CAPITAL ALLOCATION FOR INSURANCE COMPANIES WHAT GOOD IS IT? H e l m u t G r ü n d l, Berlin* and H a t o S c h m e i s e r, Berlin*

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1 CAPITAL ALLOCATION FOR INSURANCE COMPANIES WHAT GOOD IS IT? By H e l m u t G r ü n d l, Berlin* and H a t o S c h m e i s e r, Berlin* SEPTEMBER 23, SECOND DRAFT JEL-KLASSIFICATION: G22, G3, G32 *Institut für Bank-, Börsen- und Versicherungswesen Dr. Wolfgang Schieren-Lehrstuhl für Versicherungs- und Risikomanagement, gefördert von der Allianz AG und dem Stifterverband für die Deutsche Wissenschaft

2 2. Introduction In their recent article, Capital Allocation for Insurance Companies (The Journal of Risk and Insurance, 2), Stewart Myers and James Read seem to have resolved one of the major problems in the scientific discussion about capital allocation. They find a unique and non arbitrary allocation method that leads to an adding up property, i.e., the surplus (or equity capital) allocated to the single lines of business add up to the overall surplus (equity capital) of the insurance company. 2 Using option-pricing techniques, the allocation depends on the marginal contribution of a single line of business to the default value of the whole firm. 3 In the context of specific distributional assumptions, Myers and Read provide closed-form solutions for the allocated capital. In particular, they propose using this allocation method in insurance pricing and price regulation. 4 The purpose of this article is to focus on some problems of the Myers and Read contribution and to clarify a couple of arguments in the ongoing capital allocation discussion. 5 Myers and Read state that capital allocation is a prerequisite for pricing insurance contracts. 6 However, given the optionpricing framework of their paper, 7 in which no frictional costs appear, insurance prices can be calculated without allocating equity capital to lines of business, given a desired default value of the insurance company. 8 Having calculated the premium, the next step is to take specific risk management measures that will assure the desired safety level of the contract. One possible means of doing so is to increase the equity capital of the insurance company Myers and Read (2, p. 545). In the Myers and Read context, surplus must be distinguished from equity capital. We will discuss this below. The default value is the value of the payments the insured will forego if the insurance company defaults. Myers and Read (2, pp ). Cummins (2); Kneuer (23); Meyers (23); Mildenhall (23); Ruhm and Mango (23); Venter (23); Vrieze and Brehm (23). Myers and Read (2, p. 573). Myers and Read (2, pp , , and ). Phillips, Cummins, and Allen (998, pp ).

3 3 But again, to determine that change, no allocation of equity capital to the lines of business is necessary. Moreover, in the calculus determining this amount of new equity, the calculated premium is an input variable. Myers and Read also discuss finding a way to allocate the frictional costs of hing surplus to the single lines of business and in the end to single insurance contracts. 9 These frictional costs can include agency costs, information costs, or costs stemming from double taxation of investment income. More precisely, if a new marginal insurance contract is to be written, Myers and Read determine the additional surplus that will be needed to keep up the desired safety level of the insurance company. Therefore, when this additional surplus is known, the variable frictional costs of surplus due to writing a new contract can be calculated. The additional surplus needed for a marginal contract, summed up over all existing contracts of a line of business, yields the surplus allocated to that line; summed up over all existing contracts of the firm, it yields the existing surplus of the firm. In this marginal context, the adding up property hs for surplus and therefore also for the frictional costs of the company, driven by surplus. But, as has already become clear, when pricing a new contract, no capital allocation or cost allocation to the preexisting lines of business is necessary, and, of course, there is hardly a reason to re-price the preexisting portfolio. Furthermore, as admitted by Myers and Read, 2 if an inframarginal contract is to be priced, a marginal calculus is of limited use. To state this as simply as possible: if capital allocation to single lines of business is unnecessary for the intended purposes, certain properties of allocation methods e.g., the adding up property found by Myers and Read have no economic relevance. 9 2 Myers and Read (2, p. 546); Venter (23, p. 466). Myers and Read (2, p. 545). Myers and Read (2, p. 559). Myers and Read (2, p. 549).

4 4 In the next section, Pricing Insurance Contracts, Risk Management Costs, and Equity Capital, we derive our arguments in a situation without frictional costs, thus providing the basis for the subsequent section, Pricing Insurance Contracts and Frictional Costs, where we analyse the context in which frictional costs are integrated. An example illustrating our ideas is given in the appendix. 2. Pricing Insurance Contracts, Risk Management Costs and Equity Capital In our analysis we apply the same contingent claims approach as used in the Myers and Read article. It is a one-period option-pricing framework for pricing insurance contracts as first proposed by Doherty and Garven. 3 Let P indicate the competitive premium (paid at time t = ) of the preexisting underwriting portfolio of an insurance company that consists of several lines of business. The insurance portfolio yields stochastic claims costs L at time t =. The present value of these claims costs is denoted by PV(L ). PV( ) denotes an arbitrage-free valuation function. D stands for the present value of the default put option. If E indicates the initial equity capital of the company at time t =, and ~ r the stochastic rate of return on its investment portfolio, then the default value D is given by: D = PV(max{L (E + P )( + r~ ), }). () The competitive premium 4 of the initial insurance portfolio P is: P = PV(L ) D. (2) As in the Myers and Read article, the company s safety level can be defined by the default-value-to-liability ratio: Doherty and Garven (986). This premium should also be the basis for a regulated premium if the regulator wants sharehers and policyhers to earn a risk adequate return on their capital provided.

5 5 D d =. (3) PV(L ) The objective now is to price a new contract in line i with stochastic claims costs L. If the default-value-to-liability ratio of the preexisting portfolio is to be maintained, then for the default-value-to-liability ratio of the new contract in line i, d new,i = D / PV(L ), the following must h: 6 d : = = d d. (4) Given the assumption of needing to maintain the insurer s default-value-toliability ratio, the competitive price of the new contract P immediately follows without any allocation of equity capital to the lines of business from Equations (2) to (4): 7 P P = PV(L ) = PV(L ) ( d). (5) PV(L ) Given the default-free value of the claims, PV(L ), the price of the new contract P is influenced only by the desired safety level: obviously, the risk interdependencies between the existing portfolio and the new contract have no effect on pricing. The premium P is a fair premium only if the insurer does in fact maintain the promised safety level via adequate risk management measures. One example of risk management, focused on by Myers and Read, is changing the amount of equity capital. 8 The scope of the equity capital change E can Myers and Read (2, p. 557). See also Butsic (994). Myers and Read (2, p. 559). Phillips, Cummins, and Allen (998, pp ); Gründl and Schmeiser (22, pp ). Myers and Read look only at the scope of the specific risk management measure equity capital. For arbitrage reasons, the competitive price of any risk management measure (e.g., equity capital, reinsurance, financial hedging) with the identical effect on the default value is the same and depends on the risk interdependencies within the insurance company. The way to figure this price is as follows. If the insurance company received

6 6 be (implicitly) calculated by setting the overall equity capital of the insurance company equal to the present value of the future payments to its sharehers. The competitive premiums for the preexisting portfolio and the new contract are given by Equations (2) and (5). E E PV(max{(E E P P )( ~ + = r ) (L + L ), }). (6) From Equation (6) it can be seen that for calculating the additional equity capital requirements, the competitive premium P is an input variable. Hence we do not agree with Myers and Read, who claim that to set the premiums for a policy, an insurance company must estimate the surplus required to support that policy. 9 However, Myers and Read do not use the equity formulation shown in Equation (6) directly, but instead determine surplus requirements and make surplus allocations to lines of business. Surplus S (before writing a new contract) is given by: S E + P PV(L ) = E D =. (7) For marginal changes of PV(L ), and under the assumption that the variables on the right-hand side of Equation (6) are either joint-lognormally or joint-normally distributed, Myers and Read give an explicit closed-form solution for the additional surplus needed. 2 They determine the marginal change of surplus s i : P and undertook no further risk management, writing the new contract would lead to a certain (net) present value PV RM, i for the company. Thus, PV RM, i is exactly the competitive price of an (additional) risk management measure necessary to ensure the desired default-value-to-liability ratio d. In our case we get for the new contract in line i: PV RM,i = PV(max{(E + P + P )( + r ) (L + L ), }) E. ~ Myers and Read (2, p. 573). Myers and Read (2, p. 573). Myers and Read (2, pp. 559, 578).

7 7 s i,i S = (8) PV(L ) di= d E For small changes of PV(L ), the amount of new equity capital needed to ensure the company s desired (initial) safety level can be approximately calculated by transforming the surplus requirement S : E i = (s + d) PV(L ) = S + d PV(L ) (9) In our opinion, using surplus instead of equity is somewhat confusing. For example, for d < si <, a negative additional surplus coincides with a positive new equity capital. The marginal surplus requirements s i have the property: 22 M i= s i PV(L,i ) = S. () Myers and Read interpret the marginal surplus requirements of the single lines of business i multiplied by the present value of the default-free liabilities in these lines as an allocation of surplus to the single lines of business. 23 From Equation () one sees that summing up the allocated capital across all M lines of business leads to the company s present surplus. This adding up property also hs true for equity capital: M i= (s i,i + d) PV(L ) = E. () Myers and Read claim 24 that these capital allocations are unique and not arbitrary. They therefore disagree with prior literature arguing that capital Myers and Read (2, pp ). For a detailed discussion of the assumptions necessary for the adding-up property, see Mildenhall (23). Myers and Read (2, p. 554). Myers and Read (2, p. 545).

8 8 should not be allocated to lines of business or should be allocated uniformly. However, the question remains: What good is this allocation? We have seen that in Myers and Read s option-pricing formulas and examples capital allocation to lines of business is of no use in pricing insurance contracts. Furthermore, for determining the needed change in the insurance company s equity capital after it writes a new contract, neither an allocation of equity to the lines of business nor the adding up property seem to have economic relevance. 3. Pricing Insurance Contracts and Frictional Costs Frictional costs are not explicitly integrated into the analytical derivations and examples given in the Myers and Read contribution, even though considering frictional costs of surplus in insurance pricing is a motivation for their paper. 27 The idea is that if one knows the additional surplus needed for a new (and marginal) contract, then the frictional costs proportional to that additional surplus are determined as well and can be used as a loading on the net premium. Therefore, the Myers and Read approach to calculating surplus requirements leads to the correct loading in the marginal case and for frictional costs directly driven by surplus. That being said, we are puzzled as to why one should (or would) take this roundabout route when pricing a new contract. In principle, frictional costs can be directly integrated in a pricing calculus analogous to Equation (6); for the case of corporate taxes, this has already been shown by Doherty and Garven. 28 Hence, for a new contract, we get for different safety levels of the insurance company different combinations of additional surplus (or equity capital) and required premiums (including the frictional costs loading). And, once again, for determining the frictional cost loading on the premium, neither a capital allocation to lines of business nor an adding up property is needed Myers and Read (2, pp. 559, 578). For examples, see Myers and Read (2, p. 548), and Merton and Per (993, pp. 27 3). For this point, see also Venter (23, pp ). Doherty and Garven (986, p. 34).

9 9 The advantages of a direct integration of frictional costs are obvious: in contrast to the approach taken by Myers and Read, this approach not only hs for marginal changes of the underwriting portfolio, but also for inframarginal ones. Furthermore, frictional costs linked to cost drivers other than surplus can be integrated. However, an ad hoc incorporation of frictional costs in an otherwise arbitrage-free pricing model, as used in the Myers and Read article, leads to problems. First, as an example, an integration of taxes tends to result in a capital market that is without further assumptions no longer arbitrage free. Hence, a unanimously supported present value calculus may not exist. 29 Second, even if an arbitrage-free valuation exists, depending on the individual circumstances of insurance companies, different frictional costs lead to different competitive gross premiums for insurance contracts with the same safety level. Thus there will be arbitrage opportunities in the insurance market, which in turn leads to problems for an insurance regulator who sets the premiums by the described net present value calculus. Myers and Read do see this problem and therefore propose to calculate the regulated premium on the basis of an efficient (i.e., cost minimizing) risk management mix. 3 Unfortunately, no analytical framework is presented in their paper for deducing this efficient risk management mix. 3 Furthermore, under this price regulation, the (yet unknown) efficient firm structure would have to be adopted by all insurance firms if they want to survive in the long run. 4. Conclusion In the option-pricing framework used by Myers and Read (2) we found no reason to allocate capital across lines of business for the purpose of pricing insurance contracts and determining surplus requirements. This is also true in the case where frictional costs of surplus, e.g., corporate taxes, are integrated Schaefer (982, pp ); Dybvig and Ross (986); Ross (987). Myers and Read (2, pp ). For this point, see also Kneuer (23, pp ).

10 into the pricing framework. Since we do not see the necessity for allocating capital to the different lines of business, certain characteristics of the allocation method discussed by Myers and Read, especially the adding up property, have limited economic relevance. Appendix In the following, we illustrate Equations () () with an example using an insurance company consisting of two lines of business. To keep things simple, we assume a risk-neutral world and a risk-free rate of return of zero. The investment portfolio return of the insurer is risk-free. Furthermore, all random variables, i.e., the claims distributions, are normally distributed. The insurance company s default-value-to-liability ratio is set at d =.5 (see Equation (3)). The data for the example are given in the following table. TABLE Example of an insurance company running two lines of business Line Line 2 Expected claims costs per contract $ $ Standard deviation of claims costs per contract $.5 $.25 Correlation coefficient between the claims costs.5.25 Number of contracts 6, 4,,i Competitive premium ( P ) $597, $398,,i Default-free premium ( PV(L ) ) $6, $4, We assume a correlation coefficient of.25 between the claims costs of Line and Line 2, resulting in expected claims costs for the whole underwriting portfolio of $,, and a standard deviation of $25,.87. In the initial situation, a net present value of for the sharehers yields a necessary amount of equity capital E = $42, to ensure the desired default value D = $5, E follows from the condition: = E(max{E + P L, }) E. With L being normally distributed, a closed-form solution for this condition in analogy to Equation (II) below is available.

11 If the objective is to maintain the insurer s safety level after signing a new contract at d =.5, then one new contract in Line (or Line 2) must be new, new,2 priced with P = P =.995 (see Equation (5)). What is the competitive price of the additional risk management measure the insurer must take in order to provide the promised safety level? Because of the assumption that claims costs are normally distributed, there exists a closed-form solution: 33 With V = +, E + P P L = +, and L L V E(L) z =, we get: σ(l ) PV RM,i = [V E(L )] N(z) + σ(l ) N (z) E, (I) where N(.) stands for the cumulative probability function and N (.) for the probability density function. For Line (Line 2), this leads to risk management costs of PV RM, = $.3 ( PV RM, 2 = $.5). If equity capital is the chosen risk management measure, the necessary amount of equity capital can be calculated in the normal case by: E + E = [V + E E E E(L)] N(z + ) + σ(l) N (z + ). (II) σ(l ) σ(l ) new, For Line (Line 2) this leads to an equity capital increase of E = new,2 $ ( E = $.92). These values can also (approximately) be derived from the explicit formula given by Myers and Read. 34 Using Equation (), the adding up property of capital allocation approximately hs, setting s + d = and s 2 + d =.92: $6, +.92 $4, E Winkler, Roodman, and Britney (972, p. 292). For the general case, see the equation in footnote 8. Myers and Read (2, p. 578 (bottom)). The authors determine marginal surplus requirements that can be transformed to marginal equity requirements according to Equation (9). The marginal surplus requirements, in our example, are: s = and s 2 = Using Equation (), these figures lead to the exact initial equity capital E.

12 2 Calculating marginal equity requirements as proposed by Myers and Read (2) can be seen as a special case of determining equity requirements according to Equation (6). This formula clearly includes marginal and inframarginal 35 changes in the firm s structure. However, as Myers and Read point out, the adding up property hs true only for the marginal case, not for the inframarginal Merton and Per (993). Myers and Read (2, p. 547).

13 3 References Butsic, R., 994, Solvency Measurement for Property-Liability Risk-Based Capital Applications, Journal of Risk and Insurance, 6: Cummins, J. D., 2, Allocation of Capital in the Insurance Industry, Risk Management & Insurance Review, 3: Doherty, N. A., and J. R. Garven, 986, Price Regulation in Property-Liability Insurance: A Contingent-Claims Approach, Journal of Finance, 4: 3 5. Dybvig, P. H., and S. A. Ross, 986, Tax Clienteles and Asset Pricing, in: Journal of Finance, 4: Gründl, H., and H. Schmeiser, 22, Pricing Double Trigger Reinsurance Contracts: Financial versus Actuarial Approach, Journal of Risk and Insurance, 69: Kneuer, P. J., 23, Review of Capital Allocation for Insurance Companies by Stewart C. Myers and James R. Read Jr., in: The Casualty Actuarial Society Forum, Fall 23 Edition, Download (9/2/23): Merton, R. C., and A. F. Per, 993, Theory of Risk Capital in Financial Firms, Journal of Applied Corporate Finance, 6 (No. 3): Meyers, G. G., 23, The Economics of Capital Allocation, in: The Casualty Actuarial Society Forum, Fall 23 Edition, Download (9/2/23): Mildenhall, S. J., 23, A Note on the Myers and Read Capital Allocation Formula, in: The Casualty Actuarial Society Forum, Fall 23 Edition, Download (9/2/23): Myers, S. C., and J. A. Read, Jr., 2, Capital Allocation for Insurance Companies, Journal of Risk and Insurance, 68:

14 4 Phillips, R. D., J. D. Cummins, and F. Allen, 998, Financial Pricing of Insurance in the Multiple Line Insurance Company, Journal of Risk and Insurance, 65: Ross, S. A., 987, Arbitrage and Martingales with Taxation, Journal of Political Economy, 95: Ruhm, D. L., and D. E. Mango, 23, A Method of Implementing Myers-Read Capital Allocation in Simulation, in: The Casualty Actuarial Society Forum, Fall 23 Edition, Download (9/2/23): Schaefer, S. M., 982, Taxes and Security Market Equilibrium, in: Financial Economics: Essays in Honor of Paul H. Cootner, W. F. Sharpe and C. M. Cootner, eds. (Englewood Cliffs, N.J.), Venter, G. G., 23, Discussion of Capital Allocation for Insurance Companies by Stewart C. Myers and James R. Read Jr., in: The Casualty Actuarial Society Forum, Fall 23 Edition, Download (9/2/23): Vrieze, K. J., and P. J. Brehm, 23, Review of Capital Allocation for Insurance Companies by Stewart C. Myers and James R. Read Jr. Practical Considerations for Implementing the Myers-Read Model, in: The Casualty Actuarial Society Forum, Fall 23 Edition, Download (9/2/23): Winkler, R. L., G. M. Roodman, and R. R. Britney, 972, The Determination of Partial Moments, Management Science, 9:

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