As shown earlier, the Value-at-risk is a convenient risk-measure and has favourable traits. How-ever when the underlying risks are fat-tailed or when a division has a high potential for large losses, the Value-at-risk may underestimate the risk and risk manager insufficiently allocate capital among their constituents.
When dealing with heavy-tailed distributions, the Extreme Value Theory (EVT) has emerged as an useful tool in risk management. The EVT has been widely adapted and studied when considering tail-tailed distributions for risk modelling. For instance, Embrechts et al. (2001), Yamai et al.(2002).
For the risk capital allocation problem multivariate distributions are of keen importance, where the tails of the marginal distributions and dependence structure among variable are of significance
Extreme Value Theory
The EVT provides useful information on the distribution of random variable X over a certain threshold.
Let X be a random variable with distribution function F (x), for fixed u Fu(x) = P[X − u ≤ x|X > u] = F (x + u) − F (u)
1 − F (u) , x ≥ u,
is the distribution function of the variable X over the threshold u. The EVT states the approx-imation of Fu for high values of u. Nevertheless, Fu is only known if F (x) is known.
Moreover the Pickands-Balkema-De Haan Theorem states that Fuconverges to the Generalised-Pareto distribution as u tends to infinity. Hence,
Fu(x) ≈ Gα,σ(x), Gα,σ(x) = 1 − (1 −ξx σ )−1ξ.
The Generalised Pareto distribution has two parameters: ξ the tail index and σ the scale pa-rameter. The tail index specifies the fatness of the tail, so for large ξ the distribution has a fat tail. The scale parameter σ indicates how dispersed the values are.
This EVT has proven to be extremely useful in risk management, Yamai et al. (2002) de-scribes in detail how the extreme value theory is used to tackle tail-risk of risk measures such as Value-at-Risk and Expected Shortfall when dealing with fat-tailed distributions.
Copula
In risk capital allocation the correlation between divisions is paramount. When disregarding the correlation, financial institutions may allocate capital to their constituency erroneously or even has an incorrect assessment of the aggregate risk.
When modelling the dependence between random variables, copulas emerged as a mechanism that helps understand the relationships among multivariate outcomes.Frees and Valdez(1998) have done groundbreaking research into copulas and showed their applicability in the finance and insurance industry.
A copula is defined as the distribution function of a vector with standard uniform marginal distributions:
C(u1, u2) = P[U1≤ u1, U2≤ u2].
Now suppose there are N random variables, in general not assumed to be independant, with distribution functions F1(x1, F2(x2), ..., Fn(xN) then
C(F1(x1), F2(x2), ..., FN(xN)) = F (x1, x2, ..., xN). (12) The copula defines the multivariate distribution function of the random variables.
A well known copula is the Gaussian copula
C(u, v) = ΦΣ(F−1(u), F−1(v),
with Φ the distribution function of the multivariate standard normal distribution with correlation matrix Σ and F1−1, F2−1 the inverse CDF of the random variables X1and X2.
The Gaussian copula is easy to use for generating dependant variables in simulation studies.
Ross (2013) has shown how any random variable X with marginal distributions Fi and whose joint distribution is given by the Gaussian copula can be generated by the following 4 steps:
1. Generate random variable W1, W2, .. from a multivariate Normal distribution with means equal to 0, variances equal to 1 and Cov(Wi, Wj) = Cov(Xi, Xj)
2. Compute Φ(Wi) for all i.
3. Let Fi(Xi) = Φ(Wi)
4. and obtain Xi by solving Xi= Fi−1(Φ(Wi))
The Pareto distribution of the second kind
As shown by the EVT the excess-loss can be approximated by the Generalised Pareto distribution for a large enough boundary. This in turn shows that in real-world scenarios, where losses present heavy-tails, the loss distribution can be approximated by the Pareto distribution of the Second Kind. This distribution has received considerable attention is the recent years in the context of risk management, risk capital allocation, insurance pricing and optimal reinsurance retention by Park and Kim(2016),Vernic(2011),Chiragiev and Landsman(2007),Guillen et al. (2013) and Asimit et al.(2010).
A random variable X has a Pareto distribution of the Second Kind, note X ∼ P II(µ, σ, α) if its probability density function and cumulative distribution function can be written as:
f (x) = a σ(x − µ
σ + 1)−(a+1), F (x) = 1 − (x − µ
σ + 1)−a,
with location parameter µ, scale parameter σ, and shape parameter α. The shape parameter indicates the heaviness of the tail. Hence a large α indicates a fat-tail. Furthermore the random vector X has an n-variate Pareto distribution of the Second kind M PnII(Θ, α), with α > 0, and Θ = (µ, σ), µi ∈ R, σi> 0 for all i ∈ {1, ..., n} if its density and distribution function are
For the marginal moments it holds that
E(Xi) = µi+ σi
a − 1, V ar(Xi) = ( σi
a − 1)2 a
a − 2, fora > 2. (13) The tail-function F = P[X > x] is then given by
FX(x)
Asimit et al. (2010) noted that the tail function of the multivariate Pareto distribution can be expressed in terms of the Clayton survival copula, from which it follows that the dependence between the marginal risks is signified by the shape parameter α. For the Multivariate Pareto
distribution the correlation for a > 2 is given by Corr[Xi, Xj] = α1. Thus the parameter α indicates the tail-fatness as well as the dependence between the losses of the divisions. If α increases the tails get less fat and the losses are less correlated and if α decreases the tails get heavier and the losses have a stronger correlation. Additionally the Pareto distribution only allows for positively correlated variables
Park and Kim (2016) provide closed form expressions for the Value-at-Risk and ES under the Generalised Pareto distribution. as mentioned before, the second kind Pareto distribution is a special case of the GPD, hence the VaR is given by
V aRα(X) = σ[(1 − α)−1/a− 1], (15)
An interesting study by Guillen et al.(2013) has shown that the Pareto distribution of the second kind is a special case of the Beta distribution of the second kind. Guillen et al.(2013) shows that when considering the beta distribution of the second kind, risk measures such as the Value-at-Risk and expected shortfall have a closed form expression.
Consider the random variable X. Then X ∼ B2(p, a, σ) is said to have a second kind beta distribution if its density function is given as
f (x; p, a, σ) = xp−1
σpB(p, a)(1 + x/σ)p+a,
where p and a are positive shape parameters and σ a scale parameter. Furthermore if p=1, X has a Pareto distribution of the second kind. The density function with p=1 reduced to
f (x; 1, a, σ) = x1−1
σ1B(1, a)(1 + x/σ)p+a B(p,a) is the Betal function and B(1,a)=1/a. Hence
f (x; 1, a, σ) = a
σ(1 + x/σ)−(1+a)
Guillen et al.(2013) also shows that the VaR and ES have the following form.
Let X ∼ B2(p, a, σ) then the Value-at-risk for level α is given by,
Where IB denotes the Incomplete ratio Beta function, IB−1 denotes the inverse of the In-complete ratio Beta function or also known as the quantile function of the Beta distribution of the first kind and E[X] = a−1σp for a > 1
Figure 1: Probability density of second kind Pareto distribution(red) and second kind Beta distribution (black) from randomly generated values
Additionally these result can be extended to the multivariate case where.Guillen et al.(2013) Shows that the Sum of second kind Beta distributed variables, again have a Beta distribution of the second kind. These results a useful for both analytic and numerical computations of the VaR, ES and even the τ -value allocation.
Vernic (2011) and Chiragiev and Landsman(2007) have been able to provide formulas for the computation of the Euler rule by expressing the n-variate case recursively in terms of the (n-1)-variate case and by using divided differences, respectively.Vernic (2011) shows that the Euler rule can theoretically be calculated with,
KiE=E(XiI(S > V aR0.99)) FS(V aR0.99) .
If the underlying loss distribution has a multivariate Pareto distribution of the second kind, then
E(XiI(S > V aR0.99)) =
n Y
k=0
(a + n − k)
n X
k=0
(−1)k
k!(n − k)!(a + k − 1) × σa+k
(V aR0.99+ σ)a+k−1,
and
FS(V aR0.99) =
n Y
k=0
(a + n − k)
n−1 X
i=0
(−1)i (a + i)i!(n + i − 1)!
σ
V aR0.99− σ
a+i .
These formulas do make it possible to theoretically calculate the amount of capital allocated to each division, nevertheless for large n the computations become technical and require analytical methods to solve.
3 Methodology
3.1 Capital Allocation and Pareto risks
To analyse the the impact of a heavy tailed distribution on the method of allocation, the τ -value method and the Euler allocation method will be compared based on simulated losses.
The Conditional Tail Expectation with significance level α = 0.99 will be used as risk measure, since it allows for a natural allocation of capital, namely the CTE method, and has become the prescribed risk measure in the banking sector.
Emperically the expected shortfall is calculated as:
ES0.99(Xi) = mean(X1), for X > V aR0.99(Xi)
ES0.99(X1+X2+X3) = mean([X1+X2+X3]) for [X1+X2+X3] > V aR0.99([X1+X2+X3])
Intuitively, the ES is the average of the values larger than the VaR.
For a sample size of 1000, V aR0.99 equals the 990th observation of the ordered losses and the ES then equals the average of the 10 highest observations.
A firm with 3 divisions will be considered where the losses Xi of each division will be generated in the following manners:
• Multivariate Pareto distribution of the second kind: Losses will be simulated under the assumption of a multivariate Pareto distribution of the second kind. X∼ M P II((1, 1, 1, ), 3).
Hence Xi∼ P II(1, 3) for i = (1, 2, 3). In this case the divisions are identical and the cor-relation between the divisions is indicated by the shape parameter a.
• Gaussian copula Marginal losses will be generated from the univariate second kind Pareto distribution with shape parameter 3 and scale parameter 1, under the assumption that their joint distribution is given as a Gaussian copula with correlation matrix as before
in the example Σ =
To compare the effects of a heavy tailed distribution, losses will also be simulated from a multivariate Normal distribution where the mean and variance will be set equal to those of the marginal distributions second kind Pareto distribution and the correlation matrix will reflect the same dependence structure as both cases given above.
A sample size of 1000 will be generated, which corresponds to losses of approximately 40 years.
To compute the allocation base on the Euler rule of each division first the of the sum losses will be calculated X1+X2+X3. Secondly the 10 largest observations of the sum will be identified and the corresponding 10 losses of the individual divisions that yield these 10 realizations of the
sum. After this, the amount allocated to each division equals the the mean of the 10 individual losses identified.
In order to compute the τ -value allocation for each division first the Utopia allocation for each division has to be calculated with the ES as risk measure. From9it follows that
Mi= ES0.99(X
alli
Xi) − ES0.99(X
j6=i
Xi)
Secondly the worst case allocation is determined for each division. Using equation 10 and the determined utopia allocations. A matrix identical to the table given in the example above 2.2will be defined, where each row contains the result of
ES0.99(X
i∈S
(Xi) − X
i∈S\{i}
Mi
and the worst case allocation for division j equals the minimum value of row j.
To illustrate this consider division 1. The set S indicates the possible combinations of di-visions. Thus for division 1, S is each possible combination of division, including division 1.
This gives the following combinations: {1, 2, 3}, {1, 2}, {1, 3}, {1} of the set S. The worst-case allocation is then the minimum value of:
S = {1, 2, 3} →ES0.99
And lastly the τ -value allocation will be computed using11with
a = ES0.99(X1+ X2+ X3) − M1− M2− M3
m1+ m2+ m3− M1− M2− M3
The Simulation will be run 10.000 times , yielding 10.000 allocations for each division under each method and each loss generating distribution.
4 Results and Discussion
In this section the results of the simulated capital allocation proportions are presented. Rather than focusing on the actual amount allocated to each of the division, the proportion of total risk capital allocated to each division is given. In each case the Expected shortfall ( or CTE ) is used as risk measure at a probability level of 99%.
Table 1and Table2 present the descriptive statistics for the compositions of capital alloca-tions based on the Euler allocation method and τ -value method, when the underlying risk, in this case losses, have a second kind Pareto distribution and Normal distribution. Additionally Figure4.1graphically shows the comparison between Normally distributed losses and losses that have multivariate Pareto distribution of the second kind, for each of the allocation methods.
Lastly, Figure 3,4, and 5 exhibit the capital allocation compositions based on the Euler method and for the τ -value, when individual losses have Pareto distribution of the second kind and their joint distribution is given by a Gaussian copula.
4.1 Multivariate second kind Pareto losses vs. Multivariate Normal losses
From the simulation results, it is clear that capital allocation can be quite sensitive to tail risk.
In Table 1 the amount of capital allocated to each division is presented as a fraction of the whole, along with the standard deviations.
Evidently, when dealing with heavy-tailed losses, such as the second kind Pareto distribution, the amount of capital allocated to each division fluctuates strongly compared to the allocated amount of capital when losses follow a normal distribution, which generally does not have a heavy tail. When the underlying risk presents heavy tails, there is a higher probability for large losses. This causes the estimation of risk to be more volatile due to the variability in the losses.
The ES takes the mean of the losses above a certain threshold, and with large and infrequent losses, the estimated risk is also expected to vary. Regardless, as argued byYamai et al.(2002) and many that followed, when a division is prone to heavy-tailed losses the ES gives a better estimation of risk and thus results in more accurate capital allocations compared to using the VaR as risk measure.
It’s clear that both allocation methods are subject to tail-risk. Nonetheless, the τ -value method appears to be less sensitive to tail risk than the Euler allocation method
Congruent to the results presented inBoonen(2019), Euler allocation method appears to be very sensitive to measurement error. Although he compared the Euler method to the propor-tional allocation rule. The τ -value allocation method performs significantly better. The standard deviation decreases faster than that of the Euler allocation method when the thickness of the tail decreases. Figure4.1depicts the spread of the allocated amount of capital, supporting these results graphically.
ParetoII distribution Normal distribution Division 1 Division 2 Division 3 Division 1 Division 2 Division 3
Euler- mean 0.333 0.334 0.333 0.333 0.333 0.334
allocation std. dev a=3 0.0816 0.0813 0.0816 0.0310 0.0310 0.0310
std.dev a=10 0.7636 0.7572 0.07522
τ -Value mean 0.333 0.334 0.333 0.333 0.333 0.333
std. dev. a=3 0.0661 0.0658 0.0656 0.0195 0.0193 0.0193
std.dev a=10 0.4360 0.04326 0.04321
Table 1: Descriptive statistics of the capital allocation composition based on the Euler-allocation method and τ -value method with the ES as risk measure. The simulation is run 10.000 times and the mean of the portion of risk capital allocated to each division with their standard deviation is shown. a corresponds to the shape parameter of the second kind Pareto distribution
Figure 2: Comparison of allocated portion of risk capital to Division 1 based on the Euler method (left) and τ -value (right) with ES as risk measure under the Assumption that losses have a second kind Pareto distribution( gray) and Normal distribution (green)
The τ -value method allocates capital to each division by taking a convex combination of the utopia allocation and worst-case allocation which in this case basically reduces to a combination of two different values of the ES, therefore the volatility is induced by the chosen risk measure, whereas the Euler rule takes the derivative of the chosen risk measure, meaning that it is directly
impacted by small changes in the loss distribution. Thus in this case, when dealing with large and infrequent losses, the impact is expected to be even larger.
Another noteworthy result is the composition of the risk capital, which is the same for each method and each distribution. A reasonable explanation for this phenomenon is the uniformity of the divisions. The multivariate Pareto distribution of the second kind only allows for equally correlated random variables induced by the shape parameter a, meaning that the correlation between the risks of each divisions is equivalent.
However this usually not the case in real-world scenarios. In reality the correlation between divisions may differ or even be negative, which impacts the risk capital composition and thus requires different model specifications.