Business — Banking — Management — Marketing & Sales

This section provides sample simulations illustrating the generation of value distributions using the option framework of default. The simulations require three steps: summarizing the general framework, defining our set of simplifying assumptions and providing sample simulations.

Simulation of Correlated Losses from the Simulation of Asset Values

In order to obtain the portfolio loss distribution, KMV generates a large number of random standardized asset values of individual obligors from which risk migrations to the horizon derive. The risk migration varies continuously with the generated asset values. The portfolio value V_p is the summation of individual facility values V,-. The individual values V,- result from revaluation at the horizon after risk migrations corresponding to the credit state characterized by the final asset value. The model revalues the facilities according to the final distance to default of each obligor, which corresponds to a credit state characterized by a default probability specific to each obligor. The value under the default state is the exposure X, times the loss given default percentage Lgd;: V,- (default) = X, x Lgd_;. In KMV Portfolio Manager, the loss given default is a random variable, following a beta distribution, whose mode is the end-user input. This results in an additional component of the loss volatility, one resulting from the value volatility due to migrations, and another resulting from the independent random Lgd volatility. The portfolio value sums all facility values for each draw of a set of correlated asset values for all facilities V_p = J2^N₁ V,-. Generating random standardized asset values complying with the correlation structure of obligors assets results in the distribution of the portfolio values at the horizon.

The KMV model operates in full valuation mode, using either credit spreads or risk-neutral valuation to revalue any facility at the horizon. Valuation depends on modelled asset values. The modelled asset values are those of the KMV universe, if the user chooses so. If the end-users map default probabilities to internal ratings, the asset values follow standardized normal distributions with a threshold default value corresponding to the preset default probability. Cholevskys decomposition allows us to generate random correlated asset values at the horizon for each obligor. For each draw of an asset value for a particular obligor, the model revalues the facility according to its risk. The random asset value at the horizon depends on common factors and specific risk. The latter is the volatility due to the error term of the model fitting asset returns to the common orthogonal factors in the KMV universe.

With preset default probabilities, rather than the KMV modelled expected default frequency Edf, the model uses the reduced form of the full model of asset values, considering that all relevant information on asset values is in the assigned default probability. We use this standardized normal distance to default model below.

KMV Portfolio Manager requires specifying the general risk for each facility in the portfolio. The R-square of the regression—measuring general risk—is the output of the multi-factor model linking asset returns to factors. When inputting directly the default probability, general risk becomes an input rather than an output of the multi-factor model in the KMV universe. Increasing the general risk is equivalent to increasing correlation and the dependence on external factors. If this dependence were mechanical, firms credit standing would be perfectly correlated. For example, country risk creates a contagion effect on firms, increasing the correlation between defaults. A contagion scenario within a country would be reflected in a higher than usual value of the general risk of obligors.

Figure 48.1 summarizes the simulation process. What follows illustrates the process using a much simplified model, but following the rationale of the option theoretic framework.

Simplified Model (Default Mode)

The simplified example differs from the full model:

• The example uses the standardized normal distribution of the distance to default model, given the default probability of obligors.

• The model operates in default mode only, and exposures are at book value.

• There is a uniform preset asset correlation.

• There is a uniform preset default probability.

Asset Distribution and Default Probability

We use the standardized normal distance to default model, considering that the default probability embeds all information relevant to default and on unobservable asset values. With the default probability DP_X, there is a value A_X of the random asset value A_X

Multiple Simulations

A large number of trials generate a portfolio value distribution, later converted into a portfolio distribution. Note that the same procedure allows differentiated default probabilities and different losses under default for the different firms. Such discrepancies exist in the sample portfolio used to illustrate portfolio reporting in the corresponding chapters (Chapters 55 and 56). Here, we ignore such differentiation in order to focus on the correlation effect.

Since asset values trigger default, all we need is to generate correlated asset values for all obligors. When running a default model, the target portfolio variable modelled is the

portfolio loss. A random default Bernoulli variable characterizes the default event:

A_i is the threshold value triggering default, such that Prob(Z,- <A_i) is the preset default probability DP_i of firm i. Each simulated Z_i value determines a 1 or 0 value for d,-. K draws of all Z,- generate K x N values of the d_{, one for each of the N obligors, and an aggregated portfolio loss L_p summing up the individual losses for defaulting firms. Each draw of the set of correlated Z- provides a simulated portfolio loss value. K draws provide K points representing the simulated portfolio loss distribution. For each one of the K draws, we have N asset values, one for each obligor, as many as there are facilities in the portfolio, plus one portfolio value. Running 10 000 simulations results in 10 000 x N values of assets, facility migrations, plus 1000 values of V_p, from which we derive the portfolio value distribution, and finally the portfolio loss distribution.

To illustrate the process, we use the one-factor methodology, with a uniform correlation between all pairs of assets and a uniform exposure size in this example, although the technique also accommodates inequality of sizes and of default probabilities. The common factor Z conditions asset values A, of each obligor i according to the following equation:

The residuals e_{ follow a standardized normal distribution. The variance of the residual is the specific risk, which depends in a one-factor model on the correlation value. The R-square is the complement to 1. The correlation between any pair of asset values is p. To proceed, the sequential steps are:

• Generate a standardized random normal distribution of the common factor Z.

• Generate as many standardized random distributions of the specific residuals e_{ as there are obligors.

• Calculate the resulting standardized random variable Z_{, which is the asset value of each obligor.

• Transform this asset value, for each obligor, into a default 0-1 variable.

• Calculate the loss for each obligor, given the value of the default variable and the exposure.

• Sum all losses to get one value for portfolio loss for each simulation.

• Repeat the simulation as many times as desired.

In the simulation:

• The correlation coefficient p varies between 0% and 50%.

• The number of obligors N is 100.

• The default probability DP_i is uniform across obligors, at 1%, or a threshold value triggering default of assets equal to —2.33 for all obligors.

• The exposures X_i are identical across obligors and equal to 1.

• The number of simulations is 1000, so that we obtain 1000 values of the portfolio loss forming the loss distribution. This is a relatively low value, but it suffices for this example.

This is a simple but powerful way to investigate the correlated loss distributions, since both the average correlation and the average default probability are parameters.

Sample Correlated Loss Distributions

The simulations serve to visualize the distributions with increasing correlation. Generating a range of portfolio loss distributions corresponding to various uniform asset correlations makes it possible to see how loss statistics vary with correlation. Asset correlation varies from 0% to 50% in steps of 10%, generating five loss distributions.

Loss Distributions

When the average uniform correlation of assets increases, the mode of the distribution moves to the left. Simultaneously, the fat tail, grouping high losses of low frequencies, extends to the right. To highlight the differences between loss distributions, the X-axis shows only the most frequent losses, losses lower than 10, although the fat tails will extend further to the right (Figure 48.2). In fact, default losses having values higher than 10 occur in the simulation: for a 50% correlation, some simulations generate 40 to 50 defaults, close or equal to half of the total exposure.

Getting to the far end of the right-hand tail implies a much greater number of simulations to stabilize the tail and reduce simulation noise. Simulation noise is the variation of the loss values between different runs of simulations. It decreases proportionally to the square root of the number of simulations.

The lowest mode corresponds to a zero correlation and is very close to the binomial distribution since we use uniform exposure and uniform default probability. From these simulations, we derive sensitivity analyses with respect to the correlation. When correlation increases, the probability of the mode of the loss distribution increases and the tail extends to the right while getting thicker.

Sensitivity of Potential Loss Measures to Correlation

Figures 48.3 and 48.4 show how the portfolio loss volatility and the 99% loss percentile behave when asset correlation increases from 0% to 50%. Common magnitudes for asset correlation are within the range 20% to 40%. The simulation illustrates the sensitivity of loss statistics to asset correlation.

With 0% average asset correlation, uniform exposures and an average default rate of 1%, the loss volatility is 1 and the 99% loss percentile is 4, or 4/1 = 4 times the loss volatility. With 30% average asset correlation, uniform exposures and an average default rate of 1%, the loss volatility is around 2.4 and the 99% loss percentile is 11, or 11/2.4 = 4.6 times the loss volatility. Both loss volatility and 99% loss percentile increase significantly with correlation. The loss volatility at 30% asset correlation (equal to 2.4) is around 2.4 times the loss volatility under independency (1). The 99% loss percentile increases from 4 to 11, or 2.75 times the initial value. The ratio of 99% loss percentile to loss volatility is rather stable, around 4, because both loss volatility and loss percentile increase with correlation. In any case, it is much higher than the 2.33 ratio applying to the normal distribution. In summary, the loss volatility and the loss percentiles at 1% are roughly 2.5 times the values under independency. This demonstrates that it is critical to model correlations to obtain realistic orders of magnitude, and that ignoring correlations underestimates the loss statistics very strongly.