VaR STATISTICS

10.3 VaR STATISTICS

VaR statistics can be calculated in a number of ways. These include the analytic variance–covariance approach, the Monte Carlo simulation and other approaches based on extreme distributions and statistical models of various sorts. These approaches are outlined next.

10.3.1 The historical VaR approach

The historical VaR generates scenarios directly from historical data. For each day, returns and associated factors are assessed and characterized probabilisti- cally. This approach generates a number of errors, however. First, markets may

be nonstationary and therefore the scenarios difficult to characterize. This has the effect of overestimating the leptokurtic effect (due to heteroscedasticity) while ap- plication of the VaR using simulation tends to underestimate these effects because it uses theoretical distributions (such as the normal, the lognormal etc.)

10.3.2 The analytic variance–covariance approach

This approach assumes the normality of returns given by a mean and a standard deviation. Thus, if X has a standard Normal distribution, N (0, 1), then P(X >

1.65) = 5 %. If we consider again the preceding example, the standard deviation relative to the DM/USD exchange can be easily calculated. For 1995, it has a value

of 0.7495 %. Assuming a zero mean we can calculate for a risk of 5 % the value 1.645 as seen above. As a result, we obtain a critical value of 0.7495 * 1.645 =

1.233 %; and therefore a VaR of 12 330 USD. In trading rooms, VaRs are the outcomes of multiple interacting risk factors (exchange-rate risk, interest-rate risk, stock price risks etc. that may be correlated as well). Their joint effects are, thus, assessed by the variance–covariance matrix. For example, a portfolio consisting of a long position in DM and a short in yen with a bivariate normal distribution for risk factors has a VaR that can be calculated

with great ease. Let σ 1 and σ 2 be the standard deviations of two risk factors and let ρ 1,2 be their correlation, the resultant standard deviation of the position is thus:

p = 0.5 1 +σ 2 +2*ρ 1,2 σ 1 σ 2

This is generalized to more than two factors as well. Problems arise regarding the definition of multivariate risk distributions, however. Mostly, only marginal distributions are given while the joint multivariate distribution is not easily ob- servable – theoretically and statistically. Recent studies have supported the use of Copula distributions. These distributions are based on the development of

a multivariate distribution of risks based on the specification of their marginal

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distributions. Such an approach is attracting currently much interest due to the interaction effects of multiple risk sources where only marginal distributions are directly observable. There are other techniques for deriving such distributions, however. For example, the principle of maximum entropy (which we shall elabo- rate below) also provides a mechanism for generating distributions based on the partial information regarding these distributions.

10.3.3 VaR and extreme statistics

The usefulness of extreme statistics arises when the normal probability distribu- tion is no longer applicable, underestimating the real risk to be dealt with, or when investors may be sensitive to extreme losses. These distributions are defined over the Max or the Min of a sample of random returns and have ‘fatter’ tails than say the Normal distribution. For example, say that a portfolio has a loss density function F R (x) while we seek the distribution of the largest loss of n periods:

Y n = Max{R 1 , R 2 ,..., R n }

Assuming all losses independent (which is usually a strong assumption), let R (j)

be the j th order statistic, with R (1) < R (2) < R (3) <···<R (n) over n periods. Let

F j (R), j = 1, 2, . . . , n denote the density function of R (j) . Then, the distribution function of the maximum Y n =R (n) is given by:

Y n =R (n) = Max{R 1 , R 2 ,..., R n } If we set a target maximum loss VaR with risk specification P VaR , then we ought

to construct a portfolio whose returns distribution yields:

− VaR

P VaR = P(Y n < −VaR) = dF n (ξ )

In probability terms:

n (R) = P{R (n)

R} = P{all R i

R} = [F(R)] Similarly for the minimum statistic: Y 1 =R (1) = Min [R 1 , R 2 ,..., R n ], we have,

F 1 (R) = P{R (1) < R} = 1 − Pr{R (1) > R} = 1 − Pr{ all R i > R} = 1 − [1 − F(R)] n

and generally, for the j th statistic,

F (j) i (R) = n−i

i [F(R)] [1 − F(R)]

j =i

By deriving with respect to R, we have the distribution:

f j (R) =

f (R) [F(R)] j −1 [1 − F(R)] n− j ( j − 1)!(n − j)!

317 If we concentrate our attention on the largest statistic, then in a sample of size

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n , the cumulative distribution of the Max statistic is: F Y n (x) = [F R (x)] n . At the limit, when n is large, this distribution converges to Gumbel, Weibull or a Frechet distributions, depending on the parameters of the distribution. This fact is an essential motivation for their use in VaR analyses. These distributions are given

by: Gumbel: F Y (y) = exp(−e −y ), y ∈ I R

for y < 0(k < 0) Weibull: F Y (y) =

Frechet: F Y (y) = exp (−y −k ) for y > 0 (k > 0) The VaR is then calculated simply since these distributions have analytical cu-

mulative probability distributions as specified above and as the example will demonstrate.

Example: Weibull and Burr VaR

Assume a Weibull distribution for the Max loss, given by:

a e a −βx , x ≥ 0, ζ, a > 0; F(x) = 1 − e −βx where the mean and the variance are:

f (x/β) = aβx a−1

1 2 1 (x) = β Ŵ

2a +1 2 , var(x) = β Ŵ +1 −Ŵ +1

a a a Say that the parameter β has a Gamma probability distribution given by: g(β) =

δβ ν−1 e −δβ , a mixture (Burr) distribution results, given by:

VaR , we have:

VaR α −ν

P VaR = F(x ≥ VaR) = 1+ and therefore,

VaR = [δ{[P VaR ] −1/ν

The essential problem with the use of the extreme VaR approach, however, is that for cumulative risks or stochastic risk processes, these distributions are not applicable since the individual risk events are statistically dependent. In this sense, extreme distributions when they are applied to general time-varying stochastic portfolios have limited usefulness.

Example: The option VaR by Delta–Gamma

Consider an option value and assume that its underlying asset has a Normal distribution. For simplicity, say that the option value is a function of an asset price and time to maturity. A Taylor series approximation for the option price is

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,τ=T−t

The option’s rate of return is therefore:

2 p η P + θh/V t ;η=P t / V t Explicitly, we have then:

V ≈δR p η+ Ŵ R 2

V =ηδR P + 0.5ηŴ P t R P +n

V t which can be used to establish the first four moments of the rates of return R V of

the option:

Option ROR

Underlying R V R P

Mean:

Ŵσ 0.5 ˜ 2 P + ˜θn

Variance: ˜δ 2 σ 2 2 P 2 + 0.5 ˜Ŵ σ 4 P σ P Skewness:

In practice, after we calculate the Greek coefficients delta, gamma, theta and the variance of the underlying furnished by Risk Metrics, the four moments allow an estimation of the distribution. Such an approach can be generalized further to a portfolio of derivatives and stocks, albeit calculations might be difficult.

10.3.4 Copulae and portfolio VaR Measurement

In practice, a portfolio consisting of many assets may involve ‘many sources of risks’, which are in general difficult to assess singly and collectively. Empirical re- search is then carried out to specify the risk properties of the underlying portfolio. These topics are motivating extensive research efforts spanning both traditional statistical approaches and other techniques. Assume that the wealth level of a portfolio is defined as a nonlinear and differentiable function of a number of assets written as follows:

W t = F(P 1t , P 2t , P 3t ,..., P nt , B ) with P 1t , P 2t , P 3t ,..., P nt , the current prices of these assets, while B is a riskless

asset. The function F is assumed for the moment to be differentiable. Then, in a

319 small interval of time, a first approximation yields: ∂ F ∂ F ∂ F ∂ F

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W t = P 1t +

B ∂ P 1t

P 2t +···+

P nt

∂ B If we set N it = ∂ F/∂ P it to be the number of shares bought at t, we have a portfolio

∂ P 2t

nt

whose process is: W t =N 1t P 1t +N 2t P 2t +···+N nt P nt

where C is the investment in a riskless asset. Assume for simplification that the rate of return for each risky asset is α it

it / P it ) while volatility is σ 2 it

t ) of the portfolio with mean µ t

[ρijσitσjt]

i =1 ∂ Pit ∂ Pjt and, the VaR can be measured simply, as we saw earlier. This might not always be

possible, however. If the marginal distributions only are given (namely the mean and the variances of each return distribution) it would be difficult to specify the appropriate distribution of the multiple risk returns to adopt.

Copulae are multivariate distributions constructed by assuming that their marginal distributions are known. For example (see Embrecht et al., 2001), given two sources of risks (X 1 , X 2 ), each with their own VaR appropriately calculated, what is the VaR of the sum of the two, potentially interacting sources of risks (X 1 +X 2 )? Intuitively, the worst case VaR for a portfolio (X 1 +X 2 ) occurs when the linear correlation is maximal. However, it is wrong in general. The problem at hand is thus how to construct bounds for the VaR of the joint position and determine how such bounds change when some of the assets may be statistically dependent and when information is revealed, as the process unfolds over time. The Copula provides such an approach which we briefly refer to. For example, this approach can be summarized simply by a theorem of Sklar (see Embrechts

et al. , 2001), stating that a joint distribution F has marginals (F 1 , F 2 ) only if there is a Copula such that:

F (x 1 , x 2 ) = C [F 1 (x 1 ), F 2 (x 2 )]

Inversely, given the marginals (F 1 , F 2 ), the joint distribution can be constructed by choosing an appropriate Copula C. This procedure is not simple to apply, but recent studies have contributed to the theoretical foundations of this evolving area of study. For example, if the marginal distributions are continuous, then there is a unique Copula. In addition, Copulae structures can contribute to the estimation of multiple risks VaR. Embrecht, Hoing and Juri (2001) and Embrecht, Kluppelberg and Mikosch (1997) have studied these problems in great detail and the motivated reader ought to consult these references first. Current research is ongoing in specifying such functions and their properties.

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10.3.5 Multivariate risk functions and the principle of maximum entropy

When some characteristics, data or other information regarding the risk distribu- tion are available, it is possible to define its underlying distribution by selecting that distribution which assumes the ‘least’, that is the distribution with the greatest variability, given the available information. One approach that allows the defini- tion of such distributions is defined by the ‘maximum entropy principle’.

Entropy (or its negative, the negentropy) can be simply defined as a measure for ‘departure from randomness’. Explicitly, the larger the entropy the more the distribution departs from randomness and vice versa, the larger the negentropy, the greater a distribution’s randomness. The origins of entropy arose in statis- tical physics. Boltzmann observed that entropy relates to ‘missing information’ inasmuch as it pertains to the number of alternatives which remain possible to

a physical system after all the macrospically observable information concerning it has been recorded. In this sense, information can be interpreted as that which changes a system’s state of randomness (or, equivalently, as that quantity which reduces entropy). For example, for a word, which has k letters, assuming zeros

and ones, and one two, define a sequence of k letters, (a 0 , a 1 , a 2 ,..., a k ),

i = 2 and one j. The total number of configurations (or strings of k + 1 letters) that can be created is N where, N = 2 k (k + 1). The

logarithm to the base 2 of this number of configurations is the information I , or

I = log 2 N and, in our case, I = k + log 2 (k + 1). The larger this number I , the larger the number of possible configurations and therefore the larger the ‘random-

ness’ of the word. As a further example, assume an alphabet of G symbols and consider messages consisting of N symbols. Say that the frequency of occurrence of a letter is f i , i.e., in N symbols the letter G occurs on the average N i =f i N times. There may then be W different possible messages, where:

N ! W= N

i =1

The uncertainty of an N symbols message is simply the ability to discern which message is about to be received. Thus, if,

log(W ) =

p i log (1/ p i )

N →∞ N

which is also known as Shannon’s entropy. If the number of configurations (i.e. W ) is reduced, then the ‘information’ increases. To see how the mathematical and statistical properties of entropy may be used, we outline a number of problems.

(1) Discrimination and divergence

Consider, for example, two probability distributions given by [F, G], one the- oretical and the other empirical. We want to construct a ‘measure’ that makes

321 it possible to discriminate between these distributions. An attempt may be

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reached by using the following function we call the ‘discrimination information’ (Kullback, 1959):

In this case, I (F, G) is a measure of ‘distance’ between the distributions F and G. The larger the measure, the more we can discriminate between these distributions. For example, if G(.) is a uniform distribution, then divergence is Shannon’s measure of information. In this case, it also provides a measure of departure from the random distribution. Selecting a distribution which has a maximum entropy (given a set of assumptions which are made explicit) is thus equivalent to the ‘principle of insufficient reason’ proposed by Laplace (see Chapter 3). In this sense, selecting a distribution with the largest entropy will imply a most conservative (riskwise) distribution. By the same token, we have:

and the divergence between these two distributions is defined by:

F (x) J (F, G) = I (F, G) + I (G, F) = [F(x) − G(x)] log

dx

G (x) which provides a ‘unidirectional and symmetric measure’ of distribution ‘dis-

tance’, since J (F, G) = J (G, F). For a discrete time distribution ( p, q), we have similarly:

q i For example, say that q i , i = 1, 2, 3, . . . , n is an empirical distribution and say

that p i , i = 1, 2, 3, . . . , n is a theoretical distribution given by the distribution:

(1 − p)

n−i

then: n

which may be minimized with respect to parameter p. This will be, therefore, a binomial distribution with a parameter which is ‘least distant’ from the empirical

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distribution. For a bivariate state distribution, we have similarly:

q ij while, for continuous distributions, we have also:

as well as:

F (x, y) J (F, G) =

[F(x, y) − G(x, y)] log

dx dy

G (x, y) This distribution may then be used to provide ‘divergence-distance’ measures

between empirically observed and theoretical distributions. The maximization of divergence or entropy can be applied to constructing a multivariate risk model. For example, assume that a nonnegative random vari- able {θ} has a known mean given by ˆθ, the maximum entropy distribution for a continuous-state distribution is given by solution of the following optimization problem:

Max H = −

f (θ ) log [ f (θ )] dθ

subject to:

f (θ ) dθ = 1, ˆθ = θ f (θ ) dθ

The solution of this problem, based on the calculus of variations, yields an expo- nential distribution. In other words,

f (θ ) = e −θ/ ˆθ

When the variance of a distribution is specified as well, it can be shown that the resulting distribution is the Normal distribution with specified mean and specified variance. The resulting optimization problem is:

subject to:

f (θ ) dθ = 1, ˆθ = θ f (θ ) dθ, σ 2 = 2 (θ − ˆθ) f (θ ) dθ

323 which yields (as stated above) the normal probability distribution with mean and

VAR STATISTICS

variance θ,σ ˆ 2 use the divergence to minimize the distance between these distributions by the appropriate selection of the theoretical parameters. This approach can be applied equally when the probability distribution is discrete, bounded, multivariate distri- butions with specified marginal distributions etc. In particular, it is interesting to point out that the maximum entropy of a multivariate distribution with specified mean and known variance–covariance matrix turns out also to be a multivari- ate normal, implying that the Normal is the most random distribution that has a specified mean and a specified variance. Evidently, if we also specify leptokurtic parameters the distribution will not be Normal.

Example: A maximum entropy price process:

Consider a bivariate probability distribution (or a price stochastic process)

h (x, t), x ∈ [0, ∞) , t ∈ [a, b]. The maximum entropy criterion can be written as an optimization problem, maximizing:

[h(x, t)]

Subject to partial information regarding the distribution h(x, t), x ∈ [0, ∞) , t ∈ [a, b]. Say that at the final time b, the price of a stock is for sure X b while

initially it is given by X a . Further, let the average price over the relevant time interval be known and given by ¯ X (a,b) , this may be translated into the following constraints:

h (X a , a ) = 1, h(X b , b )=1 and

xh (x, t) dx dt = ¯X (a,b)

b−a

to be accounted for in the entropy optimization problem. Of course, we can include additional constraints when more information is available. Thus, the maximum entropy approach can be used as an ‘alternative rationality’ to constructing proba- bility risk models (and thereby modelling the uncertainty we face and computing its VaR measure) when the burden of explicit hypothesis formulation or the jus- tification of the model at hand is too heavy.

Problem: The maximum entropy distribution with specified marginals

Define the maximum entropy of the joint distribution by specifying the marginal distributions as constraints. Or

Max H = −

F (x, y) log [F(x, y)] dy dx

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subject to:

F 1 (x) = F (x, y) dy; F 2 (y) = F (x, y) dx

and apply the multivariate calculus of variations to determine the corresponding distribution.

10.3.6 Monte Carlo simulation and VaR

Monte Carlo simulation techniques are both widely practised and easy to apply. However, applications of simulation should be made critically and carefully. In simulation, one generates a large number of market scenarios that follow the same underlying distribution. For each scenario the value of the position is calculated and recorded. The simulated values form a probability distribution for the value of a portfolio which is used in deriving the VaR figures. It is clear that by using Monte Carlo techniques one can overcome approaches based solely on a Normal underlying distribution. While such distributions are easy to implement, there is an emerging need for better models, more precise, replicating closely market moves. But where can one find these models? Underlying distributions are essen- tially derived from historical data and the stochastic model used to interpret data. The same historical data can lead to very different values-at-risk, under differ- ent statistical models, however. This leads to the question: what model and what distribution to select? Monte Carlo simulation does not resolve these problems but provides a broader set of models and distributions we can select from. In this sense, in simulation the GIGO (garbage in, garbage out) principle must also be carefully apprehended. The construction of a good risk model is by no means

a simple task, but one that requires careful thinking, theoretical knowledge and preferably an extensive and reliable body of statistical information.