The co-TVaR report in MetaRisk gives analysts easy access to valuable information to illuminate the Enterprise Risk Management (ERM) and capital implications of reinsurance. Using co-TVaR for risk decomposition, one can develop a fuller view of the factors that contribute to company risk, ultimately delivering a more compelling case for risk-related decision-making.
Risk Decomposition, Co-Measures and Co-TVaR
Measures for quantifying company risk and its components have become increasingly useful to insurers. Understanding risk sources can help them measure performance, develop strategic plans, price products, and communicate with regulators, rating agencies and securities analysts. Decomposing underwriting risk among the constituent business components that generate risk can have several applications, including the allocation of capital. Other ways to compute business component performance without allocating capital can also benefit from being able to quantify the contributions to company risk.
Guy Carpenter prefers to use the term “decomposition” rather than “allocation” to focus on how the risk is generated within the business units rather than assigning risk to units in proportion to some measure.
The decomposition of risk measures starts with a risk pool (i.e., a sum of random variables) Y = SXj and a risk measure R(Y) that expresses some aspect of the risk inherent in Y. The composition of this risk measure by the components Xj is a function r(Xj) with R(Y) = Sr(Xj). For an insurer, Y could be, for example, total claims, change in capital, underwriting profit.
The use of “co-measures” is among the better-known risk decomposition approaches. It is based on the premise that risk is defined and measured at the portfolio level – i.e., by looking at “bad outcomes” for the company. Further, contribution by component should use the event (scenario) level detail for those bad outcomes.
The goal of the co-measures approach is to have additive allocation or attribution of portfolio risk to the components (e.g., lines of business, loss causes). In this context, “additive” means the sum of the separate allocations to components A and B will equal the allocation to a combined component AB.
The process starts with the selection of a company’s risk measure. Most companies start with Value at Risk (VaR), the finance term for probable maximum loss (PML) or percentile. So, a 100-year PML would be called VaR99 – the 99th percentile VaR on loss. VaR can be calculated using any insurance financial variable, such as pure loss (gross or net), underwriting result, discounted underwriting result, combined ratio, net income or change in surplus.
Tail VaR (also called TVaR or tail conditional expectation, TCE) is the average result beyond (i.e., worse than) a certain VaR – the conditional expected value. TVaR99, for example, would be the average of the worst 1 percent of events for the company based on the selected financial variable. If one has 100,000 simulated scenarios, all equally likely, one would calculate TVaR by sorting on the selected financial metric, then taking the worst 1,000 rows and averaging the amounts for those rows only.
Co-TVaR is the contribution of a component to the total risk in those scenarios used to determine the VaR or TVaR. One would calculate co-TVaR99 by looking at the same 1,000 “bad rows” and, for each component, calculating its loss or underwriting result for each of the 1,000 bad rows. Then, one would average those amounts over all 1,000 bad rows. This exercise is basically a “fault finding” mission, with the objective being to discover which components of company risk are most responsible for the outcome.
Uses of Co-TVaR
Guy Carpenter recommends co-TVaR as a sound method for risk attribution, assessment of capital usage and capital allocation. It has many theoretical and computational advantages that drive practical results.
- Capital and risk can be allocated to any level of detail in a completely additive fashion
- The actual contribution of the component to the company risk is calculated
- The contribution measured by co-TVaR is also the marginal impact on the company risk of removing a small percentage of the component business, grossed up to the volume of the whole component; thus, it is the incremental marginal impact of each small segment of the component
- Riskiness only needs to be defined on the portfolio total and can be done so intuitively
- TVaR is a well-established risk measure for company capital evaluation
- It is easy to do in simulation situations, and it is built into MetaRisk
- MetaRisk’s co-TVaR report can be run using Gross Loss, Net Loss, Gross Underwriting Result or Net Underwriting Result
Co-TVaR for remote probabilities (i.e., the extreme tail) provides insights into contributions to severe impairment and ruin. These insights can be valuable for rating agency and regulatory discussions of extreme event management and mitigation.
However, co-TVaR need not be thought of only in an extreme tail context. It can be calculated excess of any threshold, even a relatively low one such as zero underwriting income, an underwriting loss equating to zero net income or the expected underwriting profit. Such analyses highlight which product lines contribute to earnings misses and impairment (e.g., the capital level at which a company might be put on ratings watch). Earnings volatility is an important metric to both stock and mutual insurers. There are many firms with capital adequacy far in excess of their rating due in large part to track records of unacceptably high earnings volatility.
Several insurers have noted that using co-TVaR excess of a high probability threshold finds much of the risk arising from catastrophe-exposed lines, while running it above a low threshold attributes the majority of the risk to more routine business. Sometimes neither of these thresholds seems to capture the intuitive sense of relative risk with which insurance professionals feel comfortable. Part of the problem is that, among losses excess of the selected threshold, TVaR is a linear measure – just the average, in fact. Most risk-bearers tend to feel that risk is not linear as losses become very large. Thus, while it makes sense to use a low threshold since missing targets even a little can be painful to the company, the linear nature of TVaR does not always give reasonable risk decompositions when the threshold is low.
One helpful alternative is to use a risk measure that is the average or weighted average of TVaRs at different thresholds. The largest losses will be in all of the TVaRs and thus will get higher weights, while the low but still painful losses will continue to have some influence. Another alternative is to use risk-adjusted TVaR (RTVaR), which is the conditional excess mean plus a portion of the conditional excess standard deviation, excess of a low threshold. Like the first alternative, this approach includes the smaller losses while giving more weight to those that are larger. The details of how to do it are beyond the scope of this briefing.
Similarly, one can calculate XTVaR and co-XTVaR – with “X” standing for excess of some threshold value (typically the expected value). Generally, XTVaR is the preferred metric when using loss to decompose risk. In contrast, the choice of TVaR or XTVaR when using underwriting loss is dependent on the company’s definition of “capital.” Using TVaR for underwriting loss, one can end up with no capital allocated to lines that have low volatility or high profitability. As long as the company is willing to accept volatility down to a zero underwriting profit, this conclusion is fine. If, however, the company looks at the next year’s underwriting profit as part of capital, it needs the anticipated positive contribution of the highly profitable lines. In that case, an underwriting result that is lower than the mean for a highly profitable line will consume (next year’s) capital. In these situations, XTVaR is a better match to the company’s objectives than TVaR.
Co-TVaR is a balanced, credible measurement of relative capital usage by component. The term capital usage is carefully chosen to avoid strictly saying capital allocation, a term itself not uniquely defined in clients’ minds. Clients may use co-TVaR as a means of allocating capital, but that is not its sole meaning or purpose.
Whatever the intended use, the ease of running numerous co-TVaR reports also facilitates sensitivity and performance testing, an essential step in the adoption of any such approach.
Comparison with Contributions Report
One could think of co-TVaR as a risk contributions report. When run on net results, co-TVaR is complementary to the contributions reports, which focus on usage of the reinsurance (cessions). Net co-TVaR measures relative contribution to the net risk profile. In this respect, it is valuable in showing which loss causes (e.g., business units, lines) receive a reduction in capital (as measured by co-TVaR) from reinsurance, and which receive an increase.
Running the Co-TVaR Report
Co-TVaR reports have their own tab in the MetaRisk reports dialog box. The user selects the underwriting variable and an amount threshold. The choices are Gross Loss, Gross U/W Loss, Ceded Loss, Net Loss, and Net U/W Loss.
Note: In order to run a TVaR99, one first runs MetaRisk to determine the amount of the VaR99, then uses the result in the co-TVaR report request and re-simulates. Also, to run either a net loss or net underwriting loss report, it is critical that the user make a reasonable allocation of reinsurance premium to loss causes, as the default allocation in MetaRisk is proportional to the number of loss causes covered by the contract. The contribution report is an easy way to improve on the default MetaRisk allocation.
To log all the individual values that satisfy the co-TVaR criterion into a database, click “Details by Realization.” This option allows the user to calculate a whole host of possible risk measures.
Note: Choosing this option will slow simulation performance.
Reinsurance Program Evaluation
A co-TVaR report on the net loss under each reinsurance variation demonstrates the relative contribution of loss causes (e.g., business units, lines) to the company’s net risk position. In this example, the company’s stated risk tolerance amount of USD157 million of net loss was used as the co-TVaR threshold, corresponding to the company’s risk tolerance criteria of a 1-in-40 year event (i.e., TVaR 97.5 percent). The comparison table shows both co-TVaR amounts and proportional shares by business unit for the current program and each of several proposed aggregate covers with progressively higher limits.
Focusing on the bottom table, note that, as the aggregate limit increases, the percentage shares of business units C, D and E stay relatively constant. In contrast, the share for unit B decreases at the expense of unit A. With more reinsurance in place, the loss events that exceed the probability threshold are different scenarios. In this case, protecting unit B means that the company as a whole is more likely to have bad results when unit A has a bad year. This illustration highlights the true beneficiary of the increased protection, and also suggests how reinsurance costs should be allocated. The current program is different from any of the alternatives, especially for unit E.
Catastrophe Capital Usage
This example shows a multi-region catastrophe portfolio with co-TVaR reports above several key PML values (USD3 billion, USD4 billion and USD5 billion), both gross and net.
The implied allocation percentages are not particularly sensitive to threshold, suggesting that business line contribution to the risk profile of the portfolio remains fairly similar as the size of loss increases. They are also not particularly sensitive to reinsurance, suggesting that each business line receives roughly equivalent benefit from the reinsurance program.
Capital Allocation Using Co-TVaR and Co-XTVaR
The charts below illustrate the difference in capital decomposition for the same book of business when the metric used for decomposition of underwriting loss varies between co-XTVaR and co-TVaR. Recall that, currently, MetaRisk only has built-in co-TVaR reporting. Co-XTVaR must be calculated outside the tool by subtracting the mean (loss or underwriting loss) from the co-TVaR for each component.
As discussed previously, there can be significant differences in decompositions and interpretation of the indicated capital allocation.
Although there are many ways to evaluate profitability by segment, decomposition of capital is a convenient one. Using the co-XTVaR decomposition above and mean economic profits by line, returns on capital can be derived for each line. The following charts are two ways of illustrating returns on capital.
The estimated average (mean) return on decomposed capital is shown above; it can be determined for each business segment using MetaRisk. When divided by the capital derived for each segment based on the co-TVaR or co-XTVaR (often referred to as required capital), the returns on equity can be calculated, compared with each other and with the company target.
Required capital adds another element to the evaluation of required return on capital, as shown in the chart on the bottom of the previous page. In this illustration, the very high return on auto physical damage (APD) and the very low return on workers’ compensation (WC) are shown to have relatively little impact on the total return because only a small amount of capital is required by those segments.
- For a more thorough treatment of decomposition, see “Marginal Decomposition of Risk Measures,” by Venter, Major and Kreps, ASTIN Bulletin November 2006. www.casact.org/library/astin/vol36no2/375.pdf
- Other decompositions can include investment and credit risks separately from underwriting risk. These decompositions will be added to a future MetaRisk release and thus are not addressed in this briefing.
- This approach is also known as “RMK algorithm,” since it was developed by David Ruhm, Don Mango and Rodney Kreps.
- For example, when using underwriting loss, one can have negative co-TVaR amounts for some segments if a segment has a positive underwriting result on average across the scenarios in which company has a negative underwriting result. Negative results can make application in capital allocation problematic.
- Joan Lamm-Tennant, Global Chief Economist
- Donald Mango, Chief Actuary
- Eddy Vanbeneden, Managing Director
- Gary Venter, Managing Director
- Steve White, Managing Director
- Mike Wynne-Jones, Managing Director
- Andreas Durig, Senior Vice President
- David Flandro, Senior Vice President
- Michael Owen, Senior Vice President
- Jeff Bellmont, Senior Vice President
- Benoit Butel, Vice President
- Sebastien Portman, Vice President