*ETH ZURICH ^UNIVERSITY OF FIRENZE

THE AEP ALGORITHMfor the fast computation of the distribution of

the sum of dependent random variables

PHILIPP ARBENZ*, PAUL EMBRECHTS*, GIOVANNI PUCCETTI^

P[X1 + X2 + ... + Xd ≤ s]

Mathematical Problem

(X1, X2, . . . , Xd) ! H

- Why do you need this:

- calculation of the VaR-based capital charge associated with a risk portfolio

- general problem in applied probability

- How you can calculate it:

- under independence: closed form (very few cases), Fast Fourier Transforms, ...

- under comonotonicity: closed form

- general dependence: MC and QMC methods

The idea behind AEP, d=2

s

s

0_

+

+

_

VH[S] =∫

SdH = P[X1 + X2 ≤ s]

S

+ + . . . +!

AEP in practice, d=2

s

s0

P1(s) =

P2(s) = P1(s) +∑

−

Pn(s) = Pn−1(s) +3n−1∑

k=1

sknVH[Qk

n

]

P3(s) = P2(s) +∑

−∑

Main theoretical result, arbitrary d

VH[Sk

n

]= VH

[Qk

n

]+

2d−1∑

j=1

mjVH[SNk−N+ j

n+1

]

α = 1/d

Remarks:

- The simplexes generated at each iteration are NOT disjoint for d>3

- This volume decompositon holds in arbitrary dimensions for every choice of alpha in [1/d,1). Which alpha is the best?

Pn(s) = Pn−1(s) +(2d−1)n−1∑

k=1

sknVH[Qk

n

]

10α ! 1

Convergence of the algorithm

d=4, 15 new simplexes

VH[S] − Pn(s) =(2d−1)n∑

k=1

skn+1VH

[Sk

n+1

]

alpha=2/(d+1)

- It minimizes the total Lebesgue measure produced at each iteration

- Reduces computational complexity for odd d,

e.g. for d=3: 4 simplexes (instead of 7); for d=5 21(31).

- It allows the use of extrapolation

Convergence

- If H has a bounded density, proof of convergence for d<=5

- Continuity is required only on a neighborhood of dS={x1+...+xd=s}.

- AEP does not like atoms on dS (very simple counterexample!)

- Non-convergence restricted to pathological cases

- It works!

Extrapolation

! (d + 1)d

2dd!×

Convergence of Pn*

- If H has a C2 density, proof of convergence for d<=8

- Smoothness is required only on a neighborhood of dS={x1+...+xd=s}

P∗n(s) = Pn−1(s) +(d + 1)d

2d d!

(2d−1)n−1∑

k=1

sknVH[Qk

n

]

VH[S] − Pn(s) =(2d−1)n∑

k=1

skn+1VH

[Sk

n+1

]

linear density

Extra-accuracy

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+

+

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2 independent Pareto(1,2) 3 independent Pareto(1,2,3)

- Extrapolation works (many other examples in the paper)!

2 4 1e-53 16 4e-54 64 1e-45 256 4e-46 1024 1e-37 4096 0.028 16384 0.109 65536 0.4110 262144 2.211 1048576 6.612 4194304 31.013 16777216 118.5

n

Computational facts

d=2: 10 digits in 0.06 sec. (n=10)d=3: 7 digits in 6 secs (n=11)d=4: 4 digits in 7 secs (n= 6)d=5: 3 digits in 4 secs (n= 5)d=6:

Pareto(1)-portfolios (no moments)!

This MAC (2GB RAM)!

Computational times (d=3)

malloc: *** failedmalloc: *** failedmalloc: *** failedmalloc: *** failed

14 malloc: *** failedmalloc: *** failedmalloc: *** failedmalloc: *** failed

Asymptotic convergence rates of the AEP, standard MC and QMC methods.

d 2 3 4 5

AEP (upper bound) M−3 M−1.5 M−0.54 M−0.34

MC M−0.5 M−0.5 M−0.5 M−0.5

QMC (best) M−1 M−1 M−1 M−1

QMC (worst) M−1(log M)2 M−1(log M)3 M−1(log M)4 M−1(log M)5

Comparison with MC and QMC methods

VH[S ] =∫

SdH ! 1

M

M∑

i=1

vH(xi)

d=2

d=4

DEMO

- AEP:

- provides accurate estimates in a more than reasonable time

- is based solely on a geometrical decomposition and thus can handle any joint distribution (no density in closed form required)

- is deterministic (no confidence intervals!)

- is extremely easy to use and do not need any adaptation to the model under study

- can be used to compute VaR for risk portfolios

- Future research:

- proof of convergence (if any) in arbitrary dimension

- coming soon: a different decomposition to handle more general aggregating functionals

Conclusions and Open Problems

Acknowlegments

I would like to thank FIM and RiskLab, ETH Zurichfor warm hospitality and financial support.