Assessing VaR with a causal model and bootstrapping

Suppose we have a set of historical operational risk losses for instance in a form of daily time series. Let us now imagine that we can create a causal model relating the amount of losses (yt) for a day “t” to the value of some explanatory variables related to the frequency and severity of events (xi,t). As we have both frequency and severity variables it is better to take the logarithm of loss of these variables[1]. Also in order to avoid calculating the logarithm of zero let us add 1 to the daily amount of loss. We should also add an intercept (b) in order to balance the model (the stated explanatory variables may be insufficient for correctly modelling the amount of losses). Finally as the model is stochastic let us add an error term εt. Our final model will be:



A simple way to calculate a daily loss is to estimate the value of the parameters (ai, b) and to discover the distribution of εt (check if they are normally distributed) or at least calculate the historical residuals.


I. Modelling futures values of xi,t (supposing a trend evolution):

There are now 2 cases: we have some information about the futures values of xi,t or not.

1) If we have the expected expected value of xi,t we can write (supposing a continuous monotonic evolution):

Using the log normal property



where is related to the evolution of xi,t (a sort of continuously compounded “rate of growth”) μ and σ are respectively the expected mean and volatility (the first one calculated by dividing the future expected value of xi,t by the actual value and annualized, the second estimated on historical data).

The futures value of xi,T can then be written in the following way:


2) In the opposite case we use historical data in order to assess the parameter μ (linear progression) and proceed as in point 1).

Getting these data we can transform equation 1 into

Getting the error term
From the historical data after assessing the parameters (ai, b) we can get the historical error terms:

Simulating a value of yt+T

(5a)

where εt+T is a random error drawn from historical data for instance in time t+T (in the bootstrapped approach of errors) or simulated directly if we have modelled the distribution function of εt.

Calculating VaR
With equation (5a) it is now possible to simulate a path of futures value of yt+T. and calculate their sum:


After simulating (a multiple of 1000) different paths and sorting them in ascending order according to the final outcome, it is possible to calculate the VaR at the confidence level of 99.9% (path 0.999*n where n is the number of simulations).

II. Modelling futures values of xi,t (supposing an erratic evolution)

We suppose now that we cannot discover any trend for the explanatory variables. Unless we can discover another pattern we will be bound to based ourselves on the past evolution. In this situation we can use a bootstrapped approach that is:
1) to pick up for a randomly selected “j” historical value for the first xi,t-j
2) to pick up for a randomly selected “j” historical value for the second (third, etc.) xi,t-j
3) to pick up for a randomly selected “j” historical value for εt-j
4) to calculate for these values the corresponding value of yt+1
5) to repeat 364 time steps 1-3 in order to calculate the corresponding value of yt+2 , yt+3, .., yt+365

6) to sum up the different values of y t+T that is calculate:

7) to simulate a multiple of 1000 different paths, sort them in ascending order and select the outcome of the “y” path number 0.999*n where n is the number of simulated paths. The result will correspond to VaR at 99.9% confidence level.

III. Modeling futures values with censored data

Several situations can emerge. If we know the total amount of losses but not their specific values the model should still work. If the losses are censored from below and we do not know their financial impact our historical data will represent only losses over the threshold. In other words the proposed model will permit to calculate the VaR for losses over the threshold.

The first solution consist in executing a classical LDA approach that is in fitting conditional severity and unconditional frequency distributions to the data over the threshold and using the “true” or empirical frequency parameters in calculating VaR. The following step will be to calculate the VaR at 99.9% confidence level but with an upper individual loss equal to the threshold. Summing up the VaR results for data over the threshold and below one will give an assessment of VaR. However one should previously make an adjustment for the frequency distribution in order to not use losses over the threshold.

The second solution will consist in using the truncation point as a conservative assessment of individual losses below the threshold.


IV. Modeling futures values with truncated data

Still if we know the total amount of losses but not their specific values the model should still work. If the losses are truncated from below and we do not know their financial impact our historical data will represent only losses over the threshold. In other words the proposed model will permit to calculate the “VaR” for losses over the threshold.

The solution will be to execute a classical LDA approach that is to fit conditional severity and unconditional frequency distributions to the data over the threshold and adjust the frequency parameters. The next step will be to calculate the VaR at 99.9% confidence level but with an upper individual loss equal to the threshold. Summing up the VaR results for data over the threshold and below one will give an assessment of VaR. However one should previously make an adjustment for the frequency distribution in order to not use losses over the threshold.

Ph.D. Robert M. Korona

[1] If we have only one explanatory variable it is useless to take the logarithm and add one to the amount of losses.