24.2 Stress Testing in Credit Risk
24.2.5 Importance Sampling for Stress Testing
The procedure above gives as a very successfull approach for a very efficient variance reduction technique based on shifting the underlying factor model. Since the expected shortfall can be viewed as an expected loss under a severe, but very specific scenario, for the portfolio, namely that the loss exceeds a specific quantile, this approach is good starting point for importance sampling techniques in stress testing.
24.2.5.1 Subsampling
As a first remark, one can say that the shifted factor model produces also more losses in those factor scenarios which finally hurts the portfolio most. Hence many
stress testing procedures which are also hot spots of the portfolio might be found by the simple subsampling technique. That is choose now those scenarios under the shifted measures, where also the constraint of the stress test are fullfilled. I.e. for each functionalT of the loss variableLwe computeEŒT .L/Q by
EŒT .L/Q DEŒT .L/jX < C D.P ŒX < C /1EŒT .L/1fX<Cg (24.22) EŒT .L/1fX<CgD
Z
T .L/1fX<Cgn0;
nM;
dPM (24.23)
1X
iD1
T .L.xi;zi//1xi<C n0;
nM;.xi/; (24.24)
where .xi;zi/; i D 1; : : : ; are k simulations of the factor model and the idiosyncratic asset risk vectors. is the original covariance matrix of the factor model andMis the optimal drift for the Expected Shortfall calculation (unstressed) as above.
Remark
There might be functionalT ofLwhich give in combination with the restriction a very fast calculation by the above approach. For example ifT .x/ D x and we restrict only the most sensitive factor for the portfolio. Like for a bank lending mainly to European customer we only restrict the European factor. For other pairs of functionalsT and restriction vectorCthis might be not efficient. E.g. calculation of the Expected Loss in normal timesCi D 1alli andT .x/ D x importance sampling with the optimal expected shortfall shift is not efficient.
24.2.5.2 Stress Specific Shifts
Lead by the successfull application of the importance sampling scheme to Expected Shortfall calculation and the observation, that expected shortfall can also be viewed as an Expected Loss under a very specific downturn scenario, namely the scenario
“loss is larger than quantile”, we will now propose some specific importance sampling schemes for stress testing. First the optimisation equation for the general portfolio level stress testing defined by the restriction vectorC and a functionalT, whose expectation should be related to risk measures is
The Variance Reduction Problem for Stress Testing: compute a minimumM D .M1; : : : ; Mm/of the variance
T2.PM/D Z
T .L/1X<C n0;
nM;
2
dPM Z
T .L/1X<CdP 2
(24.25)
Also this minimization problem is not feasible and we propose several approaches to improve the efficiency:
Infinite Granular Approximation
We can proceed as in the section on unstressed importance sampling by taking the infinite granular approximation as in the first part leading to Proposition1. Then we have to minimize over the drift vectorM
Z
T .L1m//1X<C n0;
nM;
2
dNM; (24.26)
If we consider now a matrixAwithAAT D we can re-formulate this Z
Rm
T .L1m.AxCM //1AxCM <C n0;
nM;.AxCM / 2
…miD1n0;1.xi/dx (24.27) If one wants to solve the normal integral by a Monte-Carlo simulation one just has to generate vectorsx.j /D.x1.j /; : : : ; x.j /m /T ofmindependent standard normal random numbersx.j /i and minimize
X
jD1
T .L1m.Ax.j /CM //1Ax.j /CM <C n0;
nM;.AxCM / 2
: (24.28)
This is a feasible optimization problem. Of course a reduction to a one-factor optimization as for the calculation of unstressed expected shortfall is not always straight forward and subject to future research.
Remark
(i) In this approach we have avoided to work with the probability measureP, butQ did all the analysis under the originalP. The reason was that the restricted prob- ability measure is not normal anymore and we expect less analytic tractability of the optimization required in importance sampling approach. In the next section we will therefore replace the restricted probability measure by a shifted one.
(ii) We have so far assumed that the risk characteristic of interest can be written as an expectation of a functionalT ofL. Also in the unstressed case we have approximated the expected shortfall by .1 ˛/1EŒLjL > c with c the somewhere known or approximated quantil ofL.
For the derivation of the quantil of the unstressed loss distribution can be also carried out under the transformation toPM: Generate the Monte-Carlo sampleL.i/; i D
1; : : : ; underPM. Then calculate the sumPn
iD1.Œi/nnM;0;.x.Œi//, whereŒiis the index of theŒi-largest loss, until this sum equals1˛. If we denote the index of last summand byn˛thenL.n˛/corresponds to the quantile underP. In stress testing we face the additional problem that we want to know such type of risk characteristics, in particular Value-at-Risk and Expected Shortfall, which can not be written as an integral of a function of the loss distribution, underP.Q