M. Dubrovich1 1

2

M. Terraneo1

UniCredit S.p.A.

Faculty of Mathematical, Physical and Natural Sciences, Universit` a 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:1408.2462, 2014. Spadafora, Dubrovich, Terraneo (UniCredit S.p.A.)

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

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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

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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

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

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

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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.)

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XVI WQF, Parma, Jan 29-30, 2015

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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: Z VaR(α,t) 1−α= p(x(t))dx(t)

(1)

−∞

General VaR-scaling approach: find h(·) such that, given α2 6= α and T 6= t VaR(α2 , T ) = h(VaR(α, t))

(2)

For EC, i.e. VaR at CL 1 − α2 = 99.93% √ over horizon T = 1y, commonly done assuming normality of PDF and applying T -rule: VaRN (99.93%, 1y) =

√ Φ−1 (0.01%) 250 N −1 VaRN (99%, 1d) ΦN (1%)

(3)

where ΦN denotes CDF of N(ormal) distribution We propose a generalization: 1 2 3

Fit short-term P&L distribution and choose PDF with best explanatory power Calculate long-term P&L distribution (analytically or numerically), given chosen PDF Compute EC as the desired extreme-percentile

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

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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): pN (x; µ, σ, T ) = √

2

(x − µT )2 exp − 2σ 2 T 2πσ 2 T 1

Student’s t-distribution (ST, power-law decay): " #− ν+1 2 ) ( x−µ )2 Γ( ν+1 2 σ pST (x; µ, σ, ν) = √ 1+ σ νπΓ( ν2 ) ν

3

(4)

(5)

Variance-Gamma distribution (VG, exponential decay):

θ(x−µT ) r|x−µT | 2σ 2 pVG (x;µ,σ,k,θ,T )= 2e T √ 2σ 2 +θ 2 σ πk k Γ( T ) k k √

Spadafora, Dubrovich, Terraneo (UniCredit S.p.A.)

T −1 r k 2 2σ 2 +θ 2 |x−µT | k KT 1 − σ2 k 2

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

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(6)

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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. 100

100

10-1

10-2 0.025

CDF

CDF

10-1

0.020

0.015 0.010 Return

Actual Data N ST VG 0.005 0.000

10-2 10-3 10-4

0.12

0.10

0.08

0.06 0.04 Return

0.02 0.00

0.02

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.)

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Time Scaling

Convolution and the CLT (1) Long-term PDF can be calculated (analytically or numerically) by convoluting the short-term PDF p(xk (t)): Xk (t) RV (with values xk (t)) describing P&L over horizon t at time (day) k P&L over the long horizon T = nt is given by: P&L0→T = x1 (t) + x2 (t) + ... + xn (t) =

n X

xk (t)

(7)

k=1

The PDF of the sum of two independent (as we assume the P&Ls) RVs is given by: Z +∞ p(y ) = p(y − x1 (t))p(x1 (t))dx1 (t) (8) −∞

where RV Y = X1 + X2 with values y = x1 + x2 . 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.)

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Time Scaling

Convolution and the CLT (2) Is it possible to obtain an asymptotic behaviour? (n → ∞) Yes! Use CLT! Given RV X distributed as pD (x; ·), with E (X ) = µ∆t and Var(X ) = σ 2 ∆t, the n-times convoluted distribution satisfies (for all finite α and β): ( ) Z n β X (x−µn∆t)2 1 − √ lim P(α < xi < β) = e 2σ2 n∆t (9) n→+∞ 2πσ 2 n∆t α 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 xα (considered for VaR estimation) falls into central region of PDF in the sense of CLT after n = T /∆t convolutions, the normal approximation holds: √ VaRD (α, T ) ' Φ−1 (10) N (α; µT , σ T ) = VaRN (α, T ) Otherwise (convergence not achieved) long-term P&L distribution to be computed by explicit convolution (n times) Spadafora, Dubrovich, Terraneo (UniCredit S.p.A.)

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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 2 3

4

Define critical value x ∗ beyond which the two PDFs become substantially different Intuitively take x ∗ as the point where the two PDFs intersect After some math we find that, as expected, region where CLT holds enlarges slowly: √ p (11) x ∗ = σ ν T log(T ) Using Eq. (11) we estimate the percentile at which convergence condition is satisfied after exactly 1 year, as a function of ν: (ν + 1)Γ ν+1 √ p 1 2 P(σ ν T log(T ) < x < +∞) = (12) √ ν−2 πΓ ν2 2 T (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 purposes2 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, 1998 2 Recall that, for ν → ∞ the ST is a Normal. Spadafora, Dubrovich, Terraneo (UniCredit S.p.A.)

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Time Scaling

The Normal limit: VG distribution 0.05

|FVG(x)−FN (x)| / FN (x)

0.04 0.03 0.02 0.01 0.00

5

4

3

x/σ

2

1

0

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 PN (x < 4σ) ' 0.006%) is smaller than 2% Spadafora, Dubrovich, Terraneo (UniCredit S.p.A.)

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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) VaR-scaling calculation:

3

Normal VaR: through application of CLT, the 1-year P&L distribution is Normal with µ(T ) = µ(t)T = 0 2

(by assumption) 2

σ (T ) = σ (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 pD (x; ·), convolve it n = 250 times to extract the long-term PDF: If pD is VG, the long-term PDF is given by Eq. (6)3 If pD is ST, the long-term PDF can be estimated numerically by explicitly convolving pD . 4

3

Repeat steps (1-3) for 10000 different (random) portfolio weight combinations to derive the statistical properties w.r.t. asset allocation As mentioned before, in our case it always reaches convergence to the normal limit.

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VaR scaling on a real portfolio

600

5

500

4

400

VaRST / VaRN

Number of Observations (ν)

The methodology test: outcomes

300 200

2 1

100 0

3

2

3

4

5

ν

6

7

8

0 2.0

2.5

3.0 ν

3.5

4.0

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 VaRST /VaRN ' 1 When ν < ν ∗ , the scaled ST-VaR is greater than the scaled N-VaR, and, in the ν → 2 limit, VaRST /VaRN ∼ 4: the assumption of normal convergence underestimates risk! Spadafora, Dubrovich, Terraneo (UniCredit S.p.A.)

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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

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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 scaling VaR (even if the short-term distribution is not Normal)

√

T -rule is safe for

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

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