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Risk contributions always refer to a facility of obligor i and to a reference portfolio P. The portfolio P is made of N facilities. Each facility i relates to a single obligor. The notations apply to both default mode and full valuation mode models. However, all examples use calculations in default mode only for simplicity.

The losses are random variables. For the single facility i, i — 1 to N, the random loss is L. The exposures, Xp i = 1 to N, are supposedly deterministic. The loss under default for each facility or L., equals the product of exposure and loss given default, LGD.. The random loss takes values 0 or L.— X. x LGD.

For introducing randomness in the default loss for individual facilities, L., i — 1 to N, we model default as a Bernoulli variable D taking the value 1 in case of default and zero otherwise, or an indicator function of the default event. The random loss is L. — L. x D. The default probability is the probability of D having value 1, equal to d. in default mode. Consider a single exposure with loss under default having the value L. From the formula deriving the expectation, variance and volatility for such an exposure (see Chapter 49):

Since all risk contributions refer to a common portfolio P, it is convenient to use a superscript P for designating such reference portfolio in the subsequent notations.

The random portfolio loss LF sums up arithmetically all random individual losses L of each obligor. The aggregated random portfolio loss is LF = XL. for the TV obligors. Since we sum up random individual losses, the expectation of the sum is E(LF) — Y,^ E(L^.

The loss volatility is the standard deviation of a loss. It is o for a single facility and Cp for the entire portfolio. For a single facility, it is the standalone loss volatility. For the portfolio, the portfolio loss volatility is the volatility of a sum of the random individual losses and depends on correlations between individual losses L..

The "unit exposure" loss volatility of a single facility is the loss volatility for an exposure equal to one unit. In default mode, the unit exposure loss volatility is the usual a = (1 - <i) ^ld. The exposure weighted loss volatility is:

The correlation coefficients between individual losses L and L are p.. — p... The standalone loss volatilities are a. calculated with the above standard formulae. Hence, p Cov(L., L)/ap.

In default mode, the loss results from default only. In the full valuation model, the loss for one facility follows a distribution and we should replace each d. unique default probability by a discrete variable of which values are not limited to 0 and 1, but takes as many values as there are credit risk classes, including the default state, or by a continuous variable such as the distance to default. In both the discrete case and the full valuation mode, we should assign a probability to each final credit state. Such probabilities are transition probabilities from the initial credit state to the final credit state, including the default state, in the full valuation matrix model. They define the distribution of asset values under the structural default model, given initial asset value.

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