Lets consider the situation where a financial institution comprised of various divisions faces risk.
If this risk is defined as suffering a loss then the aggregate loss the institution faces is defined as S =
n
X
i=1
Xi,
where Xidenotes the loss of division i. The risk capital allocation problem arises when the insti-tution wishes to distribute the total amount of risk capital given by ρ(S) among its subdivisions.
In the literature a multitude of methods of capital allocation exist, varying from performance measurement approaches, capital allocations by optimisation and even in the context of coop-erative game theory.
An allocation method is called a risk capital allocation when it satisfies:
X
1∈N
Ki= ρ(S),
with Ki being the amount of capital distributed to division i for i = (1...N ) and ρ(S) the amount of risk capital based on the aggregate risk
The Euler allocation method
The Euler allocation method or Auman-Shapley value has been gaining popularity as a solution to the risk capital allocation property.
The Euler allocation method is denoted as:
KiE= ∂ρ(P
i∈NλiXi)
∂λi |λ(1...1), (3)
with risk measure ρ for which the partial derivative exists and random loss variable X ∈ X The Euler allocation principle is the gradient of the risk measure ( recall that a risk measure was defined as a mapping 2.1) and translates to the marginal contribution of division i to the aggregate risk of the firm. Therefore taking into account the dependence structure of the divisions and hedging the risk of the firmBoonen(2019)
Furthermore when the loss variable X has a Gaussian distribution X ∼ N (µ, σ) the Value-at-risk can be expressed as
V aRα(X) = µ + Φ−1(α)σ. (4)
Suppose that the losses of each division {Xi}i∈N follow a multivariate Gaussian distribution, it holds thatP
i∈NXi follows a Gaussian distribution as well, and as shown byBoonen(2019) the Euler rule is reduced to
KiE= µi+ Φ−1Cov(P Xi, Xi)
pvar(P Xi) , i ∈ N. (5)
Denault (2001) introduced the notion of coherence for risk capital allocation methods.
Definition 2.2. An allocation principle Π is coherent if for every allocation problem (N, ρ), the allocation Π(N, ρ) satisfies three properties:
1. No Undercut:
3. Riskless allocation For Π(N, ρ and i ∈ N such that Xi= c for c ∈ R, it hold that
Ki= c.
Denault (2001) shows that the Euler allocation method is coherent,as it satisfies the three properties given, and also results in the most preferable allocation method. ConverselyVan Gulick et al.(2012) andBoonen(2019) recognise that the Euler allocation principle is not continuous, which is a significant flaw.Boonen (2019) analyzes the volatility of the Euler rule under mild assumptions, such as normally distributed losses, and shows that the Euler rule can be very sensitive to measurement error due to the lack of continuity
The CTE allocation method
The CTE allocation method is a natural method which can be obtained from several theoretical backgrounds. Let the aggregate risk the firm faces be S =P
i∈NXi, the amount of capital Ki
allocated to division i is given by
Ki= E[Xi|S ≥ FS−1(α)]. (6)
From a game theoretical framework the CTE allocation principle was found to be a special case of the Euler allocation method byDenault(2001), hence it is a coherent allocation method and similarly regards the dependence between the divisions.
Additionally Dhaene et al.(2008) demonstrated that the allocation of risk capital based on the CTE method is justified in the sense of profit maximisation andDhaene et al.(2012) argues that the CTE method is optimal.
Dhaene et al. (2008) considers the CTE method for elliptical distributed risks and provide closed-form expressions.
Consider the case where the random variable X is normally distributed; X ∼ N (µ, σ) the CTE or expected shortfall is denoted as
ESp(X) = µ + σφ(Φ−1(α))
1 − α . (7)
The CTE allocation method as shown in Dhaene et al.(2008) can be expressed as
Ki= µi+φ(Φ−1(α)) 1 − α
Cov(P Xi, Xi)
pvar(P Xi) , (8)
where X:=(X1...Xn) has a multivariate Normal distribution with parameters (µ, σ, Σ), Lastly the CTE allocation method has also been studied for several heavy-tailed distributions regarding tail-risk. Vernic (2011), and Asimit et al. (2013) consider the Multivariate Pareto Distribution of the Second Kind for heavy tailed risks and capital allocation and Park and Kim (2016) study estimation methods for heavy tailed risk measures with Generalized Pareto distribution. These will be discussed further later on.
The τ -value method
In a more recent study Boonen (2020) presented a new allocation method called the τ -value method. Originally proposed byTijs et al.(1981) for the class of transferable utility games, the τ -value can be applied to risk capital allocation. As described in Boonen (2020) the τ - value allocation is a convex combination between the utopia allocation and worst-case allocation, with the utopia allocation being the best case allocation a certain division expects or rather the lower bound for the amount of capital allocated to division i and the worst-case allocation being the upper-bound for the amount of capital allocated to division i.
Let the utopia-allocation of division i be given by
Mi(X) = ρ X
Then the τ -value allocation is given by
τi(X) = (1 − a)Mi(X) + ami(X), ∀i ∈ N, α ∈ R. (11)
Recall that for any allocation principle to be a risk capital allocation method P
i∈Nτi =
Moreover Boonen (2020) argues that the τ -vale allocation method is the only allocation method to satisfy the following six properties: Translation Invariance,Scale Invariance, Mono-tonicity, Riskless portfolio, No Diversification, Continuity. Further detail and proof are found inBoonen (2020).
The following example demonstrates how each method of allocation previously discusses is calculated.
Example 1.
Consider a financial institution with 3 sub-divisions where the total amount of risk capital has to be distributed among these divisions. Hence N = {1, 2, 3} and set α = 0.995 when using the VaR and α = 0.99 when using the Expected shortfall or CTE. Let Xi denote the loss variable of division i and assume that X has a multivariate Gaussian distribution with
µ = (0, 0, 0), σ = (1, 1, 1) and Σ = are the stand alone risks of the division. The aggregate risk is then computed by: V aRα(X1+ X2+ X3). As mentioned earlier, if Xi∼ N (0, 1) then it holds thatP
iXi∼ N (0, σs), only now
σ2s= σ12+ σ22+ σ32+ 2σ1,2+ 2σ1,3+ 2σ2,3. Thus
with φ the density function of the standard Normal distribution and as before Φ−1 the inverse CDF function of the standard Normal distribution
Evidently, from these calculation the property of sub-additivity can be seen in clear effect.
The aggregate risk is remarkably lower than the sum of all stand-alone risks. Furthermore, the level of the chosen α also impacts the risk evaluation, especially when dealing with heavy tailed risks deviations between the Value-at-Risk and CTE are expected to be larger.
Secondly the Euler-allocation will be derived, with the VaR as risk measure, with α = 0.995.
Hence formula5 is used.
where Φ−1(0.995) denotes the 99.5-quantile of a standard normal distribution,pσ2s=√ 3 and
For the CTE-allocation method, which can be considered a special case of the Euler-allocation method when using the CTE as risk measure, α is set at 99% and equation 8 is used.
KiCT E= 2.665 ·cov(P3
i=1Xi, Xi) pσ2s , and accordingly:
K1CT E= 2.665 ·1.5√
3 = 2.308, K2CT E= 2.665 ·0.5√
3 = 0.769, K3CT E= 2.665 ·√1
3 = 1.538.
Lastly the τ -value allocation for each division will be calculated step-wise, for the reason that this method consists of components that have to be calculated separately. The τ -value allocation with both the VaR and expected shortfall will be calculated, respectively
1. The Utopia allocation: To calculate the utopia allocation of each division9 will be used.
First the Value-at-Risk is determined for each combination of divisions:
ρ(
2. The worst-case allocation: To calculate the worst case allocation10will be used.
Define S as a subset of N = {1, 2, 3}, thus the sets {∅}, {1},{2},{3},{1, 2},{1, 3},{2, 3}
and {1, 2, 3} are considered. In the next table all combinations of ρ(P
j∈SXj) −P
The worst-case allocation for division i is the minimum value of each row of the table above. Hence
m1= 2.575, m2= 1.753, m3= 2.573.
3. To τ -value allocation: In order to calculate the τ -value allocation, a has to be determined first. and by11the following are obtained:
τ1= 2.175, τ2= 0.733, τ3= 1.553.
Lastly the τ -value allocation is computed when the conditional tail expectation is used as risk measure. By following the same steps, the amount of capital allocated to each division is:
τ1CT E= 2.248, τ2CT E= 0.749, τ3CT E= 1.600.
For each of the allocation methods illustrated, the sum of the amount allocated to each division equals the aggregate risk capital (P
iXi). This equality is by definition of an allocation method. Secondly the proposed correlation between divisions is also reflected in the allocated amount of capital. Each method shows the same pattern of allocation amongst division 1,2, and 2. Since division 2 is uncorrelated to division 1 and negatively correlated to division 3, it gets rewarded with a low amount op capital for being a good hedge. Division 3 is positively correlated to division 1, but negatively correlated to division 2, due to this relationship its risk is diversified.
Furthermore it has been shown that the Value-at-risk is a coherent risk measure when risks are normally distributed, however this does not specifically hold when the underlying distribution is non-Gaussian and adding to this, the Value-at-risk has also been criticised for underestimating risk when risks present heavy tails. Additionally as mentioned earlier, Boonen (2019) showed that under smooth conditions, such as normally distributed risks, the Euler-allocation method can be quite volatile, which implies that this method could produce even more unstable results in the presence of tail risk. And lastly the τ -value method is still to be studied for extreme situations.