On the Basel Liquidity Formula for Elliptical Distributions

A justification of the Basel liquidity formula for risk capital in the trading book is given under the assumption that market risk-factor changes form a Gaussian white noise process over 10-day time steps and changes to P&L are linear in the risk-factor changes. A generalization of the formula is derived under the more general assumption that risk-factor changes are multivariate elliptical. It is shown that the Basel formula tends to be conservative when the elliptical distributions are from the heavier-tailed generalized hyperbolic family. As a by-product of the analysis a Fourier approach to calculating expected shortfall for general symmetric loss distributions is developed.


Introduction
As a result of the fundamental review of the trading book (FRTB) (BCBS, 2013) a new minimum capital standard for the trading book has emerged (BCBS, 2016). Under this standard banks are now required to calculate a liquidity-adjusted expected shortfall risk measure on a daily basis. This calculation is carried out at both the level of the whole trading book and the level of individual desks using an aggregation formula that is based on the concepts of liquidity horizons and square-root-of-time scaling.
Every risk factor affecting the value of positions in the trading book or desk is assigned to a unique liquidity horizon LH j which may be 10, 20, 40, 60 or 120 days. These horizons are conservative estimates of the amount of time that would be required to execute trades that would eliminate the portfolio's sensitivity to changes in these risk factors during a period of market illiquidity. For example, risk factors for the equity price risk of largecap stocks are assigned to the shortest horizon of 10 days; equity volatility risk factors for large-cap stocks are given a risk horizon of 20 days; risk factors for structured credit instruments have the longest liquidity horizon of 120 days.
The liquidity formula reflects the prevailing method of risk calculation in the banking industry in which changes in P&L for trading book positions are modelled in terms of sensitivities to risk factors. Expected shortfall charges are calculated with respect to shocks to risk factors with particular liquidity horizons while other risk factors are held constant. To make the calculation explicit, we give the formula and notation as published on page 52 of the revised capital standard (BCBS, 2016).
• let T = LH 1 denote the so-called base liquidity horizon of 10 days.
• Let ES T (P ) denote the expected shortfall at horizon T for a portfolio P with respect to shocks to all risk factors to which the positions in the portfolio are exposed.
• Let ES T (P, j) denote the expected shortfall at horizon T for a portfolio P with respect to shocks to the risk factors which have a liquidity horizon of length LH j or greater, with all other risk factors held fixed.
The liquidity-adjusted expected shortfall is (1) The first objective of this paper is to provide a principles-based derivation of this formula that relates it to the concept of expected shortfall as a risk measure applied to a loss distribution or P&L distribution. Most practitioners know that an assumption of normality underlies the formula. We make it precise that the formula can be justified by assuming that risk-factor changes over time steps equal to the base liquidity horizon form a multivariate Gaussian white noise with mean zero and portfolio losses are all attributable to first-order (delta) sensitivities to the risk-factor changes.
The second and major objective of the paper is to analyse the formula under the more general assumption that risk-factor changes have a multivariate elliptical distribution. This allows us to consider some particular cases with heavy tails and tail dependencies that might be considered more realistic models for market risk-factor changes.
Many results in QRM continue to hold when multivariate normal assumptions are generalized to multivariate elliptical assumptions. In particular, when losses are linear in a set of underlying elliptically-distributed risk factors, aggregation of risk measures across different business lines, desks or risk factors can generally be based on a common formulaic approach, regardless of the exact choice of elliptical distribution; see Chapter 8 of McNeil et al. (2015). The difference in the current paper is that aggregation takes place, not only across risk factors, but also across time and therefore a 'central limit effect' takes place. We will show that (1) is in fact a conservative aggregation rule for the popular generalized hyperbolic family of heavier-tailed elliptical assumptions and we will give a generalization of the rule that holds for all elliptical distributions.
As a by-product of our analyses we also demonstrate a new approach to calculating VaR and expected shortfall for symmetric distributions with a known characteristic function. This approach is particularly useful in cases where we take convolutions of elliptically distributed random vectors and lose the ability to write simple closed-form expressions for their probability densities.
We present all ideas in terms of the standard probabilistic approach to risk measures. Losses (or P&L variables) are represented by random variables L. Expected shortfall (ES α ) and value-at-risk (VaR α ) at level α are risk measures applied to L. If F L denotes the distribution function of L and q α the corresponding quantile function, they are given by VaR α (L) = q α (F L ) and ES α (L) = 1 1−α 1 α q u (F L )du. If F L is continuous then the formula ES α (L) = E(L | L VaR α (L)) also holds.

Justifying and extending the Basel liquidity formula
Let (X t ) be a d-dimensional time series of risk-factor changes for all relevant risk factors and assume that these are all defined in terms of simple differences or log-differences. We interpret X t+1 as the vector of risk-factor changes over the time step [t, t + 1]. In practice this time step will be equal to the base liquidity horizon of 10 days.
For h ∈ N, the risk factor changes over the time step [t, t + h] are given additively by Without loss of generality let the risk calculation be made at time t = 0. We make the following assumptions.
Assumption 1. (i) The risk-factor changes (X t ) form a stationary white noise process (a serially uncorrelated process) with mean zero and covariance matrix Σ.
(ii) Each risk factor may be assigned to a unique liquidity bucket B k defined by a liquidity horizon h k ∈ N, k = 1, . . . , n.
(iii) In the event of a portfolio liquidation action the loss (or profit) attributable to risk factors in bucket B k is given by where b k is a weight vector with zeros in any position that corresponds to a risk factor that is not in B k .
Assumption 1(iii) contains the linearity assumption and adopts the pessimistic view that the full liquidity horizon h k is required to remove the portfolio's sensitivity to all the risk factors in liquidity bucket B k .
Under these assumptions we compute the portfolio loss L over the maximum time horizon h n , which is the time required to remove the portfolio's sensitivity to all risk factors. It follows from Assumption 1(ii) and (iii) that where β k = n j=k b j and h 0 = 0. The vector β k contains the weights for all risk factors in the union of liquidity buckets B k ∪ · · · ∪ B n .

Let us write
. . , n for the summands in the final expression in (3). These are uncorrelated by Assumption 1(i) and we may easily calculate that where the final step follows because (2) implies that We now introduce random variables for k = 1, . . . , n. These represent losses attributable to all risk factors in the union of liquidity buckets B k ∪ · · · ∪ B n over the liquidity horizon h 1 . Note that the L k and L (k) variables differ (unless k = 1). Since var(L (k) ) = h 1 β ′ k Σβ k , we obtain from (4) the formula It may be noted that the presence of positive correlation between the variables L k in (4), caused by serial correlation in the underlying risk-factor changes X [h k−1 ,h k ] , would tend to lead to the left-hand side of (6) being larger than the right-hand side. Negative correlation would lead to it being smaller.

The Gaussian case
Suppose that (X t ) is a Gaussian process; in this case (X t ) is actually a strict white noise (a process of independent and identically distributed vectors). It follows that L k ∼ N (0, (h k − h k−1 )β ′ k Σβ k ) and the L k are independent for all k. Thus, by the convolution property for independent normals, Moreover, we clearly have , φ denotes the density of the standard normal distribution and Φ −1 (α) denotes the α-quantile of the standard normal distribution function Φ (see McNeil et al., 2015, Chapter 2). It follows from (6) which is the proposed standard formula for the trading book (1) rewritten in our notation.

An extension to the formula for elliptical distributions
In this section we assume a centred elliptical distribution for the risk-factor changes, which subsumes the multivariate normal distribution as a special case. In addition to Assumption 1 we assume that the following holds.
Assumption 2. (i) The process (X t ) is a multivariate strict white noise (an iid process).
Assumption 2(i) may seem strong but in practice we assume that (X t ) is a process of 10day returns so that the iid assumption, while unlikely to be true, is less problematic than for daily financial returns. The assumption is required in order to analyse convolutions of elliptically distributed random vectors with different characteristic generators.
Assumption 2(ii) means that X t = AY t for some matrix A ∈ R d×d satisfying Ω = AA ′ and some random variable Y t with characteristic function given by φ(s) = E(e is ′ Yt ) = ψ(s ′ s) for a function of a scalar variable ψ. Y t is said to have a spherically symmetric distribution, which is written Y t ∼ S d (ψ). It is important to note that Ω is not the covariance matrix of X t unless the covariance matrix of Y t is the identity matrix; in general we have Σ = var(Y )Ω where Y ∼ S 1 (ψ). The class of elliptical distributions contains a number of particular distributions which are popular models for financial returns including the multivariate Student t and the symmetric generalized hyperbolic distributions. See Fang et al. (1990) and McNeil et al. (2015) for further details of these distributions.
We need three key properties of an elliptical distribution for our calculation. Let X ∼ E d (0, Ω, ψ) andX ∼ E d (0, Ω,ψ) be independent elliptically-distributed variables with the same dispersion matrix Ω and possibly different characteristic generators ψ and ψ.
We will use (9) and (11) to find the characteristic functions of elliptical ramdom vectors under linear combinations and convolutions respectively. The property in (10) shows that we have some discretion in how we represent the characteristic generator of an elliptical random variable in terms of its characteristic generator and its scaling.
For α > 0.5 the expected shortfall of L is related to the expected shortfall of the variables where c α,ψ L represents the ratio of expected shortfall to standard deviation for L and c α,ψ 1 is the equivalent ratio for a univariate spherical variable Z ∼ S 1 (ψ 1 ).
It may be easily verified that when ψ(s) = exp(−s/2) (the Gaussian case), the characteristic function φ(s) = ψ L (s 2 ) implied by (12) is the characteristic function of the normal distribution in (7). In this case the constants c α,ψ L and c α,ψ 1 are identical.
When the risk factors have a heavier-tailed distribution than normal we expect that c α,ψ L c α,ψ 1 , due to the central limit effect, so the Basel liquidity formula should give an upper bound.

Calculating the scaling ratio in practice
We turn to the problem of calculating the ratio r α := c α,ψ L /c α,ψ 1 when the underlying risk factors have an elliptical distribution with generator ψ. To compute c α,ψ 1 we calculate the ratio ES α (Z)/ sd(Z) for a univariate spherical random variable Z with characteristic generator ψ 1 = ψ h 1 . To compute c α,ψ L we calculate the ratio ES α (L)/ sd(L) for a univariate spherical variable L with characteristic generator given by (12).
In general we will not be able to calculate ES α (Z) and ES α (L) from the probability densities of Z and L, since these typically do not have simple closed forms for the distributions of interest. In the following section we give results that can be used to compute expected shortfall directly from the characteristic function of a spherical random variable.

Calculating expected shortfall by Fourier inversion
A univariate spherical random variable Y ∼ S 1 (ψ) is symmetric about the origin with a real-valued even characteristic function given by φ Y (s) := ψ(s 2 ). We give a general result that applies to univariate random variables that are symmetric about the origin.
Theorem 2. Let Y be symmetrically distributed about the origin with an integrable char-acteristic function φ Y (s). Let −∞ < a < b < ∞. Then the following formulas hold: bs sin(bs) + cos(bs) − as sin(as) − cos(as) Proof. The characteristic function φ Y (s) of a random variables that is symmetric about the origin is real-valued and even. If φ Y is integrable then the density exists and the standard Fourier inversion formula for the characteristic formula yields The formula (16) for the distribution function is obtained from a well-known representation of the distribution by Gil-Pelaez (1951). To derive (17)  and (17) follows.
These formulas permit the accurate evaluation of VaR α (Y ) and expected shortfall using one-dimensional integration. Calculation of VaR α (Y ) for α > 0.5 is accomplished by numerical root finding using (16). If E|Y | < ∞ for the distribution in question, then expected shortfall is defined and it can be calculated by setting a = VaR α (Y ) and computing the limit

The case of generalized hyperbolic distributions
We will apply Theorem 2 to the family of symmetric generalized hyperbolic (GH) distributions. This is a very popular family for modelling financial returns and there are many useful sources for the properties of these distributions including Barndorff-Nielsen (1978), Barndorff-Nielsen and Blaesild (1981), Eberlein (2010) and McNeil et al. (2015).
is a vector of independent standard normal variables and W is an independent positive random variable with a so-called generalized inverse Gaussian (GIG) distribution W ∼ N − (λ, χ, κ); see formula (A.1) in the Appendix for the density of this distribution. The vector Y has a spherical distribution Y ∼ S d (ψ), and any component Y has a univariate spherical distribution Y ∼ S 1 (ψ), for a characteristic generator ψ that depends on the particular choice of the parameters λ, χ and κ. An elliptical model of the kind described in Assumption 2(ii) is obtained by taking X = AY for A ∈ R d×d and satisfies X ∼ E d (0, Ω, ψ) where Ω = AA ′ . X is said to have a d-dimensional symmetric generalized hyperbolic (GH) distribution.
To carry out our calculations it suffices to consider the single component Y . The variance of Y satisfies var(Y ) = E(W ) and an explicit formula for the case where χ > 0 and κ > 0 is given in (A.3). A formula for the characteristic function φ Y is given in (A.4) and the characteristic generator of the elliptical family can be inferred from the identity ψ(s 2 ) = φ Y (s).
We consider four special one-parameter cases of this distribution resulting from particular choices of the parameters λ, χ and κ of the GIG distribution: 1. The student t distribution with degree of freedom ν. This corresponds to the case where κ = 0, λ = −ν/2 and χ = ν or where W has an inverse gamma distribution W ∼ IG(ν/2, ν/2). In this case var(Y ) = ν/(ν − 2), provided ν > 2, and the characteristic function is given by (A.5) in the Appendix.
2. The variance gamma (VG) distribution. This corresponds to the case where χ = 0 or where W has a gamma distribution W ∼ Ga(λ, κ/2). Without loss of generality we set the scaling parameter κ = 2 so that var(Y ) = λ. The corresponding characteristic function is given by (A.6).
3. The normal-inverse-Gaussian (NIG) distribution. This corresponds to the case where λ = −1/2. The distribution can be reparameterized in terms of θ = √ χκ and χ; the latter parameter can be treated as a scaling parameter and set to one. The variance is then var(Y ) = θ −1 and the characteristic function is given by (A.7).
4. The hyperbolic (Hyp) distribution. This corresponds to the case where λ = 1. The distribution can be reparameterized in exactly the same way as the NIG distribution. The variance is var(Y ) = θ −1 K 2 (θ)/K 1 (θ) and the characteristic function is given by (A.8).

Summary of the steps in the calculation
We return to the problem of calculating the scaling ratios r α = c α,ψ L /c α,ψ 1 in (13) when the underlying risk-factor returns have symmetric distributions in the multivariate generalized hyperbolic family.

Design of experiments
In order to calibrate our model distributions, we use 2132 observations of adjusted daily closing prices for the S&P500 index, from 17.7.2007 to 31.12.2015, which have been converted to two-weekly log-returns (conforming approximately to 10 trading days, the base liquidity horizon required under FRTB).
We fit the various distributions discussed in Section 3.2 to the 10-day return data using the R package ghyp. Table 1 gives the estimated shape parameters for the distributions of interest; scale parameters are not required in our analysis. Note that we also confirm that the calculations for the Gaussian case yield a ratio of 1, as a check on our implementation.
We carry out two experiments: • In the first, we consider two risk factors, one in B 1 with a liquidity horizon of 10 days (h 1 = 1) and the other in B 2 with a liquidity horizon of 20 days (h 2 = 2). The dispersion matrix Ω is either taken to be the identity Ω = I 2 (no correlation) or a correlation matrix with correlation ρ = 0.5.
We present values of c α,ψ 1 , c α,ψ L as well as the scaling ratio r α for various confidence levels α. The case of two risk factors is reported in Table 2 and the case of five risk factors is reported in Table 3.

Results
In both tables it is clear that the scaling ratios are less than one for all non-Gaussian cases meaning that the Basel liquidity formula is indeed conservative when the risk factors have a multivariate elliptical distribution from one of the four generalized hyperbolic sub-families considered in Section 3.2 and Table 1.
The second experiment with five liquidity buckets leads in general to smaller values for the scaling ratios than the first experiment with two buckets. Thus the degree of conservatism of the formula increases with the number of liquidity buckets. This is in line with the increase in the central limit effect as we aggregate over more time periods.
Introducing correlation leads to an increase in the constants c α,ψ L and hence an increase in the scaling ratio. In other words, the weaker the correlation, the more conservative the liquidity formula. To understand why this is the case, note that the constants c α,ψ L depend on the characteristic generator ψ L in (12) and hence on the set of values {β ′ k Ωβ k , k = 1, . . . , n}. By considering formula (4) we can think of these as the relative weights attached to each of the n liquidity buckets. When ρ = 0 these weights are (5, 4, 3, 2, 1) but when ρ = 0.5 they are (15,10,6,3,1). The intuition is that, in the second case, the first few liquidity buckets dominate more in the convolution calculation and the central limit effect is mitigated.
Considering the different generalized hyperbolic special cases we see that the ratios are usually largest for the t distribution followed by the other three distributions; the exact ordering depends on the confidence level α used in the calculation. In other words, use of the Basel liquidity formula is least conservative in the case of t and more conservative for the other distributions.
When we look at the confidence level of α = 0.975 which is the level used in the new capital standard (BCBS, 2016) the normal inverse Gaussian (NIG) distribution leads to the highest level of conservatism. This distribution is often a plausible model in market risk applications. The ratio in the case where n = 5 and ρ = 0 is 0.837 which means that the Basel liquidity formula would tend to overstate capital by around 19.4%.  Table 2: Constants c α,ψ 1 , c α,ψ L and ratios rα in the experiment with 2 risk factors.

Conclusion
We have presented evidence that the Basel liquidity formula tends to lead to conservative capital charges when financial risk factors come from heavier-tailed elliptical distributions.
The Basel formula is clearly a heavily stylized formula and makes a number of crude assumptions. We have concentrated on the effect of changing the underlying distribution of the risk factors when portfolio sensitivities are linear. However, there are other important effects we have not considered which will have an influence on the ability of the formula to capture risk. In particular, the true effect of risk-factor changes on portfolio risk is likely to be highly non-linear over the kind of time horizons we consider. Moreover, as we  Table 3: Constants c α,ψ 1 , c α,ψ L and ratios rα in the experiment with 5 risk factors.
have already noted, positive serial correlation between losses over different sub-intervals [h k−1 , h k ] of the overall liquidity horizon [0, h n ] will tend to lead to a tendency towards underestimation which may counteract the central limit effect.
It should also be noted that there are many further layers of conservatism built into the new system of risk charges for the trading book, such as the requirement to calibrate the model to stress periods and the requirement to adjust the calculation to understate the possible diversification effects across risk factors.
Nonetheless it is important to be clear about the workings of the formula and the extent to which it may be interpreted as a principles-based approach to the measurement of market risk. Our study should be understood as a contribution to the clarification of this issue. density of the latter is where K λ denotes a Bessel function of the third kind. The characteristic function of Y is given by and the variance by var(Y ) = E(W ).
We first consider the case where χ > 0 and κ > 0. In this case the variance of Y is and the characteristic function is We next consider the case of a Student t distribution which corresponds to κ = 0, λ = −ν/2 and χ = ν. In this case W has an inverse gamma distribution W ∼ IG(ν/2, ν/2) and var(Y ) = E(W ) = ν/(ν − 2), provided ν > 2. The characteristic function should be interpreted as the limit of (A.4) as κ → 0. Substituting the density of an inverse gamma distribution into (A.2) yields φ Y (s) = ∞ 0 e − 1 2 s 2 w ( 1 2 ν) ν/2 Γ( 1 2 ν) w − ν 2 −1 e − 1 2 νw −1 dw = (νs 2 ) ν/4 2 ν/2−1 Γ( 1 2 ν) The special case of variance gamma (VG) corresponds to χ = 0; without loss of generality we set the scaling parameter κ = 2. In this case W has a gamma distribution W ∼ Ga(λ, 1) and var(Y ) = E(W ) = λ. The characteristic function in this case should be interpreted as the limit of (A.4) as χ → 0. Substituting the density of a gamma distribution W ∼ Ga(λ, 1) for f W in (A.2) we obtain φ Y (s) = Two further special cases are the normal inverse Gaussian (NIG) and hyperbolic distributions. In both cases we fix the parameter λ and reparameterize the GH distribution in terms of θ = √ χκ and κ; the latter then appears only as a scaling parameter and can be set to one.