Co-xTVaR has almost the opposite advantages and disadvantages of standard deviation. It is calculated as the amount by which each risk is worse than its expectation in those situations in which the totality of risks being modeled exceeds its expectation. Co-xTVaR can be viewed as the amount by which a segment is “over budget” in those scenarios in which the company as a whole is “over budget.” In other words, co-xTVaR looks at the average amount by which each segment exceeds its mean in the scenarios in which the company result exceeds some threshold.
Co-xTVaR looks only at those situations in which the company does worse than some threshold. It therefore looks only at downside risk and reflects the correlations among the different sources of risk.
A related metric is co-TVaR. It measures the average amount of each segment in those scenarios in which the company is “over budget.” It is similar to co-xTVaR in that it uses the same scenarios in the calculation (and therefore has the same favorable properties). It differs from co-xTVaR in that the expected value for each segment is included in co TVaR so it reflects the total value for the segment, not just the amount by which it exceeds its expectation.
One of the key decisions to be made in using co-xTVaR is the combination of statistic and threshold to be used. For the allocation of capital for determination of leverage ratios, situations either in which net income is negative or in which net income is less than expected are likely to be most important. Other combinations of statistics — including change in surplus, BCAR and RBC — and thresholds can be used.
The importance of the current year’s profit to surplus determines whether zero or expected net income should be used as the threshold. If the current year’s profit is considered as part of the surplus needed to support next year’s business, then a threshold of expected net income is a consistent choice. If the current level of surplus is the primary focus, a threshold of no net income would be consistent. Of course, any determinant of an adverse scenario can be used. Note that the total amount of current surplus will be allocated regardless of which scenarios are selected to determine the percentage allocation.
There is an area of common confusion between co-xTVaR and co-TVaR. The difference between the two metrics is that the former is in excess of a threshold and the latter is not. Co-TVaR of ceded losses is often used to allocate the reinsurance premium among business segments because it includes expected losses and the risk margin, both of which are of interest in allocating the entirety of reinsurance premium. When allocating capital or a risk or profit margin, only deviations from expectations are of interest because expected losses are already considered as part of premium or the financial plan. Confusion arises because, under certain circumstances , co-TVaR of underwriting profit is equivalent to co-xTVaR of losses (i.e., claim costs) when the threshold is expected losses. With the greater ease of changing the threshold, the rest of this discussion will focus on co-xTVaR of losses rather than co-TVaR of underwriting profit.
Figure 4 shows the capital allocated using co-xTVaR using each of expected profit and zero profit as thresholds.
The co-xTVaR for casualty using the expected profit threshold is USD32 million, or exactly the sum of the co-xTVaRs for the three segments. This result illustrates one of the key benefits of using a coherent metric for capital allocation, as different levels of aggregation (e.g., by line as compared to by line and state) will produce the same allocations.
The co-xTVaR values are much higher for the zero-income threshold than the expected income threshold because the average is being taken over a smaller subset of the total population of scenarios and the included scenarios are those that are at the worse end of the distribution. Thus, the greater volatility of property causes its capital allocation to increase when the lower zero-income threshold is used.
Co-xTVaR is coherent, focuses on adverse deviations and allows for flexibility in selecting a threshold. Its disadvantage, though, is that the metric is not broadly understood outside actuarial circles.
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