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12.5 Scenario generation II: Conditional versus
unconditional risk measures
We start with the most obvious way to build a scenario matrix, i.e., we build it
from historical returns. Under the assumption of stationary returns (means and
covariances do not change over time), each observation period (e.g., a month)
corresponds to a scenario. Think of it as downloading data from a data pro-
vider and arranging them in a spreadsheet, where rows correspond to scenarios
and columns to assets. With this approach, we will not only face considerable
approximation error and will need to limit ourselves to a few assets, but we
have essentially chosen an unconditional and nonparametric approach to sce-
nario generation: unconditional, because we do not make our views of future
risk conditional on risk factors or a particular (e.g., most recent) market envi-
ronment, and nonparametric, because we do not impose any assumptions
on return distributions. The nonparametric nature of the historical approach
(and its limited ability to generate a sufficient number of scenarios for all but
extremely small problems) can easily be dealt with. 14 We could, for example,
select the best fitting distributions for each marginal asset distribution and
“ glue ” these distributions together using a copula function of your choice.
Given that we can now draw a large number of scenarios from this setup, we
can make the approximation error very small. However, what remains is the
unconditional nature of the employed distribution. Sudden shifts in volatility
regimes cannot easily be married with nonnormal return distributions. One
of the few methods used by practitioners to overcome this issue is combining
GARCH models with measures of nonnormality. First, we fit a GARCH model
to a single return series. This model will provide us with a series of stand-
ardized residuals that hopefully lost their autocorrelation in squared returns.
However, these residuals might still exhibit serious deviations from normality.
We can now use the forecasted volatilities from our GARCH model (which will
be very responsive to recent market events) to scale our residuals (which con-
tain information about nonnormality of returns) up or down. This represents
a nonparametric way to deal with deviations from normality but nothing stops
us to here to fit a nonnormal distribution or an extreme value model to our
residuals. This allows us to simulate a large number of scenarios for each asset
combining risk updates conditional on the current environment with an uncon-
ditional interpretation of nonnormality. Of course, we could deal with both
issues by calculating the risk neutral probability density function from option
markets (capture the conditional nature of the return distribution completely,
i.e., dispersion and nonnormality). Marginal distributions can then be “ tied
14 One might be tempted to use bootstrapping multiperiod returns (monthly from weekly data) for
scenario generation to construct a large number of scenarios. Given the well-known problems to
maintain the original nonnormality in bootstrapped data, this does not look like a viable idea.
Bootstrapping will eventually create normal returns (central limit theorem) and, hence, leave
CVaR optimization with no advantage over mean – variance alternatives.