The next example consists of approximating the Conditional Value-at-Risk (a.k.a. Expected Shortfall) associated with the portfolio of
m defaultable assets in a multi-factor model. The total loss
L on such a portfolio up to the time horizon
T is the sum of the individual losses
, where the contribution of the loss of the
i-th asset is of the form
, where
is the weight of the asset in the portfolio and
is the default time of the
i-th obligor. Whereas the expected loss is independent from the possible correlation across defaults, it is a key driver of the Value-at-Risk, and hence of the economic and regulatory capital. Most credit risk models introduce such dependency by relying on latent variables, like
, where the
s are correlated random variables with cumulative distribution function
F and
is the marginal probability that
under the chosen measure. The most popular choice (although debatable) is to rely on multi-factor Gaussian models, i.e., to consider
, where
is a
n-dimensional vector of weights with norm smaller than 1,
is the vector of
n i.i.d. standard Normal systematic factors and the
are i.i.d. standard Normal random variables independent from the
s representing the idiosynchratic risks. Computing the Expected Shortfall in a multi-factor framework is very time-consuming as there is no closed-form solution and many simulations are required. A possible alternative to the plain Monte Carlo estimator is to rely on the ASRF
model of Pykhtin, which can be seen as the single-factor model that “best” approximates the multi-factor model in the left tail, in some sense (see
Pykhtin (
2004) for details). The ASRF
model thus deals with a loss variable
relying on a single factor
Y, but such that
where
L is the loss variable in the multi-factor model. By the law of large numbers, the idiosynchratic risks are diversified away for
m large enough, so that conditional upon
, the portfolio loss in the ASRF
model converges almost surely, as
, to
with
. Moreover,
is a monotonic and decreasing function of
x. Consequently, the Value-at-Risk of the ASRF
model satisfies
for
m large enough. The asymptotic analytical expression
is known as the
large pool approximation (see e.g.,
Gordy (
2003)). In the derivation of the ASRF
analytical formula, Pykhtin implicitly models
Y as a linear combination of the factors
appearing in the multi-factor model, i.e.,
s.t.
. One can thus draw samples for the
s by using Algorithm 2 as follows: (i) draw a value for the standard Normal factor
Y conditional upon
, i.e., set
with
U a Uniform-
random variable so that
, and then (ii) sample
conditional upon
from the joint density derived in the paper. Therefore, we use
as a proxy for the condition
involved in the expected shortfall definition (observe that both
L and
depend on the same random vector
), but use the
actual (multi-factor) loss to estimate the expected loss under this condition. In other words, we effectively compute
as a proxy of the genuine Expected Shortfall, defined as
, leading to a drastic reduction of the computational cost.