# Fwd: LO 2.3: Com pare and contrast the age-weighted, the volatility-weighted, the

LO 2.3: Com pare and contrast the age-weighted, the volatility-weighted, the

correlation-weighted, and the filtered historical sim ulation approaches.

The previous weighted historical simulation, discussed in Topic 1, assumed that both
current and past (arbitrary) n observations up to a specified cutoff point are used when
computing the current period VaR. Older observations beyond the cutoff date are assumed
to have a zero weight and the relevant n observations have equal weight of (1 / n). While
simple in construction, there are obvious problems with this method. Namely, why is the
wth observation as important as all other observations, but the (n + 1) th observation is so
unimportant that it carries no weight? Current VaR may have “ghost effects” of previous
events that remain in the computation until they disappear (after n periods). Furthermore,
this method assumes that each observation is independent and identically distributed. This
is a very strong assumption, which is likely violated by data with clear seasonality (i.e.,
seasonal volatility). This topic identifies four improvements to the traditional historical
simulation method.

Age-weighted Historical Sim ulation

The obvious adjustment to the equal-weigh ted assumption used in historical simulation is
to weight recent observations more and distant observations less. One method proposed by
Boudoukh, Richardson, and Whitelaw is as follows.1 Assume w(l) is the probability weight
for the observation that is one day old. Then w(2) can be defined as \w (l), w(3) can be
defined as X2w(l), and so on. The decay parameter, X, can take on values 0 < X < 1 where
values close to 1 indicate slow decay. Since all of the weights must sum to 1, we conclude
that w(l) = (1 — X) / (1 — Xn). More generally, the weight for an observation that is i days
old is equal to:

x 1- 1 (1 – X)

1 – X n

The implication of the age-weighted simulation is to reduce the impact of ghost effects and
older events that may not reoccur. Note that this more general weighting scheme suggests
that historical simulation is a special case where X = 1 (i.e., no decay) over the estimation
window.

Professor's Note: This approach is also known as the hybrid approach.

1. Boudoukh, J., M. Richardson, and R. Whitelaw. 1998. “The best of both worlds: a hybrid

approach to calculating value at risk.” Risk 11: 64-67.

Page 17

Topic 2
Cross Reference to GARP Assigned Reading – Dowd, Chapter 4

Volatility-weighted Historical Sim ulation

.Another approach is to weight the individual observations by volatility rather than
proximity to the current date. This was introduced by Hull and White to incorporate
changing volatility in risk estimation.2 The intuition is that if recent volatility has increased,
then using historical data will underestimate the current risk level. Similarly, if current
volatility is markedly reduced, the impact of older data with higher periods of volatility will
overstate the current risk level.

This process is captured in the expression below for estimating VaR on day T. The
expression is achieved by adjusting each daily return, r j on day t upward or downward
based on the then-current volatility forecast, ct • (estimated from a GARCH or EWMA
model) relative to the current volatility forecast on day T.

where:
rt j = actual return for asset i on day t
<Tt i = volatility forecast for asset i on day t (made at the end of day t — 1)

= current forecast of volatility for asset i

Thus, the volatility-adjusted return, rt -t , is replaced with a larger (smaller) expression if
current volatility exceeds (is below) historical volatility on day i. Now, VaR, ES, and any
other coherent risk measure can be calculated in the usual way after substituting historical

There are several advantages of the volatility-weighted method. First, it explicitly
incorporates volatility into the estimation procedure in contrast to other historical methods.
Second, the near-term VaR estimates are likely to be more sensible in light of current
market conditions. Third, the volatility-adjusted returns allow for VaR estimates that are
higher than estimates with the historical data set.

Correlation-weighted Historical Sim ulation

As the name suggests, this methodology incorporates updated correlations between asset
pairs. This procedure is more complicated than the volatility-weighting approach, but it
follows the same basic principles. Since the corresponding LO does not require calculations,
the exact matrix algebra would only complicate our discussion. Intuitively, the historical
correlation (or equivalently variance-covariance) matrix needs to be adjusted to the new
information environment. This is accomplished, loosely speaking, by “multiplying” the
historic returns by the revised correlation matrix to yield updated correlation-adjusted
returns.

2. Hull, J., and A. White. 1998. “Incorporating volatility updating into the historical simulation

method for value-at-risk.” Journal of Risk 1: 5-19.

Page 18

Topic 2
Cross Reference to GARP Assigned Reading – Dowd, Chapter 4

Let us look at the variance-covariance matrix more closely. In particular, we are concerned
with diagonal elements and the off-diagonal elements. The off-diagonal elements represent
the current covariance between asset pairs. On the other hand, the diagonal elements
represent the updated variances (covariance of the asset return with itself) of the individual
assets.

f

/
0" • •
1,1
(T • •
{ b1

\
0" • •
1,J
CT • •
b))

Variance(Xj)

C o v (X ;, Xj )'

C ov(X j, X j ) Variance(Xj)
' /

Notice that updated variances were utilized in the previous approach as well. Thus,
correlation-weighted simulation is an even richer analytical tool than volatility-weighted
simulation because it allows for updated variances (volatilities) as well as covariances
(correlations).

Filtered Historical Sim ulation

The filtered historical simulation is the most comprehensive, and hence most complicated,
of the non-parametric estimators. The process combines the historical simulation model
with conditional volatility models (like GARCH or asymmetric GARCH). Thus, the
method contains both the attractions of the traditional historical simulation approach with
the sophistication of models that incorporate changing volatility. In simplified terms, the
model is flexible enough to capture conditional volatility and volatility clustering as well as a
surprise factor that could have an asymmetric effect on volatility.

The model will forecast volatility for each day in the sample period and the volatility will
be standardized by dividing by realized returns. Bootstrapping is used to simulate returns
which incorporate the current volatility level. Finally, the VaR is identified from the
simulated distribution. The methodology can be extended over longer holding periods or
for multi-asset portfolios.

In sum, the filtered historical simulation method uses bootstrapping and combines the
traditional historical simulation approach with rich volatility modeling. The results are then
sensitive to changing market conditions and can predict losses outside the historical range.
>From a computational standpoint, this method is very reasonable even for large portfolios,
and empirical evidence supports its predictive ability.

A d v a n t a g e s a n d D i s a d v a n t a g e s o f N o n -Pa r a m e t r i c M e t h o d s