pTAS distributions with application to risk management
- Matthias Fischer^{1}Email author and
- Kevin Jakob^{2}
https://doi.org/10.1186/s40488-016-0049-9
© The Author(s) 2016
Received: 2 October 2015
Accepted: 16 June 2016
Published: 18 July 2016
Abstract
The family of positive tempered α-stable (pTAS) or sometimes also tempered one-sided α-stable distributions dates back to Tweedie (1984) and Hougaard (1986) who discussed it in the context of frailty distribution in life table methods for heterogenous populations. The pTAS family generalizes the well-known gamma distribution and allows for heavier tails depending on the parameter α. Because of this property, pTAS distributions appear to be useful in the context of risk management. Against this background, the contribution of his work is three-fold: Firstly, we summarize the properties of the pTAS family. Secondly, we describe its numerical implementation and illustrate the functions by means of R examples in the Appendix. Thirdly, we empirically demonstrate that this family can be successfully applied in risk management. Concretely, applications to credit and operational risk are given.
Keywords
Mathematics Subject Classification
Derivation and properties of the pTAS family
1.1 Evolution of the pTAS family
1.2 Properties of pTAS distributions and some remarks
Different Parametrizations: \(\mathcal {P}(K)\leftrightarrows \mathcal {P}(T)\) with β=α, α=θ ^{ α }·δ/(1−α), λ=(1−α)/δ. \(\mathcal {P}(H)\leftrightarrows \mathcal {P}(T)\leftrightarrows \mathcal {P}(P)\) using \(\delta =\mu \left (\frac {1-\alpha }{\mu \nu ^{2}}\right)^{1-\alpha }\), \(\gamma =\left [\frac {\mu \cos (\pi \alpha /2)}{\alpha }\right ]^{\frac {1}{\alpha }}\left [\frac {1-\alpha }{\mu \nu ^{2}}\right ]^{\frac {1-\alpha }{\alpha }}\) and \(\theta =\frac {1-\alpha }{\mu \nu ^{2}}\)
Reference | Parameter | Abbreviation |
---|---|---|
(α,δ,θ) | \(\mathcal {P}(H)\) | |
(α,γ,δ) | \(\mathcal {P}(T)\) | |
(α,μ,ν) | \(\mathcal {P}(P)\) | |
(β,α,λ) | \(\mathcal {P}(K)\) |
The pTAS family can be considered as an alternative to the generalized inverse Gaussian (GIG) family (see Koudou and Ley (2014) for a review of this family) which itself arises from the inverse Gaussian (IG) distribution by exponential tilting. Barndorff-Nielson and Shephard (2001) introduce the four-parameter, so-called modified stable (MS) distributions which nests both GIG family and pTAS family (setting κ=0.5 and ν=−κ in their notation), where ν denotes the fourth (additional) parameter.
Implementation issues
This chapter describes numerical algorithms which can be used for the standard functions (e.g. density and distribution function, random number generation and quantile function) of a pTAS random variable. The algorithms are described independently of a specific programming language. However, we give some examples how to use the pTAS R-package, which contains the described functions, in Appendix 4. The R-package is not yet published but can be obtained from the authors by request.
2.1 Density and distribution function
Since the pdf and cumulative distribution function (cdf) of the pTAS family are in general not available in closed form, but only via the Laplace transform, we have to use numerical inversion techniques to calculate values of the density or distribution function. Therefore, we follow a general approach given by Abate et al. (2000), which will be described briefly in this section. For further details please refer to the mentioned literature.
where \(i=\sqrt {-1}\) and b is a real number greater than all singularities of \(\hat {f}\). The second transformation is achieved via a change of variables.
This equation gives rise to three kinds of error, a discretization error caused by the step size h, a truncation error and a round-off error because of the numerical operations performed by calculating the series.
The discretization error can be reduced by interpreting (2.1) as a Fourier series of a function g(t):=e ^{−b t } f(t), whereby the coefficients of the series itself can be expressed via the values of the Laplace transform. The error then can be reduced by increasing the ratio b/h. The round-off error can be controlled by choosing \(h=\frac {\pi }{lt}\) and \(b=\frac {A}{2lt}\) for some \(l\in \mathbb {N}>0\) and \(A=\frac {2l}{2l+1}m\ln 10\), where in turn m>0 should be chosen such that 10^{−m } is close to the machine precision. In order to improve the truncation error, the so called Euler summation technique can be applied, which gives us another parameter \(n\in \mathbb {N}\) determining the number of coefficients being (fully) respected for the summation. Similarly to Abate et al. (2000) we recommend the following default values: l=1, n=38 and m=11.
Therefore, with just a minor modification, the same method used to approximate the pdf can be used to approximate the cdf of a pTAS distribution.
2.2 Quantiles and random numbers
To calculate the quantile for a given level p∈(0,1), Ridout (2008) proposed a modified version of Newton’s method to invert the cdf. Because, especially in the upper and lower tail of the distribution, the pdf may be close to zero, which may lead to an iteration step outside a given interval \([t_{\min },t_{\max }]\). In this case, the Newton step is replaced by a bisection method.
Given a tolerance ε>0, a maximum number of iterations \(N_{\max }\in \mathbb {N}\) and a closed interval \([t_{\min },t_{\max }]\) such that \(\exists t\in \ [t_{\min },t_{\max }]:\,\left |F(t)-p\right |\leq \epsilon \) the following algorithm can be used to calculate the p-quantile of a pTAS distribution:
In order to determine the initial values of the lower and upper bounds, we propose to store some values for t and F(t) on a grid t _{ i=1,…,N } depending on mean and standard deviation. Using precalculated values for \(t_{\min }\) and \(t_{\max }\) increases performance especially when quantile transformations should be performed very often.
If multiple quantiles should be calculated at the same time for different probabilities p _{ m=1,…,M }, the probabilities should be sorted in ascending order before the transformation. Assuming that p _{1}≤…≤p _{ M } we can apply Algorithm 1 on the first element p _{1} and use the resulting value t _{1}=F ^{−1}(p _{1}) as starting point (and also as lower bound) for the next element p _{2}.
For random number generation the inverse probability integral transform method can be applied very easily by using Algorithm 2 based on sorted uniform drawings.
2.3 Estimating parameters
In cases with no unique solution, the resulting parameters may lead to very different distributions (e.g. with different tails) which, in the context of risk management, can cause different risk figures. Therefore, we recommend to choose the estimation method with respect to the proposed area of application and to compare estimation results between different methods if they are not unique. Otherwise, using inappropriate parameter estimators may lead to an increase of model risk.
Application to risk management
After showing how the pTAS distribution can be implemented and parameters can be estimated, we provide two extensive risk management applications of the pTAS family.
3.1 Credit risk
Given these counterparty specific information, a crucial task of a credit portfolio model is to model the changes of PD_{ i } over the specified time horizon by taking into consideration systematic influences due to country or business dependencies as well as idiosyncratic changes. For our example we use the CreditRisk^{+} model, which is available via the GCPM R-package on CRAN. We give a short introduction on those model parts, which are necessary for this example. For detailed information please refer to Credit Suisse First Boston International (1997) or Gundlach and Lehrbass (2004).
As mentioned above, the gamma distribution was chosen for performance issues. From an economic point of view, this assumption may be questionable. Instead of a gamma distribution, one can also use a lognormal distribution, which is more heavy tailed and therefore more conservative compared to a gamma distribution. However, the lognormal distribution also possesses only two parameters. Therefore, using mean and variance ^{4} to parametrize the distribution of S _{ k } the heaviness of the tail can not be controlled explicitly. If we use a pTAS distribution instead, we can fit the sector distribution to mean and variance of observed PD changes and still have control over the tail via parameter α.
For our example, we use portfolio of 5,000 counterparties belonging to ten different sectors. The data used to estimate the sector distributions are monthly PD values over 10 years, which are estimated via a Merton type model (see Merton (1974)) from marked data (i.e. stock prices and liabilities) for over 20,000 corporations and aggregated on sector level. The portfolio as well as the underlying data are explained in more detail in Fischer and Jakob (2015) and Jakob and Fischer (2014). We estimate pTAS distributions for each sector by using MLE and the empirically observed skewness and compare the results on portfolio risk figures with the original setting (i.e. gamma distributions parametrized via mean and variance) and a framework using the lognormal distribution and the Weibull distribution. Because of the relatively small number of observations (10 years of monthly data), we do not use the kurtosis or quantile estimation methods in this example. For the Monte Carlo simulation of the CreditRisk^{+} model the GCPM R-package is used. The dependency between sector variables are modeled via a Student-t copula with 3.8 degrees of freedom, estimated using a maximum likelihood approach^{5}.
Skewness values and estimated parameters for different sector distributions
Sector | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
S _{1} | S _{2} | S _{3} | S _{4} | S _{5} | S _{6} | S _{7} | S _{8} | S _{9} | S _{10} | ||
Skewness | Empirical | 2.59 | 1.74 | 2.24 | 2.10 | 1.74 | 2.46 | 2.44 | 2.16 | 1.84 | 2.57 |
Weibull | 0.98 | 0.52 | 0.84 | 0.63 | 0.37 | 0.93 | 1.05 | 0.70 | 0.62 | 1.54 | |
Gamma | 1.29 | 0.97 | 1.19 | 1.04 | 0.87 | 1.25 | 1.34 | 1.10 | 1.04 | 1.69 | |
lognormal | 2.21 | 1.56 | 2.00 | 1.71 | 1.38 | 2.13 | 2.31 | 1.81 | 1.69 | 3.13 | |
pTAS (skewness) | 2.59 | 1.74 | 2.24 | 2.10 | 1.74 | 2.46 | 2.44 | 2.16 | 1.84 | 2.57 | |
pTAS (MLE) | 4.57 | 2.92 | 3.68 | 3.73 | 2.74 | 4.73 | 5.62 | 3.43 | 3.43 | 5.68 | |
α | pTAS (skewness) | 0.67 | 0.62 | 0.64 | 0.67 | 0.67 | 0.66 | 0.62 | 0.66 | 0.61 | 0.51 |
pTAS (MLE) | 0.84 | 0.80 | 0.81 | 0.84 | 0.81 | 0.85 | 0.86 | 0.81 | 0.82 | 0.83 |
The VaR is considerably higher
Figure 4 shows the pdf of the portfolio loss distribution together with vertical lines indicating the Value at Risk (VaR) for level τ, which is simply the quantile of the portfolio loss distribution. The VaR is considerably higher if we use a pTAS distribution which considers the skewness or which is estimated via MLE for the sector distribution compared to the standard case of a gamma distribution, which only accounts for mean and variance. Because financial institutions typically use higher values for τ (e.g. τ=0.999) to calculate the economic capital which is necessary to cover unexpected losses, the use of a simple gamma distribution may imply a significant amount of model risk. In our case, the VaR_{0.999} rises by around 12 % if we use pTAS distributions based on skewness and up to 28 % if we use MLE. Please note, that the effect in general depends on the sectors (i.e. business lines and countries) of the portfolio as well as the data used for estimating the sector distributions.
3.2 Operational risk
OpVaR for different confidences level and distributions
Percentile \(\rightarrow \) | 0.95 | 0.99 | 0.995 | 0.999 | 0.9995 |
---|---|---|---|---|---|
lognormal | 40,937,057 | 76,509,385 | 100,584,286 | 187,533,641 | 247,986,179 |
Weibull | 37,542,382 | 47,459,865 | 51,335,798 | 60,440,362 | 64,322,549 |
pTAS | 33,554,021 | 44,883,975 | 49,657,499 | 60,450,462 | 65,182,740 |
gamma | 32,023,064 | 41,335,245 | 45,109,838 | 53,586,965 | 57,257,060 |
gen.gamma | 24,836,520 | 31,636,053 | 34,333,082 | 40,596,749 | 43,033,552 |
Goodness-of-fit statistics
Test statistic | AD2UP | p-value | KS | p-value |
---|---|---|---|---|
lognormal | 2.6582 | 0.080 | 0.8235 | 0.060 |
Weibull | 7.8001 | 0.024 | 1.7422 | 0.005 |
pTAS | 2.7823 | 0.682 | 0.6585 | 0.739 |
gamma | 3.1287 | 0.077 | 0.8612 | 0.197 |
gen.gamma | 2.6755 | 0.167 | 0.5674 | 0.697 |
Finally, Fig. 7 also depicts the simulated portfolio loss for the OpRisk data set.
Conclusion
Within this article, we discussed the family of positive tempered α-stable distributions, which is a flexible distribution family and well suited to model both light and semi-heavy tailed data on the positive half-axis. Besides the derivation of the family and a summary of certain characteristics, which are relevant for practical applications, we provided an overview of the existing literature on this family. Furthermore, we introduced algorithms that can be used to implement the basic functionalities of the pTAS family (i.e. density and distribution function, calculation of quantiles and random numbers generation), which are also available via the pTAS R-package. By applying the pTAS distribution in the field of credit and operational risk we show that this distribution is more flexible and provides a better fit to empirical data compared to other competitors which are often used. Therefore, as in case of credit risk, the pTAS distribution can help to reduce the amount of model risk or as in case of operational risk, to reflect the risk more adequately which in turn helps banks to allocate economic capital more appropriately. Beyond the two given examples, the pTAS distribution may be also used to model (stock) returns and therefore being beneficial for market risk management as well.
Endnotes
^{1} Please refer to Schiff (1999, Theorem 4.3) for further details.
^{2} Besides counterparties’ defaults also changes in a counterparties creditworthiness (so-called rating migrations) cause a portfolio loss. However, for reasons of simplicity we restrict this example to default risk only.
^{3} In 1997, when the CreditRisk^{+} was published, this was a major advantage over simulative models, which use a Monte Carlo simulation to estimate the loss distribution. However, nowadays computers are much more powerful and Monte Carlo simulations are widely used.
^{4} The parametrization based on mean and variance is a standard method within the CreditRisk^{+} model. In order to ensure that \(\mathbb {E}\left (\overline {\text {PD}}_{i}\right)=\text {PD}_{i}\), which is a common assumption, we have the condition that \(\mathbb {E}(S_{k})=1\) for all k.
^{5} For readers interested in the topic of copulas within credit portfolio models, we refer to Jakob and Fischer (2014) and Fischer and Jakob (2015).
^{6} The Basel Committee have prescribed guidance for three types of methods for the calculation of capital requirement for operational risk. Those are the Basic Indicator Approach, the Standard Approach and the Advanced Measurement Approach. The latter is the most sophisticated of the approaches and this is what this section is about.
^{7} Please note that for the GoF-test for left-truncated samples the distribution of the statistic is not parameter-free, and the p-values and the critical values are obtained by means of Monte Carlo simulation. There remains a certain variation of the p-values - a direct comparison is not meaningful.
Appendix
Using the pTAS R-package
With the help of short examples we explain how the functions can be used within an R-session. We will not describe the several functions in all details and parameters. Please refer to the corresponding help pages within the package for more information.
Creating a pTAS distribution
The pTAS package uses an object oriented approach, which means that every distribution (e.g. with different parameters) is represented by a different object of class pTAS. Therefore, one can work with many different distributions at the same time (within the same workspace) without jeopardizing their consistency regarding distributional or numerical parameters.
Hence, the first step is to create a new pTAS distribution object.
The MYPTAS object describes a pTAS distribution with parameterization \(\mathcal {P}_{P}(\alpha,\gamma,\theta)=\left (0.5,1,1.5\right)\). The PTAS function automatically performs some plausibility checks on the given parameters and translates the given parameterization (i.e. according to Palmer et al. (2008) in this case) to the other ones. In addition, distribution figures such as mean, variance, skewness and kurtosis are calculated.
To obtain a first impression of the distribution, the density (and also the distribution function) can be shown by simply using the PLOT function.
Density, distribution function, quantiles and random numbers
Density and distribution function are available via the standard R-notation (d…/ p…). The numerical parameters described in 2.1 can be set when the object is created via the PTAS function.
Quantiles and random numbers are generated via Algorithm 2.
Estimation methods
For implementation issues, the MLE will be performed on the \(\mathcal {P}_{P}\)- parameterization always. For parameters α,μ,ν lower and upper bounds as well as fixed values can be specified. Especially for parameter α an appropriate upper bound may be helpful, because pdf calculations for parameters α close to 1 are numerically challenging. The FIT_MLE function uses the MLE function from the STATS4 package, which in turn uses R’s OPTIM optimizer.
Detailed information on the optimization results can by obtained via the OPTIM_RESfunction. This gives a list containing the number of iterations, the convergence result and the Hessian matrix for further calculations (e.g. to calculate confidence intervals). If the optimization did not converge properly, a warning is displayed automatically.
Alternatively, distribution parameters can be also estimated based on mean and variance, which determine parameters μ and ν regarding the \(\mathcal {P}_{P}\) parametrization and either the skewness, the kurtosis or a quantile to estimate parameter α. If α should be estimated based on a given kurtosis value, which possible has multiple solutions, a pTAS distribution with either the highest or the lowest value of possible α’s or a list containing all distributions will be returned.
If α should be estimated based on a given quantile, an grid search method as described in section 2.3 is applied. For given values t ^{∗}>0 and p ^{∗}∈(0,1) the algorithm terminates if \(\left |F_{\mathcal {P}_{P}(\alpha,\mu,\nu)}(t^{*})-p^{*}\right |<\epsilon \) for a prespecified ε>0, whereas values for μ and ν are fixed based on mean and variance. Similar to the estimation based on kurtosis, the solution for α (if one exists) may be not unique. Again, one can determine which distribution should be returned with an additional argument (see example below).
Declarations
Acknowledgement
The authors thank two anonymous referees and an associate editor for their helpful comments and suggestions which significantly improved the presentation of the paper.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Abate, J, Choudhury, GL, Whitt, W: An introduction to numerical transform inversion and its application to probability models. In: Computational Probability, pp. 257–323. Springer Science+Business Media, New York (2000).Google Scholar
- Barndorff-Nielson, OE, Shephard, N: Normal modified stable processes. Economics papers 2001-w6. University of Oxford, Oxford (2001).Google Scholar
- Bolancè, C, Guillèn, M, Gustafsson, J, Nielson, JP: Quantitative Models for Operational Risk: Extremes, Dependence and Aggregation. Chapman & Hall/CRC Finance Series, New York (2012).Google Scholar
- Chernobay, A, Rachev, S, Fabozzi, F: Composites goodness-of-fit tests for left-truncated loss samples. In: Lee, CF, Lee, J (eds.)Handbook of Financial Econometrics and Statistics, pp. 575–596. Springer (2015).Google Scholar
- Credit Suisse First Boston International: CreditRisk ^{+} A Credit Risk Management Framework (1997). http://www.csfb.com/institutional/research/assets/creditrisk.pdf. Accessed 27 Jan 2014.
- El Adlouni, S, Ezzahid, E, Moutazzim, J: Mixed distributions for loss severity modelling with zeros in the operational risk losses. Int. J. Appl. Math. Stat. 11(21), 96–109 (2011).Google Scholar
- Esscher, F: On the probability function in the collective theory of risk. Scand. Actuarial J. 1932(3), 175–195 (1932). doi:http://dx.doi.org/10.1080/03461238.1932.10405883.View ArticleMATHGoogle Scholar
- Fischer, M, Jakob, K: Copula-Specific Credit Portfolio Modeling. In: Glau, K, Scherer, M, Zagst, R (eds.)Innovations in Quantitative Risk Management. Springer Proceedings in Mathematics & Statistics, pp. 129–145. Springer International Publishing (2015).Google Scholar
- Gundlach, M, Lehrbass, F: CreditRisk+ in the Banking Industry. Springer Finance. Springer, Berlin Heidelberg (2004).View ArticleMATHGoogle Scholar
- Haas, M, Pigorsch, C: Financial economics: Fat-tailed distributions. In: Encyclopedia of Complexity and Systems Science vol. 4, pp. 3404–3435. Springer, New York (2009).Google Scholar
- Hougaard, P: Survival models for heterogeneous populations derived from stable distributions. Biometrika. 73(2), 387–396 (1986).MathSciNetView ArticleMATHGoogle Scholar
- Jakob, K, Fischer, M: Quantifying the impact of different copulas in a generalized CreditRisk+ framework An empirical study. Depend. Model. 2, 1–21 (2014).MATHGoogle Scholar
- Janson, S: Stable distributions. arXiv preprint arXiv:1112.0220 (2011). Accessed 2015-08-16.Google Scholar
- Jørgensen, B: Exponential dispersion models (with discussion). J. R. Stat. Soc. Series B. 49(2), 127–162 (1987).MATHGoogle Scholar
- Koudou, AE, Ley, C: Characterizations of gig laws: A survey. Probab. Surv. 11, 161–176 (2014).MathSciNetView ArticleMATHGoogle Scholar
- Küchler, U, Tappe, S: Tempered stable distributions and applications to financial mathematics (2011).Google Scholar
- Merton, RC: On the pricing of corporate debt: The risk structure of interest rates. J. Finance. 29(2), 449–470 (1974).Google Scholar
- Neslehova, J, Embrechts, P, Chavez-Demoulin, V: Infinite mean models and the LDA for operational risk. Journal of Operational Risk. 1(1), 3–25 (2006).Google Scholar
- Nolan, J: Stable Distributions: Models for Heavy-tailed Data. Birkhauser, Boston (2003). In progress available via: http://fs2.american.edu/jpnolan/www/stable/chap1.pdf.Google Scholar
- Palmer, KJ, Ridout, MS, Morgan, BJ: Modelling cell generation times by using the tempered stable distribution. J. R. Stat. Soc. Series C (Appl. Stat.) 57(4), 379–397 (2008).MathSciNetView ArticleMATHGoogle Scholar
- Ridout, MS: Generating random numbers from a distribution specified by its Laplace transform. Stat. Comput. 19(4), 439–450 (2008). doi:http://dx.doi.org/10.1007/s11222-008-9103-x.MathSciNetView ArticleGoogle Scholar
- Schiff, JL: The Laplace Transform: Theory and Applications. Springer-Verlag, New York (1999).View ArticleMATHGoogle Scholar
- Stacy, EW: A generalization of the gamma distribution. Ann. Math. Stat. 3(33), 1187–1192 (1962).MathSciNetView ArticleMATHGoogle Scholar
- Tweedie, MCK: An index which distinguishes between some important exponential families. In: Statistics: Applications and New Directions: Proc, Indian Statistical Institute Golden Jubilee International conference, pp. 579–604 (1984).Google Scholar