P(A) = 0 implies that µ(a) = 0, µ C(ν) (i.e., µ P). Haijun Li Risk, Coherency and Cooperative Game Tokyo, June 2015 26 / 30

Distortion For simplicity, consider only bounded, non-negative loss variables X defined on (Ω, F, P). Define ϱ(x) := sup E µ (X). µ C(ν) Rewrite ϱ(x) in terms of Choquet integrals, and one has ϱ(x) = 0 ν({ω : X(ω) > x})dx. If ν(ω) = 1, then ϱ(x) is a coherent risk measure. Example: Let f : [0, 1] [0, 1] be increasing and concave such that f (0) = 0 and f (1) = 1. The the set function ν(a) := f (P(A)), A F, defines a convex game. The coherent risk measure: ϱ(x) = 0 f (P(X > x))dx is known as the distortion risk measure in actuarial science (Denneberg, 1989). Haijun Li Risk, Coherency and Cooperative Game Tokyo, June 2015 27 / 30

Distortion A coherent risk measure ϱ is said to be comonotone if ϱ(x + Y) = ϱ(x) + ϱ(y) for all loss variables of the following form: X = g(z), Y = h(z) where g, h : R R + are increasing. Coherent Risks Arising from Convex Games A coherent risk measure ϱ arises from a convex game ϱ(x) = if and only if ϱ is comonotone. sup E µ (X) µ C(ν) CVaR is comonotone, and thus can be (easily) written as a distortion risk. Haijun Li Risk, Coherency and Cooperative Game Tokyo, June 2015 28 / 30

Distortion Coherent Risk Measures Coherent Risk Measures Arising from Convex Games Distortion Risk measures Haijun Li Risk, Coherency and Cooperative Game Tokyo, June 2015 29 / 30

Concluding Remarks The theory of univariate coherent risk measures is fairly complete. But multivariate (or set-valued) coherent risk measures are still an open field. For example: 1 How to establish the asymptotic expressions of coherent risks for extreme events? How to utilize limiting homogeneous structures to estimate these expressions based on observed extreme values, which are usually rare? 2 How would the dependence of loss variables among different portfolios affect systemic risk assessment, and in particular, the systemic risk arising from a cooperative game? Haijun Li Risk, Coherency and Cooperative Game Tokyo, June 2015 30 / 30