La biodiversidad: un entramado entre lo vivo y el recurso natural

In practice, computing portfolio ETL is done through the Monte Carlo method. We hypothesize a parametric model for the multivariate distri- bution of financial returns, we fit the model, and then we generate a large number of scenarios. From the generated scenarios, we compute scenarios for portfolio return. Employing formula (2.4), we calculate portfolio ETL at a specified tail probabilityǫ.

We can regard the generated scenarios as a sample from the fitted model and thus the computed ETL in the end appears as an estimate of the true ETL. The larger the sample, the closer the estimated ETL is to the true value. If we regenerate the scenarios, the portfolio ETL number will change and it will fluctuate around the true value. Figure 2.1 illustrates this phe- nomenon for the standard normal distribution with ǫ= 0.01, for which the true value ET L0.01(X) = 2.665. This stochastic variability is an issue in- herent in the Monte Carlo method and cannot be avoided. In this context, the Monte Carlo method can be viewed as a numerical method of comput- ing portfolio ETL when the hypothesized multivariate model does not allow portfolio ETL to be computed analytically. In this section, we discuss the asymptotic distribution of the estimator in (2.4) which we can use to deter- mine approximately the variance of (2.4) when the number of scenarios is large.

Before proceeding to a more formal result, let us check what intuition may suggest. If we look at equation (2.4), we notice that the leading term

8

A formal proof can be found in Rockafellar and Uryasev (2002). The reasoning in Rockafellar and Uryasev (2002) is based on the assumption that the random variable describes losses while in equation (2.9), the random variable describes the portfolio return or payoff.

500 1,000 5,000 10,000 20,000 50,000 100,000 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3 3.1 3.2 AVaR

Figure 2.1: Boxplot diagrams of the fluctuation of the ETL at ǫ = 1% of the standard normal distribution based on 100 independent samples. Thex- axis represents number of scenarios. (Reproduced from Figure 7.4 in Rachev, Stoyanov and Fabozzi (2008))

is the average of the smallest observations in the sample. The fact that we average observations reminds of the central limit theorem (CLT) and the fact that we average the smallest observations in the sample suggests that the variability should be influenced by the behavior of the left tail of the portfolio return distribution. Basically, a result based on the CLT would state that the distribution of the ETL estimator becomes more and more normal as we increase the sample size. Applicability of the CLT however depends on certain conditions such as finite variance which guarantee certain regularity of the random numbers. If this regularity is not present, the smallest numbers in a sample may vary quite a lot as they are not naturally bounded in any respect. Therefore, for heavy-tailed distributions we can expect that the CLT may not hold and the distribution of the estimator in such cases may not be normal at all.

The formal result in Stoyanov and Rachev (2007b) confirms these ob- servations. Taking advantage of the generalized CLT, we can demonstrate that

Theorem 15. Suppose thatX is random variable with distribution func- tion F(x) which satisfies the following conditions

• xαF(x) =L(x)is slowly varying at infinity, i.e. limx→∞L(tx)/L(x) =

1, _∀t >0.

• R_−∞0 xdF(x)<∞

• F(x) is differentiable atx=qǫ where qǫ is the ǫ-quantile of X.

Then, there existcn n= 1,2, ..., such that for any 0< ǫ <1,

c−_n1ET L[ǫ(X)−ET Lǫ(X)

_w

→Sα∗(1,1,0) (2.10) in which →w denotes weak limit,1< α∗= min(α,2), andcn=n1/α

∗

L0(n)/ǫ

where L0 is a function slowly varying at infinity andET L[ǫ(X) is computed

from a sample of independent copies of X according to equation (2.4).

This theorem implies that the limit distribution of the ETL estimator in (2.4) is necessarily a stable distribution totally skewed to the left. In the context of the theorem, we can think of X as a random variable describing portfolio return. If the index α governing the left tail ofX is α ≥2, then the above result reduces to the classical CLT as in this caseα∗ = 2 and the limit distribution is normal. This case is considered in detail in Stoyanov and Rachev (2007a).

Stable distributed returns

In this section, we consider the case in which portfolio return distribution is a stable law with parameter 1 < α < 2. Under this assumption, α∗ =

−60 −4 −2 0 2 4 6 0.1 0.2 0.3 0.4 0.5 0.6 0.7 n = 250 n = 1,000 n = 10,000 n = 100,000 S 1.5(1,1,0)

Figure 2.2: The density of the sample ETL as the number of scenarios increases together with the limit stable law,X _∈S1.5(1,0.7,0). (Reproduced from Figure 2 in Stoyanov and Rachev (2007b))

α and thus the limit distribution of the ETL estimator is also a stable law with parameter α. That is, we are not in the case of the classical CLT. We carry out a Monte Carlo study in order to see for how many scenarios the limit distribution Theorem 15 is sufficiently close to the real distribution of the estimator. We chooseX ∈S1.5(1,0.70) and we generate

2,000 samples from the corresponding distribution the size of which equals n= 250; 1,000; 10,000; and 100,000. Figure 2.2 shows the density of the random variable in the left part of equation (2.10) and how it approaches the limit distribution, i.e. the right part of (2.10), as the number of scenarios increases.

Student’s t distribution

If we assume that portfolio return has a Student’st distribution, then the degree of freedom parameterν determines the tail thickness. In the notation of Theorem 15, α =ν and, therefore, if ν _≥2, then the asymptotic distri- bution of the ETL estimator is normal. Even though it is normal, the tail thickness influences the convergence rate which means that the acceptance of the limit distribution as an approximate model decreases asν decreases. The rationale is that the higherν is, the more regular the random variable is.

No tail truncation With tail truncation ν ǫ= 0.01 ǫ= 0.05 ǫ= 0.01 ǫ= 0.05 3 70000 17000 12000 4000 4 60000 9000 11500 3600 5 50000 7000 11000 3300 6 23000 4500 11000 3200 7 14000 4200 10500 3100 8 13000 4100 10000 3000 9 12000 4000 10000 3000 10 12000 3900 10000 3000 15 11000 3850 10000 2950 25 10000 3800 10000 2900 50 10000 3750 10000 2900 ∞ 10000 3300 10000 2900

Table 2.1: The number of observations sufficient to accept the normal dis- tribution as an approximate model whenX has Student’st distribution for different values ofνandǫ(Reproduced from Table 1 and Table 4 in Stoyanov and Rachev (2007b).)

The first two columns of Table 2.6.4 show how the number of scenarios changes when ǫ= 0.01 and ǫ= 0.05 which correspond to the two standard choices of 95% and 99% for the confidence level of the ETL. The numbers in the table are calculated by generating 2,000 samples of a given size which we use to approximate the distribution of the left part of equation (2.10). We use the Kolmogorov distance in order to check if the hypothesis of a normal distribution can be accepted. We notice that the minimum number of scenarios increases from about 10,000 whenν = 25 to 70,000 when ν= 3 forǫ= 0.01. For additional details, see Stoyanov and Rachev (2007a).

The effect of tail truncation

In the introduction, we noted that one way to deal with the issue that a heavy-tailed model may have an infinite volatility is to apply a tail trunca- tion. This means that we truncate the tails of the distribution very far away from the center, e.g. at 0.1% and 99.9% quantiles. This is not so artificial as it may seem. Stock exchanges usually have regulations according to which trading stops if a given market index drops too much. Essentially, this reg- ulation does not allow panick in the market to result in arbitrarily large losses. If the truncation is done far away from the center of a distribution, the general shape of the distribution is preserved and the descriptive power of the model does not deteriorate.

the stochastic stability of the ETL estimator in two ways. First, if the portfolio return distribution is such that the limit law of the ETL estimator is stable, after tail truncation the limit law becomes normal. However, the convergence rate to the normal distribution depends on how far away from the center we truncate — the deeper we go into the tails, the slower the convergence rate. Second, if the limit law of the ETL estimator is normal, tail truncation increases the convergence rate. The last two columns of Table 2.6.4 illustrate this observation when portfolio returns are assumed to have Student’s t distribution for a truncation threshold equal to the 0.1% quantile. We notice that the minimum number of scenarios drops from 70,000 to 12,000 when ν = 3 and ǫ = 0.01. Stoyanov and Rachev (2007b) provide additional examples and details.

For an additional discussion of ETL including geometric interpretations, see Rachev, Stoyanov and Fabozzi (2008) and Stoyanov et al. (2008).