Value-at-Risk scaling for long-term risk estimation

Transcription

1 Value-at-Risk scaling for long-term risk estimation L. Spadafora 1,2 M. Dubrovich 1 M. Terraneo 1 1 UniCredit S.p.A. 2 Faculty of Mathematical, Physical and Natural Sciences, Università Cattolica del Sacro Cuore, Brescia XVI Workshop on Quantitative Finance, Parma, January 29-30, 2015 The views and opinions expressed in this presentation are those of the author and do not necessarily represent official policy or position of UniCredit S.p.A. Main Reference: L. Spadafora, M. Dubrovich and M. Terraneo,Value-at-Risk time scaling for long-term risk estimation, arxiv: , Spadafora, Dubrovich, Terraneo (UniCredit S.p.A.) Value-at-Risk scaling for long-term risk estimation XVI WQF, Parma, Jan 29-30, / 15

2 Outline 1 Introduction and Motivation 2 Value-at-Risk Scaling 3 Modelling P&L Distributions 4 Time Scaling 5 VaR scaling on a real portfolio 6 Summary and Conclusions Spadafora, Dubrovich, Terraneo (UniCredit S.p.A.) Value-at-Risk scaling for long-term risk estimation XVI WQF, Parma, Jan 29-30, / 15

3 Introduction and Motivation Introduction: Value-at-Risk vs Economic Capital Regulatory Capital: 99% Value-at-Risk at a short time-horizon (1 day) Economic capital (EC): capital required to face losses within a 1-year time-horizon at a more conservative percentile (we refer to 99.93%) Possible estimation approaches for EC: 1 Scenario generation (for the risk factors) and portfolio revaluation to obtain a 1-year profit-and-loss (P&L) distribution 2 Extension of the short-term market-risk measures to longer time-horizons/higher percentiles 1 The first approach has the following drawbacks: Assumptions are needed to generate scenarios at 1 year (both with historical simulation and with Monte-Carlo methods...) No rebalancing of portfolio (i.e. unrealistic assumption of freezing positions for 1 year) 2 The second approach bypasses such difficulties: Assumes hedging/rebalancing of the portfolio Relies on models already approved and used into day-to-day activities Spadafora, Dubrovich, Terraneo (UniCredit S.p.A.) Value-at-Risk scaling for long-term risk estimation XVI WQF, Parma, Jan 29-30, / 15

4 Introduction and Motivation Motivation: Economic Capital as a scaled Value-at-Risk Main idea: follow the second approach and develop a scaling mechanism for computing efficiently Economic Capital out of Regulatory Value-at-Risk measures Model short-term P&L using iid RVs distributed according to some benchmark PDFs Apply convolution theorem to subsequent time-steps and interpret scaling in light of the Central Limit Theorem (CLT), to derive conditions needed for normal convergence Main results: generalized VaR-scaling methodology to be used for calculating EC, in dependence of the short-term PDF s properties: If the P&L distribution has exponential decay, VaR-scaling can be correctly inferred using T -rule, even if starting distribution is not Normal With power-law decay, T -rule can be applied naively only if tails are not too fat. Otherwise the long-term P&L distribution needs to be determined explicitly, and EC can be significantly larger than what would have been inferred under Normal assumptions Theoretical results are integrated by a numerical simulation performed on a test equity trading portfolio. Spadafora, Dubrovich, Terraneo (UniCredit S.p.A.) Value-at-Risk scaling for long-term risk estimation XVI WQF, Parma, Jan 29-30, / 15

5 Value-at-Risk Scaling The VaR-scaling approach Given x(t) P&L over time-horizon t (e.g. 1 day) and its PDF p(x(t)), VaR at confidence level (CL) 1 α (e.g. 99%) is defined by: 1 α = VaR(α,t) p(x(t))dx(t) (1) General VaR-scaling approach: find h( ) such that, given α 2 α and T t VaR(α 2, T ) = h(var(α, t)) (2) For EC, i.e. VaR at CL 1 α 2 = 99.93% over horizon T = 1y, commonly done assuming normality of PDF and applying T -rule: VaR N (99.93%, 1y) = 250 Φ 1 N (0.01%) Φ 1 N (1%) VaR N(99%, 1d) (3) where Φ N denotes CDF of N(ormal) distribution We propose a generalization: 1 Fit short-term P&L distribution and choose PDF with best explanatory power 2 Calculate long-term P&L distribution (analytically or numerically), given chosen PDF 3 Compute EC as the desired extreme-percentile Spadafora, Dubrovich, Terraneo (UniCredit S.p.A.) Value-at-Risk scaling for long-term risk estimation XVI WQF, Parma, Jan 29-30, / 15

6 Modelling P&L Distributions Modelling real-world P&L distributions: candidates Which theoretical PDF class better fits empirical P&L data? Benchmark the basic normal assumption using leptokurtic distributions: 1 Normal distribution (N): p N (x; µ, σ, T ) = 2 Student s t-distribution (ST, power-law decay): p ST (x; µ, σ, ν) = [ ] 1 2πσ 2 T exp (x µt )2 2σ 2 T ν+1 Γ( 2 ) [ σ νπγ( ν 2 ) 1 + ( x µ σ )2 ν ] ν+1 2 (4) (5) 3 Variance-Gamma distribution (VG, exponential decay): T θ(x µt ) 1 k p VG (x;µ,σ,k,θ,t )= 2e 2σ 2 2 σ T x µt x µt 2σ 2 πk k Γ( T ) 2σ 2 K Tk k k +θ2 1 k +θ2 2 σ 2 (6) Spadafora, Dubrovich, Terraneo (UniCredit S.p.A.) Value-at-Risk scaling for long-term risk estimation XVI WQF, Parma, Jan 29-30, / 15

7 Modelling P&L Distributions Fit performances over time Test 1: fitting performance on a 250-day P&L strip (each P&L distribution made by N = 500 obs.) Test 2: fitting performance on a single P&L distribution with N = 8000 obs CDF 10-1 CDF Actual Data N ST VG Return Return N performs much worse than ST and VG in explaining empirical P&L data VG and ST: comparable performances when N = 500 Raising the number of observations clarifies which PDF better fits the P&L dataset: in the IBM example the winner is ST Takeaway: though a challenging task, the determination of the PDF to fit P&L data is crucial to implement any efficient VaR-scaling methodology Spadafora, Dubrovich, Terraneo (UniCredit S.p.A.) Value-at-Risk scaling for long-term risk estimation XVI WQF, Parma, Jan 29-30, / 15

8 Time Scaling Convolution and the CLT (1) Long-term PDF can be calculated (analytically or numerically) by convoluting the short-term PDF p(x k (t)): X k (t) RV (with values x k (t)) describing P&L over horizon t at time (day) k P&L over the long horizon T = nt is given by: P&L 0 T = x 1 (t) + x 2 (t) x n(t) = n x k (t) (7) The PDF of the sum of two independent (as we assume the P&Ls) RVs is given by: + p(y) = p(y x 1 (t))p(x 1 (t))dx 1 (t) (8) where RV Y = X 1 + X 2 with values y = x 1 + x 2. k=1 Apply n times to the short-term PDF to obtain long-term PDF What about our benchmark distributions? Normal: well-known T -rule VG: analytic expression (see Eq. (6)) ST: numerical convolution (we apply Eqs. (7) and (8) with FFT algorithm) Spadafora, Dubrovich, Terraneo (UniCredit S.p.A.) Value-at-Risk scaling for long-term risk estimation XVI WQF, Parma, Jan 29-30, / 15

9 Time Scaling Convolution and the CLT (2) Is it possible to obtain an asymptotic behaviour? (n ) Yes! Use CLT! Given RV X distributed as p D (x; ), with E(X ) = µ t and Var(X ) = σ 2 t, the n-times convoluted distribution satisfies (for all finite α and β): { } n β 1 (x µn t)2 lim P(α < x i < β) = e 2σ 2 n t (9) n + 2πσ2 n t i=1 The above holds for n. For finite n it is understood that convergence takes place only in the central region of the PDF, which needs to be quantified somehow (see next slides) Therefore, we have the crucial result: If percentile x α (considered for VaR estimation) falls into central region of PDF in the sense of CLT after n = T / t convolutions, the normal approximation holds: α VaR D (α, T ) Φ 1 N (α; µt, σ T ) = VaR N (α, T ) (10) Otherwise (convergence not achieved) long-term P&L distribution to be computed by explicit convolution (n times) Spadafora, Dubrovich, Terraneo (UniCredit S.p.A.) Value-at-Risk scaling for long-term risk estimation XVI WQF, Parma, Jan 29-30, / 15

10 Time Scaling The Normal limit: ST distribution Which conditions to define the central region of the ST PDF Normal? We propose a quantitative method following Bouchaud et al. 1 1 Define critical value x beyond which the two PDFs become substantially different 2 Intuitively take x as the point where the two PDFs intersect 3 After some math we find that, as expected, region where CLT holds enlarges slowly: x = σ ν T log(t ) (11) 4 Using Eq. (11) we estimate the percentile at which convergence condition is satisfied after exactly 1 year, as a function of ν: P(σ ν T log(t ) < x < + ) = (ν + 1)Γ ( ) ν ( πγ ν ) (12) 2 T ν 2 2 (log(t )) ν/2 Imposing P = 0.07% and T = 250 days = 1 year, we obtain ν = 3.41 Using the above-defined criterion we have a discrimination: convergence regime (ν > ν ): the ST distribution becomes sufficiently normal for our purposes 2 non-convergence regime (ν < ν ) where the ST distribution cannot be approximated with a normal. Accordingly, the lower ν, the fatter the tails. 1 J. P. Bouchaud and M. Potters,Theory of Financial Risks - From Statistical Physics to Risk Management, Cambridge University Press, Recall that, for ν the ST is a Normal. Spadafora, Dubrovich, Terraneo (UniCredit S.p.A.) Value-at-Risk scaling for long-term risk estimation XVI WQF, Parma, Jan 29-30, / 15

11 Time Scaling The Normal limit: VG distribution F VG (x) F N (x) / F N (x) x / σ In the VG case, convergence takes place in much quicker way, due to exponential decay We present a proof just by numerical example: after convolving the VG PDF a number of times, we compare its CDF with the target Normal CDF Already for n = 50 iterations, the relative deviation at 4σ (corresponding to P N (x < 4σ) 0.006%) is smaller than 2% Spadafora, Dubrovich, Terraneo (UniCredit S.p.A.) Value-at-Risk scaling for long-term risk estimation XVI WQF, Parma, Jan 29-30, / 15

12 VaR scaling on a real portfolio The methodology test: setup To assess our VaR-scaling methodology we built a test equity trading portfolio composed by 10 FTSE stocks and ATM european calls to achieve -hedging: representative of real portfolios, convex and asymmetric. 1 Perform a (1-day) historical simulation to infer the short-term P&L distribution 2 Fit the P&L distribution with the benchmarks (N, VG and ST) 3 VaR-scaling calculation: Normal VaR: through application of CLT, the 1-year P&L distribution is Normal with µ(t ) = µ(t)t = 0 (by assumption) σ 2 (T ) = σ 2 (t)t where µ(s) and σ 2 (s) are mean and variance of the PDF over horizon s Convoluted VaR: given the short-term fitted PDF p D (x; ), convolve it n = 250 times to extract the long-term PDF: If p D is VG, the long-term PDF is given by Eq. (6) 3 If p D is ST, the long-term PDF can be estimated numerically by explicitly convolving p D. 4 Repeat steps (1-3) for different (random) portfolio weight combinations to derive the statistical properties w.r.t. asset allocation 3 As mentioned before, in our case it always reaches convergence to the normal limit. Spadafora, Dubrovich, Terraneo (UniCredit S.p.A.) Value-at-Risk scaling for long-term risk estimation XVI WQF, Parma, Jan 29-30, / 15

13 VaR scaling on a real portfolio The methodology test: outcomes Number of Observations (ν) ν VaR ST / VaR N ν As in the single-stock case, VG and ST provide comparable goodness-of-fit; again, N yields the worst performance In the VG case, Normal approximation always holds (as expected) The majority ( 70%) of fitted ν values for the ST case lies below the critical value ν = 3.41: the assumption of normal convergence is often unsafe When ν > ν, ST has reached Normal convergence and VaR ST /VaR N 1 When ν < ν, the scaled ST-VaR is greater than the scaled N-VaR, and, in the ν 2 limit, VaR ST /VaR N 4: the assumption of normal convergence underestimates risk! Spadafora, Dubrovich, Terraneo (UniCredit S.p.A.) Value-at-Risk scaling for long-term risk estimation XVI WQF, Parma, Jan 29-30, / 15

14 Summary and Conclusions Summary and Conclusions (1) Derived a generalized VaR-scaling methodology for calculating Economic Capital (i.e. 1-year 99.93% VaR) Chosen as benchmarks for explaining empirical (daily) P&L data were Normal, Student s t- (leptokurtic power-law) and Variance-Gamma (leptokurtic exponential) distributions Defined long-term P&L distribution by means of convolution and explored its asymptotic properties using the Central Limit Theorem (CLT) Theoretical results are a range of possible VaR-scaling approaches depending on PDF chosen as best fit given confidence level and given time horizon Spadafora, Dubrovich, Terraneo (UniCredit S.p.A.) Value-at-Risk scaling for long-term risk estimation XVI WQF, Parma, Jan 29-30, / 15

15 Summary and Conclusions Summary and Conclusions (2) Main discriminant: reaching of Normal convergence (in the sense of CLT) by the chosen PDF If assuming exponential decay (Variance-Gamma case) CLT can be safely applied for the typical time-horizons and percentiles If assuming power-law decay (Student s t- case), CLT can be applied only if number of degrees of freedom ν exceeds a critical value ν depending on chosen percentile and time horizon Outcome of methodology test by portfolio simulation: In the VG case, Normal convergence is always reached and the T -rule is safe for scaling VaR (even if the short-term distribution is not Normal) In the ST case, Normal convergence is often not achieved. In this case, CLT cannot be applied. The naive usage of the T -rule in the non-convergence regime (ν < ν, the most likely in our simulation) can lead to severe underestimation of the risk measure Spadafora, Dubrovich, Terraneo (UniCredit S.p.A.) Value-at-Risk scaling for long-term risk estimation XVI WQF, Parma, Jan 29-30, / 15