Estimating Value at Risk of Portfolio: Skewed-EWMA Forecasting via Copula
Zudi LU
Dept of Maths & Stats Curtin University of Technology
(coauthor: Shi LI, PICC Asset Management Co.) Talk outline
¾ Why important? -- Background on financial risk modeling. EWMA: exponentially weighted moving average ¾ Why difficult? -- stylized facts on financial return series ¾ How to solve? -- Asymmetric Laplace distribution -- Skewed-EWMA and forecasting of VaR via copula -- Evaluation of VaR forecasting ¾ Outlook I. Why important? Background on financial risk modeling Financial practice / globalization poses new challenging questions on financial risks, e.g., modeling / pricing of ¾ financial risks (market risk, credit risk, etc) and hedging derivatives Financial innovation requires and produces new financial instruments: e.g., modeling/pricing of securitization of insurance risks ¾ weather risk -- weather derivatives, ¾ catastrophe risk -- catastrophe insurance derivatives Financial market risk
Risk management has truly experienced a revolution in the last few years. This was started by value at risk (VaR), a new method to measure financial market risk that was developed in response to the financial disasters of the early 1990s.
--Philippe Jorion (2001, preface): Value at risk: The new benchmark for managing financial risk Examples of financial disasters
¾ Barrings, UK, 02/1995, loss $1.33 billion ¾ Metallgesellschaft, Germany, 01/1994, loss $1.34 billion ¾ Orange county, USA, 12/1994, loss $1.81 billion ¾ Daiwa, Japan, 09/1995, loss $1.1 billion ¾ Asia’s 1997 financial market turmoil Importance of VaR modeling
¾ Group of Thirty (G-30) in 1993 advised to value positions using market prices and to assess financial risks with VaR. ¾ Benchmark risk measure for minimum reserve capital, recommended by Basel Accord, the U.S. federal reserve bank, and EU’s Capital Adequacy Directives, etc. ¾ The greatest advantage: summarises in a single, easy to understand number the downside risk of an institution due to financial market factors. ¾ Information reporting (passive), controlling risk (defensive), managing risk (active) II. Why difficult? VaR modeling VaR: expected maximum loss (or worst loss) of a financial variable / portfolio over a target horizon at a given confidence level, α. e.g., α=95% ¾ In terms of return series of the target horizon , with a distribution F(x), then the Loss prob.5% worst return at the given confidence level, α, is: VaR=F-1(1-α). Normal distribution ¾ Black-Scholes model:
dStt= S α dt + Stσ dBt -- standard Brownian motion, . Bt σ > 0 ¾ Gemoetric retern: 2 2 1(x − µ) rSStt=−log log t−1 ∼ N(µσ, ) : f(x) = exp−2 2πσ 2σ Where µα =− σ 2 /2 . ¾ VaR: VaR=μ+σФ-1(1-α) , Ф(.) is c.d.f. of N(0,1). Practically, assumeμ=0. Stylized facts of financial return
¾ Return series show little autocorrelation, but not i.i.d. ¾ Conditional expected returns are closed to zero ¾ Absolute or squared return series show profound autocorrelation ¾ Volatility appears to vary over time ¾ Extreme returns appear in cluster ¾ In reality, distribution of return series is skewed and heavy-tailed & high-peaked, departure from normality.
-- c.f., Taylor (2005): Asset price dynamics, volatility, and prediction. Princeton University Press. Changing volatility: Standard-EWMA
JP Morgan RiskMetrics: Exponentially weighted moving average (EWMA) ¾ assume conditional normality for return series，with volatility modeled as IGARCH(1,1) of Engle & Bollersleve (1986), −1 2 2 2 VaRt+1 = σ t+1Φ (1−α) , σ t + 1 = λσ t + (1− λ)rt , 0 < λ <1 or equivalently, ∞ 1 T 2 i 2 ⇐=σˆ 22r σ t+1 = ∑λ (1− λ)rt−i ∑t=1 t i=0 T geometrically declining weights on past observations, assigning greater importance to recent observations. Nelson and Foster (1994): when returns are conditionally normal，EWMA is optimal. Changing volatility: Robust-EWMA
¾ Guermat and Harris (2000): based on Laplace distribution 12|x − µ | rL∼ D(,µσ2 ):f(x)=−exp t 2σ σ σ-standard deviation heavier-tailed than normality. ¾ Robust-EWMA: assumeμ=0, σ t+1 , VaRt+1 =−ln[2(1 α)] σ t+1 = λσt + (1−λ) 2 | rt | 2 ∞ 1 T ⇔=σλi (1 −λ) 2 | r | ⇐=σˆ 2 ||r tt+1 ∑ −i∑t=1 t i=0 T ¾ It accounts for heavy tails，but no skewness. ¾ Absolute return GARCH: Taylor (1986), Schwert (1989) Importance of skewness
¾ Documents of skewness of return series, e.g. Kraus and Litzenberger (1976), Friend and Westerfield (1980), Lim(1989), Richardson and Smith(1993), Harvey and Siddique(1999, 2000), Ait-Sahalia and Brandt (2001), Chen(2001) ¾ Simaan(1993): skewness in portfolio ¾ Theodossiou(1998): generalized t distribution， too complex Two challenging questions How to account for the stylized facts of skewness and heavy tails simultaneously, which may also change with time, in modeling and forecasting of VaR?
Changing skewness and kurtosis: Harvey, C. R. and A. Siddique (1999), “Autoregressive Conditional Skewness”, Journal of Financial and Quantitative Analysis 34, 465- 487.
How to account for the complex dependence among individual securities? III. How to solve? Our work: Skewed-EWMA via copula ¾A Skewed-EWMA VaR modeling, based on Asymmetric Laplace distribution taking into account both skewness and heavy tails in financial return series ¾A varying shape parameter by EWMA, leading to changing skewness and kurtosis, adaptive to time- varying nature of financial systems. ¾ Skewed-EWMA outperforms both Standard- and Robust- EWMAs in VaR forecasting. ¾ Complex dependence between individual securities, modelled via copula Motivation: how to estimate quantile
Check function: Koenker & Bassett (1978, Econometrica)
ρpx()xx=| |{pI[0≥<]+−(1 p)I[x0]},0 fxpp()= p(1−−p)exp{ρ (x)} n qXˆ ()=−arg max f(X θ ) ppθ ∏ i=1 i c.f., Yu, Lu & Stander (2003, JRSS, series D) Quantile regression: applications and current research areas Asymmetric Laplace Distribution ALD: density function kk 11 f (xp|σ,)=−exp I[0xx><]+I[0] ||x σσ1− pp 22 where kk==()p p+(1−p) ¾ σ-standard deviation，p -shape parameter in (0,1). Laplace distribution: p=0.5. Skewed EWMA and forecasting Skewed EWMA Volatility forecasting Lu & Huang (2007): p 1−α 2 2 t+1 , k = p +(1− p ) VaRt+1 = σ t+1 ln t+1 t+1 t+1 kt+1 pt+1 kk, k T 11 ⇐=σˆ I + Ir|| σλtt+>1[=+σ(1 −λ) IIr0]+[r<0]||rt ∑[0rrii<>] [0] i 1− pptt Tpi=1 1− p 1 p t + 1 = , 1 + u t + 1 / v t + 1 1 n ⇐=ur||I ut+1 = βut + (1− β ) | rt | I[r <0] ∑ ir[0i > ] t n i =1 1 n v = β v + (1 − β ) | r | I ⇐=vr||I t +1 t t [ rt > 0 ] ∑ ir[0i < ] n i =1 Joint distribution via copula Archimedean copulas • Definition : Let φ :[ 0 , 1 ] →∞ [ 0 , ] be a continuous, strictly decreasing, convex function such that φ (1 ) = 0 and let φ [1 − ] () t be its pseudo inverse. Then Cu(,v)=+φφ[1− ]((u) φ(v)) is an Archimedean copula. • Clayton copula function fits the dataset best 1/ −θ −−θθ− θ φ()tt=−1 Cu(,v)=+()u v −1 Evaluation of VaR forecasting 7 financial return series: 1 Jan. 1992 to 31 Dec. 2001 Name mean s.d. Skew kurtosis sample-size ¾ Confidence level for VaR: α=99% ¾ Likelihood criterion: First 500 observations (estimation sample) β=0.998，λ taking 15 values between 0.85 and 0.99. Backtesting methods: Kupiec(1995) and Christofferson (1998) likelihood ratio test: Let 1,if r < VaR T I = t t t N = ∑ It 0, otherwise , t= 1 , T – sample size ¾ Unconditional coverage test: τ =1−α Assume {It} independent， H0：E(It) = τH1：E(It)≠τ TN−−N TN T 2 LRu =−2ln[(1−ττ) ]+2ln[(1−NT/ ) (NT/ ) ]~ χ(1) ¾ Independence test: H0：{It} independent H1：{It}is a first order Markov process TT00 ++10 T01 T11 T00 T01 T10 T11 2 LRin =−2ln[(1−ππ) ]+2ln[π00 π01 π10 π11 ]~ χ(1) ¾Conditional coverage test: H0： E(It) = τ H1：{It} a first order Markov process 2 LRcu=+LR LRin~(χ 2) Unconditional Coverage •Y axis-failure rate of VaR (theoretical 1%) • X axis- value of λ •St-EWMA +; R-EWMA x; Sk-EWMA • • Sk-EWMA: about 1% ; St- and R-EWMA: >>1% Unconditional Coverage test •Y axis-Likelihood ratio value • X axis- value of λ •St-EWMA +; R-EWMA x; Sk-EWMA • • Sk-EWMA: all less than 1% critical value of 6.63 (pink); St- and R-EWMA: mostly larger than 1% critical value (pink). Independence test •Y axis-Likelihood ratio value • X axis- value of λ •St-EWMA +; R-EWMA x; Sk-EWMA • • 5% critical value 3.84 (blue) ; 1% critical value 6.63 (pink) Conditional coverage test • Y axis-Likelihood ratio value • X axis- value of λ •St-EWMA +; R-EWMA x; Sk-EWMA • • Sk-EWMA: less than 1% critical value of 9.21 (pink); St- and R-EWMA: mostly larger than 1% critical value (pink) Portfolio via copula • Portfolio: Shanghai composite index and gold yield (half-half) • Empirical test: Clayton copula function fits the dataset best −−θθ−1/θ Cu(,v)=+(u v −1) Portfolio via copula: Unconditional Coverage test •Y axis-Likelihood ratio value 16 Skewed-EWMA Forecasting via Copula RiskMetrics Forecasting Model • X axis- value of λ 14 12 • Sk-EWMA via 10 copula: LR is less 8 than 1% critical 6 value of 6.63 ; 4 • RiskMetrics: LR is 2 0 larger than 1% 0.86 0.88 0.9 0.92 0.94 0.96 0.98 critical value. Portfolio via copula: Independence test •Y axis-Likelihood 9 Skewed-EWMA Forecasting via Copula ratio value 8 RiskMetrics Forecasting Model • X axis- value of λ 7 • 1% critical value 6.63 6 (blue) 5 4 3 2 1 0 0.86 0.88 0.9 0.92 0.94 0.96 0.98 Portfolio via copula: Conditional coverage test • Y axis-Likelihood 18 Skewed-EWMA Forecasting via Copula ratio value 16 RiskMetrics Forecasting Model • X axis- value of λ 14 12 • Sk-EWMA: LR is 10 less than 1% critical 8 value of 9.21(pink); 6 4 RiskMetrics: LR is 2 0 larger than 1% 0.86 0.88 0.9 0.92 0.94 0.96 0.98 critical value. IV. Further work and outlook Combining skewed-EWMA with historical simulation, Monte Carol simulation and extreme value theory ¾Hull,J. and White, A., 1998b, "Incorporating volatility updating into the historical simulation method for Value- at-Risk", Journal of Risk, 1, 5-19. Extending to risk modeling beyond market risk: ¾E.g., credit risk, insurance risks