## Monte Carlo Simulations

Category: Risk Management in Banking

The main technique to obtain correlated loss distributions is to use Monte Carlo simulation. The technique necessitates the correlation structure of the credit risk drivers. When asset values are risk drivers, it becomes a simple matter to generate random asset values such that each firm has a predetermined default probability. For each run, the asset value of each individual firm corresponds to a credit standing of this firm: the risk can improve or deteriorate, or ends up in default state. Implementing this technique requires only the correlation structure of asset values of firms. KMV Portfolio Manager and Credit Metrics (J. P. Morgan, 1997) use this technique.

With CPV, the random values of factors influence the default rates of segments. Under the econometric approach, default rates depend on economic factors that correlate. With the correlation structure, it is feasible to generate random values complying with this structure. With each set of factor values, it is possible to derive the default rates of portfolio segment through the logit function. Therefore loss distributions of portfolio segments correlate.

Modelling Joint Migrations

Another way of generating a loss distribution is the matrix approach proposed by the Credit Metrics technical document. The technique specifies the N possible migrations of each obligor, including to the default state (there are N risk classes). For each obligor, there is a value distribution with N final states, including the default state. For each pair of obligors, there are N^{2} risk migrations possible, as many as there are combinations of final credit standing for the pair of obligors.

The transitions to risk classes are sensitive to correlations. If the risks of two firms tend to deteriorate simultaneously, rather than independently, their joint migration towards higher risk will have a probability higher than if their transitions were independent. The technique generates a distribution of values for the portfolio, which includes upward risk migrations (lower risk) resulting in higher mark-to-model values and downward risk migrations (higher risk) resulting in lower mark-to-model values, down to the default states.

Capital and Provisioning

Capital results from the loss distribution. VaR is the loss percentile corresponding to the selected confidence level. It depends on the horizon of the calculation of the loss distribution, on the valuation method of exposures and on the time profile of exposures when mark-to-model applies. Capital measures are: the loss percentile; the loss percentile net of expected loss under economic provisioning; the loss percentile net of expected revenues, minus expected losses, if these can serve for offsetting losses. The final capital measure requires a consistent set of rules.

Most models use a specified horizon, such as 1 year. The loss distribution results from the distribution of asset values at the horizon, using some value as the starting point for counting losses. KMV Portfolio Manager, Credit Metrics and CPV provide a VaR calculation at the horizon, in addition to defaults between the current date and the horizon. CreditRisk+ and CPV theoretically provide loss distributions over various horizons using cumulative default rates.