Closed-form approximations for operational VaR

Lorenzo Hernández[2], Jorge Tejero[2], Alberto Suárez[1,2,3] , Santiago Carrillo-Menéndez[2,3,4]

[1] Computer Science Department, Universidad Autónoma de Madrid [2] Quantitative Risk Research (QRR) [3] RiskLab Madrid[4] Mathematics Department, Universidad Autónoma de Madrid

alberto.suarez@uam.es1

2

Extreme events in finance

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Rare events Large impact Difficult to predict their specific characteristics

(but can be explained after the fact). Amenable to modeling

Modeling of extreme events is possibleand provides insight

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Basel II

http://www.bis.org/publ/bcbs128.htmSums of heavy-tailed random variables

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The three pillars approach

First pillar: Minimum capital requirements(quantification of risk)

• Specifies the guiding principles for the estimation of regulatory/economic capital.

• Operational risk is included as a new type of risk.

Second pillar: Supervisory review process.

Third pillar: Market discipline (+ public disclosure)

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Operational Risk: Definition

[source: Basel II]644. Operational risk is defined as the risk of

loss resulting from inadequate or failedinternal processes, people and systemsor from external events. This definition includes legal risk, but excludes strategic and reputational risk.

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Operational Risk: Measurement

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Loss distribution approach

Model the distribution of the aggregate losses for a given business line & risk type

Calculate Capital at Risk (99.9% percentile) of the aggregate loss distribution per business line & risk typeand add them.

. risk type and line businessfor year in losses ofnumber theis

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Actuarial models: Frequency + Severity

Hypothesis Severities of individual losses are independent. Severities and frequencies are independent.

Model separately Frequency {Nt}

E.g. Poisson, negative binomial, Cox process,… Severity {Lnt}

E. g. Lognormal, Pareto, g-and-h, … Approximate the distribution of aggregate losses by

combining these distributions. [Panjer, FFT, MC,single-loss approximations and beyond]

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Aggregation of frequency and severity dists.

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Expected & unexpected loss

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Risk analysis

Calculate aggregate yearly loss distribution from the frequency and severity distributions.

Compute risk measures• Expected loss• Capital at Risk (CaR)

e.g. 99.9% percentile of the aggregate loss distribution.• Conditional CaR (Expected shortfall)

Expected loss, given that the loss is above CaR

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Computational issues in risk analysis

Algorithms to compute risk measures Deterministic algorithms

• Discretized approximation (FFT, Panjer).• Piecewise approximation for the distribution

• Body approximation: Not very important• Tail approximation. E.g. Single loss

Monte Carlo algorithmsEmpirical compound distribution obtained by simulation.Computationally costly.

• Use variance reduction techniques• Hardware solutions: Grid computing, GPU’s, …

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Distribution of aggregate losses

Consider L1 ,…, LN iid samples from the density f(x)For the individual losses

The aggregate loss Z = L1 +…+ LN is a random variable with density g(x)

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on]distributi [Tail )(1Prob)(

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The lognormal distribution

exp() scale tails

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xx

xLNpdf

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Sums of iid lognormal rvs

Z = L1 +L2 +...+LN {Li} ~ iid LN(,) For sufficiently large , and specially at the tails,

the sum is dominated by the maximum.

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Ncentral limit

0.5 21.0 81.5 912.0 29812.5 2683383.0 656599703.5 4.37E+104.0 7.90E+134.5 3.88E+175.0 5.18E+21

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Heavy-tailed distributions

A heavy-tailed distribution is such that the tails of the distribution decay more slowly than the exponential.

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0,)(lim xx e

xF

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Subexponential distributions

Subexponential distributions are a special class of heavy-tailed distributions.

Definition: F(x) is a subexponential distribution if

where L1 ,…, LN an iid sample from F(x)

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)(1)(1lim

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One loss causes ruin [subexponential distributions]

One loss causes ruin: If the distribution of individual losses is subexponential, large values of the aggregate loss from N events are dominated by the maximum loss in one of the N events.

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Single loss approximation [Böcker + Klüppelberg, 2005]

From the definition of subexponential distribution

From the definition of VaRα

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Single loss approximation + mean correction

Single loss approximation for the aggregate loss distribution[Böcker + Sprittulla, 2006]

Single loss approximation + mean correction for VaRα

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Beyond the single-loss approximation

For the Pareto distribution Based on [Omey & Willekens, 1986, 1987],

[Sahay et al., 2007] A. Sahay, Z. Wan, B. Keller, Operational risk capital: asymptoticsin the case of heavy-tailed severity. The journal of operational risk 2(2):61–72 (2007)[Suárez, 2010] http://www.fields.utoronto.ca/audio/09-10/forum-oprisk/suarez/[Degen, 2010] Degen, M. The calculation of minimum regulatory capital using single-loss approximations. Journal of Operational Risk 5(4):3–17 (2010)

For rapidly varying subexponential distributionsBased on [Barbe, McCormick & Zhang, 2007][Suárez & Fernández, 2011] http://www.risklab.es/es/jornadas/2011/index.html

Based on asymptotics with a shifted argument[Albrecher et al., 2010] H. Albrecher, C. Hipp, D. Kortschak, Higher-order expansions

for compound distributions and ruin probabilities with subexponential claims. Scandinavian Actuarial Journal 2010(2):105–135 (2010)

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Correction by the mean from the perturbative expansion [Hernández et al. 2011]

Using the first order approximation and assuming that Q0 >> X

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Estimation of the mean can be difficult

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Error in the sample estimate of the mean

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Pre-CLT behavior in convergence to mean

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= 0; = 3 mean = 90.0171

N = 104 N = 105N = 103N = 102N = 10

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Perturbative expansion I

Consider a random variable Z = X + Y

Let Q0 = FX-1() be the percentile of X at probability level

Let Q be the percentile of Z at probability level :

From these two equations, one obtains

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Perturbative expansion II

Expressing Q as a perturbative series in

27Sums of heavy-tailed random variables

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Perturbative expansion III

Taylor expansion

In terms of Bell polynomials

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Perturbative expansion IV

Therefore, for the nth term in the series

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30

Perturbative expansion V

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Perturbative expansion VI

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Expansion around the maximum I

Consider a random variable ZN = L1+ L2 + ... + LN = L[1]+ L[2] + ... + L[N]

where the order statistics are L[1] L[2] ... L[N]

Identify XN with L[N] = max {L1, L2, ... , LN}YN with L[1]+ L[2] + ... + L[N-1]

Define ZN() = XN + YN [ZN = ZN(=1)]

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Expansion around the maximum II

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)()()( )(ProbProb)( 1][][][ xfxFNxfxFxLxLxF N

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Expansion around the maximum III

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Stochastic frequency

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Benchmarks for comparison (finite mean)

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Benchmarks for comparison (divergent mean)

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Lévy distribution

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Lévy distribution (deterministic N)

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Lognormal distribution (stochastic N)

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Pareto distribution: dependence on

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Pareto distribution (finite mean)

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Pareto distribution (divergent mean)

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Pareto distribution (asymptotic analysis)

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Dominant for a > 1 up to order k a

/N: Expansion parameter

=1- 0+

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Pareto distribution (asymptotic analysis)

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Dominant for a < 1and for a >1 (k > a)

No recognizableexpansion parameter

=1- 0+

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Asymptotic formulas to operational VaR

The single-loss approximation is insufficient, specially with Lower percentiles. Less heavy tails. Higher frequencies.

The single loss formula corrected by the mean Can be derived from the second order asymptotics. Accurate in a wide range of cases Estimation of the mean can be inaccurate if tails are

heavy The following orders in the perturbative approximation

Improve estimate and are easy to compute.46Sums of heavy-tailed random variables

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Open questions

The perturbative approximation is a formal expansion with good numerical accuracy (at least for low orders)

Convergence properties?General asymptotic behavior?Resummation of the series?Transition to CLT regime?

Extend the analysis to Non-identically distributed random variables Dependent variables

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Thanks for your attention!

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