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risk. Violation of the sub-additivity rule may lead to several problems.

In particular, it may provoke investors to establish separate accounts

for every asset they have. Unfortunately, VaR satisfies (11.1.4) only if

the probability density function for P/L is normal (or, more generally,

elliptical) [3].

The generic criterions for the risk measures that satisfy the require-

ments of the modern risk management are formulated in [3]. Besides

the sub-additivity rule (11.1.4), they include the following conditions.

r(lA) Â¼ lr(A), l > 0 (homogeneity) (11:1:5)

r(A) r(B), if A B (monotonicity) (11:1:6)

r(A Ã¾ C) Â¼ r(A) Ã€ C (translation invariance) (11:1:7)

In (11.1.7), C represents a risk-free amount. Adding this amount to

a risky portfolio should decrease the total risk, since this amount is

124 Market Risk Measurement

not subjected to potential losses. The risk measures that satisfy the

conditions (11.1.4)â€“(11.1.7) are called coherent risk measures. It can

be shown that any coherent risk measure represents the maximum of

the expected loss on a set of â€˜â€˜generalized scenariosâ€™â€™ where every such

scenario is determined with its value of loss and probability of occur-

rence [3]. This result yields the coherent risk measure called expected

tail loss (ETL):2

ETL Â¼ E[LjL > VaR] (11:1:8)

While VaR is an estimate of loss within a given confidence level,

ETL is an estimate of loss within the remaining tail. For a given

probability distribution of P/L and a given a, ETL is always higher

than VaR (cf. Figures 11.1 and 11.2).

ETL has several important advantages over VaR [2]. In short, ETL

provides an estimate for an average â€˜â€˜worst case scenarioâ€™â€™ while VaR

only gives a possible loss within a chosen confidence interval. ETL

has all the benefits of the coherent risk measure and does not discour-

age risk diversification. Finally, ETL turns out to be a more conveni-

ent measure for solving the portfolio optimization problem.

0.45

0.4

0.35

ETL at 84%

z = âˆ’1.52

0.3

0.25

ETL at 95%

z = âˆ’2.06

0.2

0.15

ETL at 99%

z = âˆ’2.66

0.1

0.05

0

âˆ’5 âˆ’4 âˆ’3 âˆ’2 âˆ’1 0 1 2 3 4 5

Figure 11.2 ETL for the standard normal probability distribution of P/L.

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Market Risk Measurement

11.2 CALCULATING RISK

Two main approaches are used for calculating VaR and ETL [2].

First, there is historical simulation, a non-parametric approach that

employs historical data. Consider a sample of 100 P/L values as a

simple example for calculating VaR and ETL. Let us choose the

confidence level of 95%. Then VaR is the sixth smallest number in

the sample while ETL is the average of the five smallest numbers

within the sample. In the general case of N observations, VaR at the

confidence level a is the [N(1 Ã€ a) Ã¾ 1] lowest observation and ETL is

the average of N(1 Ã€ a) smallest observations.

The well-known problem with the historical simulation is handling

of old data. First, â€˜â€˜too oldâ€™â€™ data may lose their relevance. Therefore,

moving data windows (i.e., fixed number of observations prior to

every new period) are often used. Another subject of concern is

outliers. Different data weighting schemes are used to address this

problem. In a simple approach, the historical data X(t Ã€ k) are multi-

plied by the factor lk where 0 < l < 1. Another interesting idea is

weighting the historical data with their volatility [4]. Namely, the asset

returns R(t) at time t used in forecasting VaR for time T are scaled

with the volatility ratio

R0 (t) Â¼ R(t)s(T)=s(t) (11:2:1)

where s(t) is the historical forecast of the asset volatility.3 As a result,

the actual return at day t is increased if the volatility forecast at day T

is higher than that of day t, and vice versa. The scaled forecasts R0 (t)

are further used in calculating VaR in the same way as the forecasts

R(t) are used in equal-weight historical simulation. Other more so-

phisticated non-parametric techniques are discussed in [2] and refer-

ences therein.

An obvious advantage of the non-parametric approaches is their

relative conceptual and implementation simplicity. The main disad-

vantage of the non-parametric approaches is their absolute depend-

ence on the historical data: Collecting and filtering empirical data

always comes at a price.

The parametric approach is a plausible alternative to historical

simulation. This approach is based on fitting the P/L probability

distribution to some analytic function. The (log)normal, Student

126 Market Risk Measurement

and extreme value distributions are commonly used in modeling P/L

[2, 5]. The parametric approach is easy to implement since the analytic

expressions can often be used. In particular, the assumption of the

normal distribution reduces calculating VaR to (11.1.2). Also, VaR

for time interval T can be easily expressed via VaR for unit time (e.g.,

via daily VaR (DVaR) providing T is the number of days)

pï¬ƒï¬ƒï¬ƒï¬ƒ

VaR(T) Â¼ DVaR T (11:2:2)

VaR for a portfolio of N assets is calculated using the variance of the

multivariate normal distribution

X

N

2

sN Â¼ (11:2:3)

sij

i, jÂ¼1

If the P/L distribution is normal, ETL can also be calculated analyt-

ically

ETL(a) Â¼ sPSN (Za )=(1 Ã€ a) Ã€ m (11:2:4)

The value za in (11.2.4) is determined with (11.1.3). Obviously, the

parametric approach is as good and accurate as the choice of the

analytic probability distribution.

Calculating VaR has become a part of the regulatory environment

in the financial industry [6]. As a result, several methodologies have

been developed for testing the accuracy of VaR models. The most

widely used method is the Kupiec test. This test is based on the

assumption that if the VaR(a) model is accurate, the number of the

tail losses n in a sample N is determined with the binomial distribu-

tion

N!

(1 Ã€ a)n a(NÃ€n)

PB (n; N, 1 Ã€ a) Â¼ (11:2:5)

n!(N Ã€ n)!

The null hypothesis is that n/N equals 1 Ã€ a, which can be tested with

the relevant likelihood ratio statistic. The Kupiec test has clear mean-

ing but may be inaccurate for not very large data samples. Other

approaches for testing the VaR models are described in [2, 6] and

references therein.

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