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.

©2018 Kaplan, Inc.

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

returns with volatility-adjusted returns.

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

©2018 Kaplan, Inc.

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