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Chapter 4. Numerical Methods for Value-at-Risk

Chapter 4. Numerical Methods for Value-at-Risk

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240



CHAPTER 4



. Numerical methods for value-at-risk



value-at-risk



FIGURE 4.1 The probability that a loss is greater than value-at-risk, the density of the shaded region,

is equal to 1 − .



in-depth, general, algorithmic, and mathematical discussions, we have a personal preference

for [Mor96a, Hul00, DP97].2

In financial markets, risk is caused by uncertainty about the value of an investment in

the future. The value of a portfolio is a function of a set of risk factors. Risk factor is the

generic term for a financial variable related to market prices of selected reference securities,

for example, equity indices, interest rates, foreign exchange rates, and commodity futures

prices. Market risk is the risk that the value of a portfolio declines as a consequence of

changes in the risk-factor values. Therefore, to model market risk we need to understand how

risk factors evolve over time.

Consistently with the hypothesis of absence of arbitrage, we will assume that the changes

in risk factors are random. Although historical data is of limited use to predict changes in risk

factors, it can be used to estimate statistical models to model risk factors and their correlations.

In our examples, we use stocks as elementary risk factors, although the methodology applies

to a wide range of financial instruments.

A simple formula for value-at-risk can be obtained in the case where an n × 1 vector

of relative changes R in the market risk factors is a multivariate normal random variable

with mean vector and covariance matrix C, and if one assumes that the change in portfolio value can be approximated by an affine function of the relative changes in the risk

factors:





+



T



R



(4.2)



Throughout this chapter we shall use superscript T to denote the transpose. Note: We are

using

to denote the change in portfolio, i.e.,

= t − 0 for a time lapse t, whereas

in the dot product is the vector of sensitivities w.r.t. the returns (i.e., the delta Greeks of

the portfolio), as defined later.



2



The Web site www.gloriamundi.org is an excellent source for information and links to papers on value-at-risk.



Numerical Methods for Value-at-Risk



241



Since

− = T R is a sum of normal random variables, then it is itself normal. The

distribution is determined by its mean and variance,



2



So







=E







=





2



=E



=E

T



T



R =



R−



2



T



ER =



=



T



T



C +



T



(4.3)



E R−



R−



T



=



T



C



(4.4)



is the random normal variable

=z



+



T



(4.5)



where z ∼ N 0 1 . Hence, inverting equation (4.1) while using N −1 1 −

the value-at-risk

VaR = N −1



T



C −







T



= −N −1



gives



(4.6)



where N −1 · is the inverse of the standard normal cdf.

The linear model with normal relative changes has a closed-form solution, but it suffers

from two serious problems. First, real-world returns have fatter tails than normal distributions. The model will therefore underestimate the likelihood of extreme returns, which

as a consequence may lead to inaccurate estimates of value-at-risk. Second, for portfolios

with derivatives, the change in value is a nonlinear function. The local error in the linear

approximation will therefore often be unacceptable, a property that is exacerbated by dynamic

hedging strategies that use the linearization to eliminate risk locally. To compute value-at-risk

for models that take these difficulties into account is a substantially harder task.

Let St be the process for a risk factor. Returns on St over the time horizon 0 t can be

defined either as arithmetic returns

Rt =



St − S 0

St

=

S0

S0



or as the log-return,

˜ t = log St − log S0

R



(4.7)



Log-returns have the advantage that one can aggregate returns over time by addition. In the

multivariate case, St is a vector of prices and returns are taken componentwise. Of course the

two are closely related. The difference,

˜t =

Rt − R



1

2



St

S0



2



+O



St

S0



3



is typically negligibly small for estimation purposes, and either type of return can safely be

approximated by the other. In the examples that follow, we choose log-returns.

Because the return is dimensionless, i.e., the quantity does not have a unit, return models

are preferred over models for prices. We consider a model in which the returns, sampled

at equally spaced points in time, form a sequence Ri i=1 of independent and identically

distributed random variables. This means that stock prices are discrete time Markov chains

with an infinite state space [Ros00]. Choosing different distributions gives different models

in this family.



CHAPTER 4



. Numerical methods for value-at-risk

Closing prices, 1997–2001



50

45

40

35

Price (CDN)



242



30

25



CTRa



20

15

BCE

10

5

1997



1998



1999

BCE and Canadian Tire



2000



2001



FIGURE 4.2 Daily closing prices for BCE and Canadian Tire from January 1997 to December 2001.



Visual inspection of historical time series gives clues on the key statistical properties.

Figure 4.2 shows the daily closing prices over 4 years for two Canadian stocks traded on the

Toronto Stock Exchange (TSX): Bell Canada Enterprises (BCE) and Canadian Tire (CTRa).

The scatter plot in Figure 4.3 shows that the daily returns form a cloud of samples around the

origin in what resembles a multivariate unimodal distribution. The time series can be divided

into segments with the same time span as the returns in the model Ri i=1 . For each time

interval, the relative return can be computed as

Ri =



Si − Si−1

Si−1



i=1



d



(4.8)



where Si−1 and Si are, respectively, the prices at the beginning and end of the time interval. Since the returns Ri i=1 in the model are independent and identically distributed, the

computed (observed) returns ri are viewed, rightly or wrongly, as independent samples from

the same distribution. After settling on a family of distributions for the random-walk increments, the parameters of this distribution can be estimated from the time series of returns

ri di=1 .

Many generalizations of the random-walk model have been proposed to correct shortcomings revealed in empirical studies; see, for instance, [CLM97]. Over time periods of a

few days one can make the simplifying assumption that the returns Ri i=1 are independent

and identically distributed. First, for time periods spanning more than a few years, the returns

are not identically distributed. To obtain the current reading and forecast for the volatility, it is standard practice either to use only recent data or to use a weighting scheme to

attribute a lesser weight to older data or to model the intertemporal dependencies by means

of more elaborate statistical models, such as ARCH and GARCH [Eng82, Bol86, Nel91,

Hul00].



4.1 Risk-Factor Models



243



FIGURE 4.3 Scatter plot of relative returns for BCE and Canadian Tire.



4.1 Risk-Factor Models

Recall that, in the random-walk model, returns are modeled as a sequence Ri i=1 of independent and identically distributed random variables. In this section, we discuss three different

instances of this model, three different alternatives for the distribution of the random variables: the normal random walk, the asymmetric Student’s t-distribution and the nonparametric

density estimator due to Parzen [Par61]. The methods will be generalized to the multivariate

case in the next section.



4.1.1 The Lognormal Model

In the lognormal model, the distribution of log-returns

Ri ∼ N

is normal with mean



i=1 2



2



(4.9)



and volatility . The mean can be estimated using the sample returns

=



1

d



d



ri

i=1



(4.10)



244



CHAPTER 4



. Numerical methods for value-at-risk



and the variance by

2



=



1

d−1



d



ri −



2



(4.11)



i=1



See, for instance, [LM86]. Some authors advocate using estimators that give more weight to

recent returns than to old ones (see, for example, [Mor96a, Hul00]).

To illustrate the performance, we estimate the parameters

and 2 for daily returns

for the BCE time series. Figure 4.4 shows the quantile-quantile plot3 for the fitted normal

distribution. It is clear that the normal model is a good approximation for small returns, but



FIGURE 4.4 Quantile-quantile plot for the normal random walk with parameters estimated from

4 years of daily returns for BCE.



3

A quantile-quantile plot is a method for comparing two distributions. Given a set of observations, we use it to

compare the empirical distribution and a distribution fitted to this data. Sorting the observations gives the empirical

cumulative distribution functions (cdfs). Each observation, which corresponds to a quantile, and the corresponding

quantile for the fitted distribution are marked in the plot. If the two distributions are the same, the points fall on

the diagonal reference line. Deviations from the diagonal line indicate that one distribution has fatter or thinner tails

with respect to the other. To learn more about this, the reader is referred to the relevant numerical project in Part II.



245



4.1 Risk-Factor Models



for both the negative and positive tails the distribution does not fit the data. Fat tails are

typical for stock returns; to estimate value-at-risk, where we need to compute tail quantiles,

the normal model is less suitable. The next two subsections explore different approaches to

construct random-walk models with more realistic tails.



4.1.2 The Asymmetric Student’s t Model

Student’s t-distributions have fat tails. The density for a t-distributed random variable is

+1

2



=



pT x



1+











x2



+1

2



x∈



(4.12)



2



= 0, and the variance for



the mean is



> 2 is

2



=



(4.13)



−2



The normalization factor involves the gamma function · . The degrees of freedom control

the fatness of the tails; as → , the distribution converges to the normal distribution.

An alternative to the normal model is to define a random walk with t-distributed increments. Since the fatness of the tails can be different for negative and positive returns, we

generalize this idea and let each random variable in the sequence Ri i=1 be distributed as

+ −2



1−



A = m+



− −2



B T+ +



+







B − 1 T−



(4.14)



The random variables T+ and T− are t-distributed with degrees of freedom + and − ,

respectively. The random variable B is a Bernoulli random variable; B takes the value 0 or 1

with probability 5. The random variables T− , T+ , and B are independent. We say that A is an

asymmetric Student’s t-distributed random variable. The density, figuratively a density made

up of a Student’s t pdf cut in half, is



px =



⎧ √



⎨ pT



⎩ pT



x−m





− −2











+

+ −2



x−m



1−





1−







+





− −2



if x ≤ m,



+

+ −2



(4.15)



if x > m.



Since the two regions each make up half of the density, m is the median of the distribution,

and, with a little algebra, it is easy to derive moment properties relative to the median. We

then have the following result, whose proof is left as an exercise.

Proposition 4.1. Suppose that − > 4 and + > 4. Then an asymmetric t-distributed random

variable, defined by equation (4.14), satisfies the following moment properties:

(i) The expectation is



1



= E A−m =







+ +1



2

+



2



1−





+ −2

+ −1







− +1



2





2







− −2

− −1









246



CHAPTER 4



. Numerical methods for value-at-risk



(ii) The second moment is

2



= E A−m



=



2



2



(iii) The second conditional moments are, for negative values,

2





= E A−m



2



A≤m =2



2



and, for positive values,

2

+



= E A−m



2



A>m =2



1−



2



(iv) The fourth conditional moments are, for negative values,

4





= E A−m



4



A≤m =2



3+



4 2



6

− −4



and, for positive values,

4

+



= E A−m



4



A>m =2



1−



4



3+



2



6

+ −4



Once the moment properties are known, estimating the parameters in the model is straightforward. The first step is to compute the median m of the observed returns ri di=1 by sorting

the samples and taking m to be the order-k value if d = 2k + 1 is odd, or the average of

the order-k and-(k + 1) values if d = 2k is even. Then find the sample estimate for the

second moment

2



=



d



1

d−1



ri − m



2



i=1



We then estimate the contribution to the second moment from the negative and the positive

halves. Let d = d− + d+ , where d− and d+ are the number of observations less than and

greater than m, respectively. Then

=



1

2 2 d−



ri − m



2



ri ≤m



Finally, using the sample estimates for the fourth moments,

4





=



2

d−



ri − m



4



4

+



and



ri ≤m



we can solve for estimates of the degrees of freedom





6



=

2



=



4



4 2



−3



+4



and



+



+



2

d+



ri − m



and



−,



6



=



4

+



2



4



4



ri >m



1−



2



−3



+4



The advantage of the asymmetric t model over the normal model is that, as illustrated

by the quantile-quantile plot in Figure 4.5, the tails of the empirical distribution can be

reproduced more accurately. However, this improvement comes at a price, since the pdf has

a discontinuity at the center. The jump is counterintuitive and the implementation of this

model is more difficult, but in comparison to the advantage of increased accuracy these are

minor concerns.



4.1 Risk-Factor Models



247



FIGURE 4.5 BCE quantile-quantile plot for the random walk model with the asymmetric t model.



4.1.3 The Parzen Model

A nonparametric density estimator is an alternative to using a parametric method, such as

either of the first two examples. Let ri di=1 be samples from a distribution with an unknown

pdf, p x . In [Par61] Parzen develops and analyzes a family of estimates of the form

pd x =



1

dh



d



K

i=1



x − ri

h



(4.16)



initially suggested by Rosenblatt in [Ros56]. In our examples, we use the weighting function [TT90]

Kx =



15

1 − x2

16



2



for x ≤ 1



(4.17)



248



CHAPTER 4



. Numerical methods for value-at-risk



Note that K x ≥ 0 is a kernel function that integrates to unity. Parzen shows that, if p x

is sufficiently smooth, pd x is asymptotically unbiased and, for an optimal sequence of

h-values, the mean square error converges to zero as4

E pd x − p x



2



= O d− 5

4



We refer to a random walk using the Parzen estimate (4.16) for the pdf as the Parzen model.

Similar to the asymmetric t model, the Parzen model can recreate the fat tails more

accurately than the normal model, and it also seems to have a slight advantage over the

asymmetric t model, as illustrated by the quantile-quantile plot in Figure 4.6. The advantage



FIGURE 4.6 BCE quantile-quantile plot for the random-walk model with the Parzen density estimate.



4

Parzen presents a theory for density estimates of the form of equation (4.16), with general weighting functions

K x . Let hd → 0 as the number of samples d → . He shows that density estimates of the form of equation (4.16)

converge (pointwise in a mean square sense) to a continuous pdf as d → , More precisely, given a sequence of

smoothing parameters hd d=1 with limd→ hd = 0 and limd→ dhd = ,



E pd x − p x



2



→0



as d →



The sequence of smoothing parameters giving optimal rate of convergence depends on both the point x and the pdf p x

as well as the weighting function K x . See Parzen [Par61] for examples of and details about general weighting functions.



4.1 Risk-Factor Models



249



of using a nonparametric model is that it does not rely on specific assumptions about the

shape of the density. There are three disadvantages to the Parzen model. First, the optimal

smoothing parameter h is unknown. While experimenting with different stocks, we have

found that taking h equal to the standard deviation works well.5 Second, for our choice of

weighting function, the density estimate has compact support. However, the support covers

the region of interest for value-at-risk calculations, so it should have a minor influence on the

result. Third, evaluating equation (4.16) or the corresponding cumulative distribution function

(cdf) for different values of x is expensive for large samples. In our implementation, we

avoid summing over all sample points by using cubic splines to approximate the cdf and

the pdf.



4.1.4 Multivariate Models

So far we have only considered models for the return on a single risk factor. In general,

portfolios depend on many risk factors. Therefore we must extend the one-dimensional

random-walk models, presented in the previous sections, to the multivariate case.

In the multivariate random walk, Ri i=0 is a sequence of n -valued vectors of random

variables. The random vectors are independent and identically distributed. The difficulty in

constructing a realistic multivariate model is that returns on the risk factors are typically

dependent, as exemplified by Figure 4.7. To approximate the dependence structure without

introducing an overly complex model, we restrict our attention to multivariate models where

the random vectors Ri i=1 satisfy

Ri = A−1 Xi + b



(4.18)



Moreover, we assume that the random vector X has independent components and the pdf is

a product of one-dimensional density functions:

p x = p1 x1 · · · pn xn

We postpone the discussion about how to choose the linear transformation, i.e., the matrix A

and the vector b, to Section 4.3, after discussing portfolios of derivatives.

To find a stochastic process to model stock prices in continuous time is a more difficult

problem. Returns are often modeled by stochastic differential equations (SDEs). As discussed

in Chapter 1, Brownian motion is the natural continuous-time generalization of a random

walk with normal increments. In this model, the return process is a constant-coefficient SDE,

dR = dt + dWt . Like the normal model for stock prices, geometric Brownian motion

underestimates the likelihood of large returns: It does not have fat tails.

Many different types of continuous-time models have been proposed and studied in the

literature, in particular for pricing derivatives. If the returns are a stationary Markov process,

then, for example, the sequence ri di=1 of historical returns can be used to find an estimate

for the transition density — the time-dependent probability density p r t representing the

density for the return r at time t. Figure 4.8 shows the Parzen estimate for the transition

density for the stock BCE. A good model has a transition density that is close to this estimate.



5

This choice of h may work well in our examples, but it is not a satisfactory solution in general since a fixed

smoothing parameter does not give convergence as the number of samples d → . The estimate converges for

a sequence of smoothing parameters that decrease to zero as the number of samples increases (see [Par61] for

details).



250



CHAPTER 4



. Numerical methods for value-at-risk



FIGURE 4.7 Principal components superimposed on the scatter plot for the returns on BCE and CTRa.



FIGURE 4.8 Parzen estimate for BCE to the transition density.



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