# «A Survey of Capital Allocation Methods with Commentary Topic 3: Risk Control Gary G Venter Guy Carpenter Instrat One Madison Avenue, New York NY USA ...»

A Survey of Capital Allocation Methods with Commentary

Topic 3: Risk Control

Gary G Venter

Guy Carpenter Instrat

One Madison Avenue, New York NY USA

+1-212-323-1739

Fax: +1-212-390-4739

gary.g.venter@guycarp.com

**Abstract:**

A number of methods of allocating capital to business unit, e.g., line of business, profit cen-

ter, etc., are discussed. Goals of capital allocation include testing the profitability of business

units and determining which units could best be grown to add value to the firm. Methods of approaching these questions without allocating capital are included in the discussion.

Keywords: capital allocation, risk measures A Survey of Capital Allocation Methods with Commentary Capital allocation is generally not an end in itself, but rather an intermediate step in a deci- sion-making process. Trying to determine which business units are most profitable relative to the risk they bear is a typical example. Pricing for risk is another.

Return-on-capital thinking would look to allocate capital to each business unit, then divide the units’ profits by that capital. Of course if profit were negative, you would not need to di- vide by anything to know it is not sufficient. But this approach would hope to be able to dis- tinguish the profitable-but-not-enough-so units from the real value-adders.

The same issue can be approached without allocating capital, using a theory of market risk pricing. The actual pricing achieved by each business unit can be compared to the risk price needed. This would require having a good theory of risk pricing, where the previous approach would depend on having a good theory of capital allocation. Since both are addressing the same decisions, both will be included in this survey. For those who like to phrase the issue as one of return on capital, the pricing method can be put into allocation terminology after the fact by allocating capital to equalize the ratio of target return to capital across business units.

Rating business units by adequacy of return is not necessarily the final purpose of the exer- cise. The rating could be used in further decisions, such as compensation and strategies for future growth. For strategic decisions another question is important – not how much capital a business unit uses, but how much more is needed to support the target growth. In general it will be profitable to grow if the additional return exceeds the cost of the additional capital. In some cases a company might not need too much more than it already has for the target growth, in which case not much additional profit would be needed to make the growth worthwhile.

This is the marginal pricing approach, a basic tenet of financial analysis. It differs from capital allocation in that for marginal-cost pricing not all capital has to be allocated to reach a decision. Only the cost of the capital needed to support the strategy has to be determined, to see if it is less than the profit anticipated. Methods of quantifying the cost of marginal capital will be reviewed here as well, as again this is aiming at answering the same strategic questions.

Finally, another way to determine which business units are adding most to the profitability of the firm is to compare the insurer to a leveraged investment fund. Sometimes this is called the cost-of-float approach. The overall return of the insurer can be evaluated by finding the borrowing rate that would equalize its risk and return after tax to a leveraged investment fund. If the fund would have to be able borrow significant funds at a particularly low rate of interest to match the insurer’s risk and return, then the insurance business is clearly adding value. The business units can be ranked based on their impacts on this borrowing rate.

Thus while the general topic is capital allocation, this survey is looking at methods for answering questions that capital allocation is addressing. To summarize, four basic approaches

**will be reviewed:**

1. Selecting a risk measure and an allocation method and using them to allocate capital

2. Comparing actual vs. model pricing by business unit

3. Computing the cost of the marginal capital needed for or released by target strategies

4. Evaluating profitability in comparison to a leveraged mutual fund

EPD is expected policyholder deficit, i.e., the expected value of default amounts. It can also be generalized to include the expected deficit beyond some level, rather than beyond default.

If b is the target amount, the EPD beyond b is: Pr(Xb)E[(X – b)|Xb].

Tail value at risk is the expected losses in the event that losses exceed the value-at-risk target.

If the target loss level is b, this is E(X|Xb).

Proportional spread is the most direct method – apply the risk measure to each business unit and then allocate the total capital by the ratio of business unit risk measure to the sum of all the units’ risk measures. Usually the sum of the individual risks will be greater than the total risk, so this method is crediting each unit with a diversification benefit.

3 Marginal analysis measures the risk of the company with and without a specified business unit. The difference in required total capital is then the marginal capital for the business unit.

The total capital can then be allocated by the ratio of the business unit marginal capital to the sum of the marginal capital of all the units. This usually allocates more than the marginal capital to each unit. The incremental marginal method is similar, but the change in capital is calculated for just the last increment of expected loss for the unit, say the last dollar. Whatever reduction that is produced in the risk measure by eliminating one dollar of expected loss from the business unit is expressed as a capital reduction ratio (capital saved per dollar of expected loss) and applied to the entire unit to get its implied incremental marginal capital to use in the allocation.

The game theory approach is another variant of the marginal approach, but the business units are allowed to form coalitions with each other. The marginal capital for a unit is calculated for every group of units it could be a part of, and these are averaged. This gets around one objection to marginal allocation – that it treats every unit as the last one in. This method is sometimes called the Shapley method after a founder of game theory.

The Myers-Read method also uses marginal allocation. It sets the marginal capital needed to support an exposure increase equal to the additional capital it would take to make the cost of the default put, as a percentage of expected losses, the same before and after. It has the advantage over other marginal methods that the marginal increments add up to the total capital. This method is discussed in detail in Appendix 2.

Equalizing relative risk involves allocating capital so that each unit, when viewed as a separate company, has the same risk relative to expected losses. Applying this to the EPD measures, for instance, would allocate enough capital to each business unit make the EPD for every unit the same percentage of expected loss.

Co-measures were introduced by Rodney Kreps as a way of allocating capital in an additive manner that is nonetheless consistent with the overall risk measure used to define total capital. Appendix 3 discusses these in greater detail. They can be most easily thought of in terms of a scenario generator. Take the case where the total capital requirement is set to be the tail value at risk at the 1-in-1000 probability level. Then in generating scenarios, about 1 in 1000 would be above that level. The co-Tail VaR for each business unit would just be the average of its losses in those scenarios. This is its contribution to the overall Tail VaR.

Co-measures provide a totally additive allocation. Business units could be combined or subdivided in any way and the co-Tail VaR’s would add up. For instance, all the lines of business could be allocated capital by co-Tail VaR, then each of these allocated down to state level, and those added up to get the state-by-state capital levels for all lines combined. This could be done for peril or other business categories as well.

Commentary on Allocation by Risk Measure VaR could be considered to be a shareholder viewpoint, as once capital is exhausted, the amount by which it has been exhausted is of no concern to them. EPD, default option cost, X TVaR, and Tail VaR relate more to the policyholder viewpoint, as they are sensitive to the degree of default. And indeed the shareholders might do well when they consider policyholder needs. All of these measures ignore risk below the critical probability selected. VaR also ignores risk above that level, while the tail measures evaluate that risk linearly, which many consider to be an underweighting.

4 Variance does not distinguish between upward and downward deviations, and so could provide a distorted view of risk when these directions are not symmetric – which is the usual case. Semi-variance looks only at adverse deviations, so accounts for this. Taking the mean of a transformed loss distribution is a risk measure aiming at quantifying the financial equivalent of a risky position, and it can get around the problems of the tail methods. More exploration of transformations could be useful.

Allocating by marginal methods is accepted in financial theory. However, allocating more than the pure marginal capital to a unit it could lead to pricing by a mixture of fixed and marginal capital costs, violating the marginal pricing principle. Even when the total capital is the sum of the marginal increments, as in Myers-Read, there is no tie-in between the capital allocated to a line and the value of its risk. Thus it would be a great coincidence if this allocated capital were right for a return-on-capital ranking.

The co-measure approach is consistent with the total risk measure and is completely additive.

Thus if the risk measure gives the right capital need overall, the co-measure shows each line’s contribution to that. But it too could violate marginal pricing.

Myers-Read was introduced as a method of allocating the frictional costs of holding capital.

These are discussed more in Appendix 2, but as a definition I would propose that costs which arise from holding capital even if no risk is written are frictional costs. Corporate tax on investment income is an example. A more delicate issue is any lower investment income resulting from taking less investment risk in order to give policyholders greater security. I would hold that this is a frictional cost as well. Even though it results from the intent to sell insurance, this does not differentiate it from other frictional costs.

The return for actually putting the capital at risk is a different matter. This relates to the amount of risk taken, not the amount of capital allocated. In financial models beta is almost always a component of the return for bearing risk, but it is not generally a part of the frictional cost. Some actuarial pricing approaches have assumed that pricing to recoup frictional costs is sufficient, and this is encouraged by assertions that beta is zero for underwriting anyway. More recent theory, discussed below, shows that risk pricing is more than beta. This suggests that even if allocating capital by risk measure is sufficient for allocating frictional costs, there are other elements of return that will not be proportional to the amount of capital held and so should be measured in some other way.

Approach 2 – Compare Actual vs. Model Pricing A traditional use of capital allocation is to price business to equalize return on capital. However even if the allocation method is intuitively satisfying, there is no guarantee that such pricing would correspond to the market value of the risk transfer. If instead actual pricing is compared to value pricing, the profitability of business units can be evaluated without allocating capital at all (except to the degree this is necessary in the pricing to compute the frictional costs of holding capital). But for those who still prefer a target return on capital, capital could be allocated after the pricing by equalizing the return on capital from the value prices.

This method requires an evaluation of the market value of the risk transfer provided. Financial methods for valuing risk transfer typically use transformations of the loss probabilities to risk-adjusted probabilities, with covariance loadings like CAPM being one special case. This is a fairly technical calculation and to date there is no universal agreement on how to do it.

5 Some transforms do appear to give fairly good approximations to actual market prices, however. The Wang transform has been used successfully in several markets to approximate risk pricing. Finance professionals now appear to favor an adjusted CAPM approach that corrects many of the over-simplifications of the original formulation. For instance, a correlation with the insurer’s own results may be as important as correlation with the market in determining the cost of risk transfer.

To use CAPM or similar methods, costs are first identified, then a risk adjustment added.

Three elements of cost have been identified for this process: loss costs, expense costs, and the frictional costs of holding capital. The latter is not the same as the reward for bearing risk, which is separately incorporated in the risk adjustment.

The CAS Committee on the Theory of Risk is sponsoring the Risk Premium Project to look

**into how to do risk pricing right. Starting from CAPM, they are looking at are several considerations needed to get a realistic market value of risk transfer. Some issues in this area are:**

• Company-specific risk needs to be incorporated, both for differential costs of retaining vs. raising capital1 and for meeting customer security requirements.

• The estimation of beta itself is not an easy matter2

• Other factors besides beta are needed to account for actual risk pricing3

• To account for the heavy tail of P&C losses, some method is needed to go beyond variance and covariance4,5

• Jump risk needs to be considered. Sudden jumps seem to be more expensive than continuous variability, possibly because they are more difficult to hedge by replication.

Large jumps are an element of insurance risk, so need to be recognized in the pricing.