September 10th, 2009

## Optimize Capital Allocation with Co-xTVaR

Posted at 6:00 AM ET

Donald Mango, Chief Actuary, and Susan Witcraft, Managing Director
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In choosing a capital allocation method, firms must balance the sophistication of the method with calculation time and resource commitment. One approach, co-xTVaR, strikes a balance between theoretical soundness and efficiency. In a capital-constrained environment, using co-xTVaR to allocate the cost of capital can provide a clear competitive advantage.

Standard deviation and covariance are the best known approaches to capital allocation. While easy to calculate, both have limitations, and carriers need to understand that using one of these methods may compromise capital efficiency.

With standard deviation, each segment of a portfolio has its own “standalone” (i.e., independent) standard deviation calculated, and capital is allocated in proportion to the segment’s standard deviation relative to the sum of the standalone standard deviations for all segments. This method is commonly understood and easy to execute, but it ignores correlations and treats “upside” (favorable) and “downside” (unfavorable) deviations from expectations equally. True capital usage only occurs in unfavorable scenarios. Also, standard deviation is not a “coherent risk measure.” Lack of coherence becomes problematic when trying to decompose a segment into subsets. The sum of subset standard deviations will not equal the segment standard deviation, necessitating ad hoc adjustments to balance.

Covariance addresses the coherence issue. It is calculated by taking the expected value of the product of the deviations of two variables from their means. For each segment, the difference between its value and mean is multiplied by the difference between the total value and the total mean, with the average of the products being the covariance. Capital is then allocated in proportion to the covariances. Like standard deviation, though, it measures risk using both upside and downside.

Unlike standard deviation and covariance, co-xTVaR is both coherent and downside-focused. An additional advantage is its explicit reflection of the company’s specific definition of “downside.” Co-xTVaR can be viewed as the amount by which a segment’s losses are “over budget” in scenarios where the company as a whole is also “over budget.” It considers only those situations in which the company does worse than some threshold — for example, losing 10 percent of capital in one year or having zero earnings.

Co-xTVaR is related to the slightly better-known co-TVaR. The difference between the two is that co-TVaR uses pure loss amount, while co-xTVaR uses loss in excess of plan or budget. The co-TVaR of ceded losses is usually used to allocate reinsurance premium among business segments, as it includes expected losses and the risk margin. Co-xTVaR is more appropriate for capital allocation, as capital only begins to be consumed once losses exceed planned levels — i.e., reserve strengthening.

Though co-xTVaR remains fairly obscure, its performance characteristics make it well worth further investigation. This coherent, downside-focused approach to cost of capital allocation remedies critical shortcomings of standard deviation and covariance. The result is a calculation aligned with the way risk managers deploy their capital - focused on the downside impact to the portfolio of a particular risk. Also, with today’s capital modeling tools, co-xTVaR requires little additional resource commitment. The benefits will make every dollar of capital more productive.

Guy Carpenter & Company, LLC provides this text for general information only. The information contained herein is based on sources we believe reliable, but we do not guarantee its accuracy, and it should be understood to be general insurance/reinsurance information only.