Risk estimation using copulas

Catalina Bolancé, Montserrat Guillén & Alemar Padilla

Here is an example on how to calculate the risk of a portfolio using bivariate parametric copulas and Monte Carlo simulation. First, the copula parameters are estimated, then marginal distributions are fitted and value at risk (VaR) and tail value at risk (TVaR) are calculated. A methodological overview can be found in:

Data

There are 753 observed stock returns calculated from daily prices of several stocks: Merck, Novartis, Pfizer, Deutsche Bank, ING, Santander, NASDAQ and S&P500. The example is done using the returns from Novartis (NVS) and Pfizer (PFE).

Copulas

Let L be the total loss in a risky situation, due to the losses that are caused by two subrisks (risk sources) L1 and L2, so that L=L1+L2. The dependence between those two individual risks affects the aggregated loss L.
Copulae allow the modelization of a dependence structure between variables (sources of risk). So, given (L1,L2) a random vector of loss components with F1 (l1)=P(L1=l1) and F2 (l2)=P(L2 =l2) their respective continuous distribution functions, there is a unique copula Cθ:[0,1]2 ∈ [0,1] that depends on parameter θ, so that it holds that:
\begin{equation*} F(l_{1},l_{2})=C_{\theta}(u_{1},u_{2}),\forall l_{1},l_{2}\in \Re, \end{equation*}
where u1=F1 (l1) and u2=F2 (l2) are the values of two random variables U1 and U2 that are uniformly distributed on the interval (0,1).
Examples of copulae are Gaussian, t-Student, Gumbel, Clayton and Frank. They are all used in this example.

Parameter estimation

Copulae are fitted on the basis of one or more specific parameters. For this propose, according to the type of copula one or more initial values are proposed and then final parameters are estimated.
Estimation of parameters is carried out by the method of pseudo-maximum likelihood, which consists in maximizing numerically the following expression: \begin{equation*} l(\hat{\theta})= [ \sum_{i=1}^{N}ln\{c_{\hat \theta (\tilde U_{i1}, \tilde U_{i2})}\} ] \end{equation*} where the arguments of the copula density c are called pseudo-observations: \begin{equation*} \tilde U_{i1} = \dfrac{N}{N+1} \sum_{j=1}^{N}I(L_{j1}\leq L_{i1}) \end{equation*} and \begin{equation*} \tilde U_{i2} = \dfrac{N}{N+1} \sum_{j=1}^{N}I(L_{j2}\leq L_{i2}) \end{equation*}

Goodness of fit

With the aim to establish a comparison between the adjusted models, the AIC (Akaike’s Information Criterion) and the BIC (Schwarz's Bayesian criterion), are estimated where:
\begin{equation*} AIC=-2l(\hat \theta)+2k, \end{equation*} \begin{equation*} BIC=-2l(\theta \hat)+kln(n) \end{equation*}

Simulation

The quantification of the risk is obtained by the method of simulation called Monte Carlo. First, losses are randomly generated from the adjusted copula and then later estimates of the measure of risk are obtained.

Estimation of Marginal distributions

The estimate of the marginal distribution functions depends on the type of data, thus, in this example only the Normal distribution and the t-Student distribution are fitted.

Simulation of loss sources

Losses L1 y L2 are quantiles coming from the adjusted copulae, with marginal distribution functions F1 (l1) and F2 (l2).

VaR

Let L be the random variable that represents the total loss with distribution function FL, then Value at Risk is a risk measure defined as: \begin{equation*} VaR_{\alpha}(L)=inf\{l\ \in R:F_{L}(l)\geq \alpha\}=F^{-1}_{L}(\alpha), \end{equation*} where α is the desired level of confidence.

TVaR

Tail Value at Risk is another measure of risk that is defined as: \begin{equation*} TVaR_{\alpha}(L)=\dfrac{1}{1-\alpha}\int_\alpha^1 VaR_{u}(L)du \end{equation*} Whenever L is continuous like in the current example, the previous expression is simplified so that: \begin{equation*} TVaR_{\alpha}(L)=E[L|L>VaR_{\alpha}] \end{equation*}

REFERENCES

[1] Bargès, M., Cossette, H., Marceau, É. TVaR-based capital allocation with copulas. Insurance: Mathematics and Economics 2009; 45: 348 – 61.

[2] McNeil A. J., Frey R., Embrechts P. Quantitative Risk Management: Concepts, Techniques, and Tools. Princeton University Press 2006.

[3] Nelsen, R. An Introduction to Copulas. Springer, New York 2006.

[4] Sklar A. Fonctions de Repartition á n Dimensions et leurs Marges. Publications de l’Institut de Statistique de l’Université de Paris 1959; 8: 229-31.

[5] Weiss, G.N.F. Copula parameter estimation by maximum-likelihood and minimum-distance estimators: a simulation study. Computational Statistics 2011; 26: 31-54.

• Universitat de Barcelona - Last Updated: 02-19-2015