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In order to get a value distribution for the portfolio, using the joint migration probability matrix, we consider all combinations of risk classes of X and Y, including default. When a facility defaults, its value drops to zero. When it migrates to the risk class 1 or 2, its value discounts the final 100 flow at the corresponding risky yield to maturity. We assume that credit spreads at Eoy 1 are identical to those of current date. For each combination of final state, we calculate the values as of Eoy 1, of X and Y, and of the portfolio. The probabilities are the joint migration probabilities of the above correlated matrix, arranged in column format (Table 49.4).

The table provides the value distribution of the portfolio in the last column. The expected value at Eoy 1 is 180.476. It is above the current value of 173.619 because the final flows gets closer (they are 1 year away from the horizon date) and because of migrations and the corresponding discounted values. Note that the loss compared to the current value is negative, since the expected value is higher. Higher default probabilities would make it lower. The probability distribution of the portfolio values is given in Table 49.5. Figure 49.1 shows the usual fat tail to the left of the current value.

Value and Loss Percentiles

The portfolio value volatility is 22.750. The value at 0.03% confidence level value is 93.128, and the 2.78% value percentile is 93.215. These represent losses of around 80.5 from the current value, or 46% of this current value. These values are important because we have a two-obligor portfolio. The loss distribution would be more continuous and loss percentiles would increase more progressively with a diversified portfolio.

Loss Volatility

When the number of obligors increases, the number of horizon values gets very large. Instead of proceeding from the loss distribution to get the loss volatility, Credit Metrics calculates the resulting variance of the distribution from a formula making the variance of

The formula expresses the portfolio variance in terms of the variances of each facility value and the variances of all two-obligor subportfolios. Using the variances of pairs of obligors allows us to use the same set of data of joint migrations for two-obligor portfolios as above. It facilitates the calculations over a large number of obligors.

Credit Metrics also uses the Monte Carlo simulation technique rather than the joint migration matrices, as KMV does. Joint migration matrices are useful to better understand the migration framework. However, the simulations allow us to achieve the same goal.

Capital and Credit Risk VaR

The measures of portfolio risk are statistics, expectation, volatility and percentiles, derived from the portfolio value distributions. Losses derive from setting a value, such as current value, as the zero point for losses. Most portfolio models use a single future horizon to construct the value and the loss distributions and calculate the credit risk Value at Risk (VaR). The loss percentile is L(a), at confidence level a, and the economic capital at the same confidence level is K(a).

Once the value distribution is available, there are still some issues pending to complete the process, including: What is the horizon of the calculations? What are the alternative measures of capital, given economic provisioning policy and portfolio revenues? What are the alternative options for calculating the Risk-adjusted Return on Capital (RaRoC) ratios?

The horizon depends on the portfolio management philosophy. Models capture risk up to a preset horizon that should allow sufficient time to either alter the risk of the portfolio or adjust the capital. The simplest compromise is to use a horizon of 1 to 2 years, together with simple rules for dating default and losses within and beyond this horizon. Note that setting up a horizon shorter than facility maturities does not ignore maturity or actual cash flow dates, because the revaluation at the horizon includes all flows up to maturity. In addition, early losses have more value than late losses, due to discounting.

Once the horizon is set, there are alternative options to determine expected loss and capital, whose choice is a management decision. When looking forward, the portfolio earns income and loses the expected loss. Therefore, capital should be the loss in excess of revenues net of expected loss. Netting expected losses makes sense if economic provisioning is in force. Considering accrued income as a shield against losses assumes that it remains effectively available at the horizon. Modifying the assumptions would require calculating the capital differently.