# Portfolio Risk Return Profile

The "all-in spread" (AIS) is the annualized revenue from each obligor, including interest margin and fees, both upfront flat fees and recurring fees, averaged over the maturity of the facility. The aggregated contractual spread, or total AIS under no default, is 401, while expected loss is 101. The expected spread is the contractual AIS minus the EL, or 401-101 — 300 (rounded value).

TABLE 57.1 **Portfolio overview**

The portfolio Sharpe ratio is its overall RaRoC. The calculation with capital in excess of expected loss is:

RaRoC = (AIS - EL)/£(1%) = 31.48%

The RaRoC is equal to the Sharpe ratio, because the spread AIS is net of the risk-free rate. The ROA is the spread in percentage of exposure. The SVA is the difference between the AIS, the EL, and the cost of the economic capital of the portfolio, pre-tax and pre-operating costs, set at 25%. Since RaRoC = 31.48% > 25%, the SVA is positive and equal to 62 (rounded). The book return on asset, ROA, is the ratio of contractual AIS to exposure, and equals 3.65%, gross of expected loss. The ratio is the overall spread above the risk-free rate, in percentage of exposure. The book return on equity, ROE, is not risk adjusted, and is the contractual AIS divided by capital:

(401-300)/954 = 31.6%

It is quite close to the Sharpe ratio.

# Portfolio Concentration and Correlation Risk

Both correlation risk and concentration risk are related measures of portfolio risk. Correlation risk relates to the loss dependency. High correlation results in a large number of losses occurring simultaneously. The effect is similar to a large loss. Concentration designates the effect of size discrepancies. A high size concentration implies that some lines have a much bigger size than other lines.

Concentration risk is the risk of large losses due to default of large transactions. Concentration characterizes size discrepancies. The individual loss given default weights, ratios of the individual loss given default to the total loss given default (6000), measure exposure sizes. It is common to report the lines having the largest weights in the portfolio. The alternate synthetic views of portfolio size concentration include the diversity score and the concentration curve.

The diversity score is an index synthesizing the discrepancies of exposures of individual facilities. There are as many concentration indices as there metrics for risk. Alternate metrics include exposure, losses given default, standalone risk measured by individual loss volatilities, or capital allocation. Concentration implies significant discrepancies of the weights of the individual facilities to the total portfolio size (whatever metric is used for measuring risk, size, loss given default, etc.). A concentration index, or diversity score, can be defined for each metric. The diversity score is a number that is always lower than the actual number of facilities. The ratio of diversity score to the number of exposures is the concentration index, always lower than one, except when the portfolio has uniform exposures. The concentration risk is higher when the diversity score is lower.

The diversity score is the number of equal size exposures equivalent to the weight profile of individual exposures. The diversity score (DS) is the following ratio:

The *w.* are the weights of facilities, using one risk metric, the most common one being the size of exposure. For interpreting the diversity score, we consider the case where all weights are equal to *lln, n* being the number of obligors. In this case, the ratio would be:

or *l/(n/n2) — n.* This allows the interpretation of the diversity score as the number of uniform exposures "equivalent" to the number of actual unequal exposures.

The ratio of the diversity score to the actual number of exposures is always lower than 1 whenever there are size discrepancies, and the gap measures the size concentration. As an example, consider the simple two-obligor portfolio, with exposures 100 and 50 respectively. The exposure weights are 66.667% (100/150) and 33.333% (50/150). The diversity score is:

It is the "equivalent" number of equal size exposures. The concentration index is the ratio of the diversity score to the actual number of exposures, 2 in this case. This is 1.8/2 *—* 0.9. The ratio is lower than 1 because the exposures are unequal.

TABLE 57.2 **Concentration risk**

FIGURE 57.4 **Gini concentration curves of exposures**

Table 57.2 provides the diversity scores for exposure and capital allocation, equal to 35.02 and 33.22 respectively. Both numbers compare to the actual number of exposures, 50. The ratio of the diversity scores to the actual number of exposures measures the concentration in terms of weight discrepancies. Note that the capital diversity score combines both effects of size concentration and of diversification, since capital allocations capture the retained risk post-diversification effect.

A second measure of concentration risk is the "concentration" curve, or "Gini" curve. The curve shows the cumulated exposure, or any alternate risk metric, such as capital, as a function of the number of exposures. The curve cumulates the exposures starting with the largest exposures and ranked by descending values. A uniform exposure portfolio would have a straight-line concentration curve. The higher the curve is above the straight line, the higher is the concentration risk.

In the exposure concentration curve (Figure 57.4), the first five biggest obligors represent 21% of the total portfolio exposure; the first ten biggest obligors represent 40% of the total portfolio exposure, and so on. The curve hits 100% when all 50 exposures cumulate. The slope is steeper at the beginning of the curve because the largest exposures are the first along the x-axis.

# From Portfolio Risk to Individual Facilities

The loss volatility of each facility is LGDV[<i(l - *d)], d* being the default probability. The standalone loss volatility of a facility is intrinsic to that facility and does not depend upon any portfolio effect. For a uniform correlation portfolio, allocating capital to each facility follows the simple rule of the pro-rata of the standalone facilities. For each facility, the ratio of the individual standalone loss volatility, calculated as above, to the sum over the entire portfolio, is the percentage of total capital allocated to each facility.

Together with the all-in spread, the capital allocation allows one to calculate the RaRoC and SVA of each individual facility as well. Once the allocation is available, all facilities have a risk-return profile. The subsequent sections provide detailed reports on individual facility risk and return.