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Latest Research
An X-Ray for Risk
by David L. RuhmIn this article, a method will be described that measures the contributions to risk (adverse scenarios) from business segments, lines, and large individual contracts. The method is analogous to an x-ray because, once risk scenarios are identified at the total-company level, a simple technique is applied to examine each of the company's parts (lines, segments or even large individual contracts), revealing where risk is residing and how much is present in each area.
This method is based on the philosophy that the company's total results are the most meaningful basis for the measurement of risk. Line level risk measurements are less meaningful since they effectively treat each line as if it were written on a stand-alone basis, which does not account for correlation and diversification effects. Also, all lines are not equal when viewed through the total company perspective: a 1-in-100-years event in a larger, catastrophe-exposed line can have more impact on the company's financial condition than a 1-in-100-years event in a smaller, less volatile line.
The concept that total-company results are the ideal basis for defining risk has intuitive appeal, but there have been difficult technical problems with modeling and measuring risk across several segments, and also with attributing such a risk measurement down to individual lines in an accurate manner. The first problem is starting to give way, through recent advances in modeling and computing power. The method described in this article addresses the second problem.
The Method
The risk x-ray is simply a weighted average of possible outcomes, where the weights are a function of total-company results. More severe outcomes receive greater weight. The risk x-ray method is as follows:
1. Start with a table of potential outcome scenarios for the total company's results and for the lines of business being analyzed (see the table below for an example).
2. Identify those particular scenarios that define risk for the company, from the viewpoint of company management. The risk definition will be based on a measure of total-company results, such as net income or change in equity.
3. Assign non-negative weights to each outcome scenario, reflecting the degree of risk associated with each outcome. Assign higher weights to the risk-scenarios identified in Step 2, based on their severity and the secondary consequences that can be expected. For example, a ten million dollar loss scenario that would also destroy future business opportunities worth five million dollars would receive a weight of 1.50, reflecting the lost opportunities' value in addition to the direct loss amount. Some judgment will be necessary, since these secondary impacts are usually difficult to forecast precisely.
4. Multiply each scenario's weight from Step 3 by the scenario's probability. This produces the "risk x-ray vector."
5. To determine any line's contribution to total-company risk scenarios, calculate the line's expected value using the x-ray vector in place of probabilities. In spreadsheet terms, take the "sum-product" of the line's net outcomes with the x-ray vector, across all scenarios. A positive value shows that, overall, the line provides relief in risk scenarios (it probably helps in some scenarios and adds to risk in others), while a negative value shows that the line is contributing to total-company risk. The individual lines' values will sum to the company's risk value as shown below, making it possible to determine each line's percent contribution to the company's overall risk.
An Example (simplified for clarity of exposition)
We are going to analyze a hypothetical company with three lines of business. There are ten possible scenarios for the current underwriting year, as shown in Table 1. The number of lines and the number of scenarios in this example are both intentionally tiny to more clearly illustrate the ideas and method, which would apply in the same way to thousands of scenarios generated by an actual DFA model.
When generating the individual line results using a DFA model, the common parameters, such as realized interest rates, are the same across lines within each scenario. This allows actual correlating effects such as interest rate movements to be reflected in the simulated results for the total company, and in the assessment of risk.
This hypothetical company's management views "risk" in terms of net income, described by two factors: the probability of negative income for the year, and the potential severity of such a net loss if it occurs. If a very severe net loss were to occur, it could trigger secondary financial consequences, so those loss scenarios are disproportionately detrimental to the company and will receive more weight.
By looking across each row, we can easily see the lines' contributions to each scenario. For instance, Line A is contributing much of the risk for scenario #3, _10 out of the _15 total, while Line B is actually helping to hedge the risk for scenario #3 by contributing positive income of +5. At the bottom of each column, expected returns can be compared. As compensation for the risk it contributes across scenarios, Line A provides four times the expected return of the other lines (2.00 vs. 0.50).
The objective is to identify which lines are contributing the most to total-company risk, and how the lines compare in terms of risk vs. return contributed.
To make the process more quantitative and objective, we follow the steps described above. First, we assign the weights. Weights are assigned as follows: zero for non-loss scenarios, one for small loss scenarios (between 0 and _10), and two for large loss scenarios worse than _10. Then we calculate the risk x-ray vector, yielding the results in Table 2.
Next we apply the x-ray to the total company and the individual lines, which gives the results in Table 3.
The "% of income" row shows the respective contributions of the lines to total-company expected income. The "X-ray result" row shows the results of applying the risk x-ray column's values to the lines' outcomes. All lines contribute to risk, with the most risk contributed by Line B. The proportions contributed by each line to total-company risk are shown in the "% of risk" row. Line A contributes one-third of the risk, Line B contributes close to half and Line C about one-fourth of the total.
The "Risk/Return" row shows the ratio of the risk percentages to the expected income percentages. For example, Line A contributes 67 percent (about 2/3) of the company's total expected income but only 33 percent (about 1/3) of the total risk, so it has a risk/return ratio relative to the overall company of 0.49 (about 1/2). Based on the idea that expected income should be commensurate to risk assumed, this statistic shows whether a line's risk/return profile is better or worse than the company average, and by how much.
If this company's management wanted to improve the overall risk profile, one approach might be to expand slightly in Line A and contract slightly in Line B. The results of a 20 percent increase in Line A and a 20 percent decrease in Line B would be as follows in Table 4.
Expected income for the company has increased 10 percent, from 3.00 to 3.30. The most likely risk scenario (#10), which has a probability of 20 percent, has had its severity cut in half, from _8 to _4. One of the severe scenarios (#3) has been increased slightly. In total, risk is a bit lower, as reflected in the slightly better X-ray result for the company (_7.72 vs. the previous _7.90).
Applications
The example shown above is highly simplified. In practice, a more refined set of weights can be chosen to reflect risk in greater detail. Also, the weights can be chosen so as to implement a risk metric of the actuary's choice, such as tail-value-at-risk.
The concepts discussed in this article have been written about in more detail by Rodney Kreps in his work on co-measures (a term coined by Dr. Kreps) and by Donald Mango in several papers and presentations. Dr. Kreps' work includes formulas for the weights needed to implement several well-known risk metrics. Related work has also been published by Gary Venter and John Major. Those interested in a more thorough technical discussion of these ideas can consult their papers, available from the CAS online library.
