PRICING DOUBLE-TRIGGER REINSURANCE CONTRACTS: FINANCIAL VERSUS ACTUARIAL APPROACH

Transcription

1 The Journal of Risk and Insurance, 2002, Vol. 69, No. 4, PRICING DOUBLE-TRIGGER REINSURANCE CONTRACTS: FINANCIAL VERSUS ACTUARIAL APPROACH Helmut Gründl Hato Schmeiser ABSTRACT This article discusses various approaches to pricing double-trigger reinsurance contracts a new type of contract that has emerged in the area of alternative risk transfer. The potential coverage from this type of contract depends on both underwriting and financial risk. We determine the reinsurer s reservation price if it wants to retain the firm s same safety level after signing the contract, in which case the contract typically must be backed by large amounts of equity capital (if equity capital is the risk management measure to be taken). We contrast the financial insurance pricing models with an actuarial pricing model that has as its objective no lessening of the reinsurance company s expected profits and no worsening of its safety level. We show that actuarial pricing can lead the reinsurer into a trap that results in the failure to close reinsurance contracts that would have a positive net present value because typical actuarial pricing dictates the type of risk management measure that must be taken, namely, the insertion of additional capital. Additionally, this type of pricing structure forces the reinsurance buyer to provide this safety capital as a debtholder. Finally, we discuss conditions leading to a market for double-trigger reinsurance contracts. INTRODUCTION Risk transfer between primary insurance and reinsurance companies today takes place against a backdrop of major changes in international insurance and financial markets. 1 One of these changes is the trend for the two markets to converge. This trend manifests itself in a growing number of product innovations, subsumed under the terms alternative risk transfer 2 (ART) and financial reinsurance. 3 Certain aspects of these innovations Helmut Gründl and Hato Schmeiser are at Humboldt-Universität zu Berlin, Germany. Germany, Dr. Wolfgang Schieren Chair for Insurance and Risk Management, Sponsored by Allianz A6. The authors wish to thank Neil A. Doherty and two anonymous referees for helpful comments. 1 Bochicchio et al. (1998); Laster and Raturi (2001). 2 Baur and Schanz (1999). This would include, in particular, the various forms of risk securitization, derivative financial instruments on insurance index positions, contingent capital, and reinsurance swaps. 3 Schanz (1997). This would include, in particular, time and distance contracts, loss portfolio transfer, spread loss treaties, and finite risk contracts. 449

2 450 THE JOURNAL OF RISK AND INSURANCE have been examined by others, 4 but this is not the case for double-trigger reinsurance contracts. 5 Double-trigger contracts differ from ordinary reinsurance contracts. The potential coverage from a double-trigger contract depends on both actuarial risk development and financial risk development. 6 Thus, a double-trigger contract will pay only in situations where poor underwriting results can no longer be offset by good capital investment returns (and vice versa). In this sense, the various forms of trigger products fulfill the function of asset-liability management and balance sheet protection, especially in jurisdictions that require the use of current market value in firms financial statements. In practice, essentially two basic types of trigger reinsurance contracts are in the reinsurance market: (1) traditional reinsurance contracts 7 that make the reinsurer liable for payments only when a capital market index of a level i is below a certain threshold value at the end of a given period, and (2) contracts specifying that the reinsurer must make further compensation payments (in addition to those specified above) depending on the extent of the shortfall level [Y i] +. In practice, both contracts can occur with limited (financial reinsurance form) or full risk transfer. We will discuss the first contract type with full risk transfer. The way this type of double-trigger contract affects the primary insurer s portfolio was recently discussed in a Swiss Re publication, 8 which revealed that the use of such contracts considerably lowered the downside risk associated with the insurer s operating results (measured on a value-at-risk basis). The impact on volatility, however, is relatively small. This means that this type of product is especially suitable for hedging rare and extreme events. However, Doherty points out that limitations on the use of double-trigger products exist. 9 Although the external index trigger helps prevent moral hazard problems, a substantial basis risk may emerge. In this article, we will consider in particular the issue of pricing a specific doubletrigger contract. Not much literature exists on pricing double-trigger products; 10 therefore, we will discuss such pricing within the context of different types of insurance pricing models. We analyze the effects of typical double-trigger contract parameters in different pricing environments and evaluate the appropriateness of each such envi- 4 Cox and Schwebach (1992); D Arcy and France (1992); Cummins and Geman (1995); Han and Lai (1995); Canter et al. (1996); Litzenberger et al. (1996); Doherty (1997); Smith et al. (1997); Doherty (2000). 5 Chichilnisky and Heal (1998, pp ); Bochicchio et al. (1998, pp ); Doherty (2000, pp ). 6 Bochicchio et al. (1998, p. 7). For moral hazard reasons, the financial trigger is typically based on one or more capital market indices. Doherty (2000, p. 533). 7 Typically, stop-loss or aggregate excess-of-loss whole account coverage. 8 Bochicchio et al. (1998, pp ). 9 Doherty (2000, pp. 437, , ). 10 Bochicchio et al. (1998, pp. 15, 21); Doherty (2000, pp ).

3 PRICING DOUBLE-TRIGGER REINSURANCE CONTRACTS 451 ronment. We also consider the circumstances under which demand for such contracts may arise. In general, insurance pricing models are either actuarial (or statistical) models or financial models. 11 The point of departure for the actuarial models is the risk characteristics displayed by the insurance contract either by the isolated contract or by the contract as embedded in the portfolio of the (re)insurance company. The isolated valuation is carried out by using one of the usual premium principles, 12 e.g., the standard deviation principle. In the portfolio context, insurance premiums typically are determined by two objectives: (1) no reduction should occur in the expected gain, and (2) a certain ruin probability for the (re)insurance company should be maintained. 13 Financial models of insurance pricing differ from actuarial models in that they are preference-independent. The most important financial models are associated with capital market equilibrium (e.g., the insurance Capital Asset Pricing Model (CAPM)) and option pricing theory. 14 Their economic foundation is the pricing of insurance contracts so as not to worsen the financial position of the (re)insurance company s shareholders. In addition, financial insurance pricing reveals the reservation price of insurance buyers, assuming that the buyers pricing method is the same as that used by the shareholders. Our article begins by looking at a particular type of double-trigger contract in isolation to show how this contract works in general. For pricing the contract we use a oneperiod contingent claims approach. In our model, the extreme case, in which the double-trigger contract is not subject to default, coincides with the valuation under the insurance CAPM. We then compare the results of this pricing scheme with the price determined using a portfolio-oriented actuarial valuation model. We conclude with a discussion of the conditions likely to lead to a market for double-trigger reinsurance contracts. We obtained the following results. In the actuarial model, the reinsurer s specific situation (size, safety level, etc.) and the correlations between the existing portfolio and the double-trigger contract were of considerable significance in pricing. The premiums calculated can be characterized as high because of the profound influence that double-trigger contracts have on the reinsurer s insolvency level. Using the contingent claims approach demonstrated that should contracts be closed at actuarial reservation prices, a redistribution of wealth from the insurance buyers to the reinsurance shareholders would often occur. In competitive markets, in which the primary insurer bases its calculations on a net present value approach, premiums calculated under an actuarial model may not often result in a demand for double-trigger contracts. Hence, the actuarial pricing model might lead to market failure. 11 Cummins (1990, pp ); Cummins (1991). 12 Bühlmann (1970, pp ); Daykin et al. (1994, pp ). Hereby, the insurance premium consists of the expected claims plus a safety loading based on risk parameters of the claims distribution and the insurer s risk aversion. 13 Bühlmann (1970, pp. 131 ff); Straub (1997, pp. 36 ff). 14 Fairley (1979); Doherty and Garven (1986); D Arcy and Doherty (1988); Cummins (1991, pp ).

4 452 THE JOURNAL OF RISK AND INSURANCE The results of the comparison between actuarial and financial pricing models are not restricted to double-trigger products but can be applied to the valuation of insurance risks in general. INDIVIDUAL CONTRACT LEVEL In this section, we approximate the loss distribution of a double-trigger contract, which is the basis for traditional actuarial premium principles. The objective is to establish a benchmark using a single contract level for comparison to the portfolio-oriented valuation that will be discussed later. We examine a traditional stop-loss (SL) reinsurance contract (type (1), see Introduction) that pays only if the capital market index of level i at time t = 1 falls below an agreed threshold Y. Let S represent the (whole account) loss distribution (including incurred, but not reported IBNR claims) of the cedent 15 for the period under consideration (t = 0;1). Assume that the corresponding premium income is a constant deterministic value. If we have a reinsurance protection M excess of A, payoff by the reinsurer S SL for an SL contract is: S SL = min [ M A; max(s A;0) ] : (1) In the case of the type (1) double-trigger product, its contingent payments S DT at the end of a single period are equal to: S DT =1 {i<y} S SL : (2) An analytical expression of Equation (2) is possible only for a number of distributional assumptions as to the capital market index level (i) and the loss distribution of the cedent (S ). For example, this is the case if both variables i and S are normally distributed. 16 We use a simulation model to determine the payoff distribution S DT of the double-trigger contract. This approach makes it possible to graphically illustrate the effects of the different correlations between the random values analyzed. In the following example, 17 assume that the loss distribution faced by the cedent (S ) before it closes the trigger contract is log-normally distributed with a mean of $75,000,000 and a standard deviation of $10,000,000. Fix the attachment point A at a level of $80,000,000 and let the upper layer limit M be $100,000,000. The mean value of the SL-contract (S SL ) payoff distribution is thus $2,014,130 and the standard 15 To meet the function of balance sheet protection, claims are typically defined on a calendaryear basis. The existing moral hazard problems regarding the reserving policy of the primary insurer can be offset by setting a regulating top limit with respect to the coefficient losses incurred/losses paid in the observed period. 16 Johnson and Kotz (1972, pp ). 17 This example is in essence based on a contract that the Tempest Re closed with the California State Automobile Association (CSAA) in March This example has, like all subsequent examples, been calculated using a Latin Hypercube simulation of 1,000,000 iterations. In general, with a given number of iterations, the approximation from the Latin Hypercube technique is superior to the Monte Carlo simulation, as explained by McKay et al. (1979, pp ). To ensure that the simulation results can be accurately compared with each other, we used the same sequence of random numbers for all simulations.

5 PRICING DOUBLE-TRIGGER REINSURANCE CONTRACTS 453 TABLE 1 Approximate Values of the Compensation Payments Made Under the Double-Trigger Contract (S DT ) With Different Correlations () Between the Capital Market Index Level (i) and the Cedent's Original Loss Distribution (S ) 0:6 0: E(S DT ) (mean of S DT ) $960,263 $600,279 $322,418 $131,932 $22,485 (S DT ) (standard deviation of S DT ) $3,418,743 $2,672,762 $1,881,537 $1,118,773 $381,024 prob(s DT > 0) 10.6% 7.4% 4.6% 2.4% 0.6% deviation is $4,329, The probability that at time t = 1 payments by the reinsurer will occur stands at 29.0 percent (pure SL contract). Assume further that the capital market index level at time t = 1 has a truncated normal distribution, with i R +, mean E[i] = 5;000 points, and standard deviation [i] = 1;000 points. The threshold trigger value y is 4,000 points. Table 1 shows the influence of the correlation coefficient between the capital market index level (i) and the original loss distribution (S )on the payment distribution of the double-trigger contract under these conditions. As expected, the correlation between the cedent s loss distribution and the capital market index is of considerable importance for the loss distribution of the doubletrigger contract. Where the capital market index is widely diversified, we can assume that no significant correlations exist. If, however, the index is composed exclusively of insurance companies, then negative correlations may exist between the losses suffered by the primary insurer and the index level. In such cases, the probability that the reinsurer will have to pay will tend to rise, all else being equal. In general, we can view the double-trigger contract as a derivative of the SL contract. Hence, one can see a typical example of a risk-leverage effect. If, for example, =0, the coefficient of variation of the double-trigger contract is almost three times as large as that of the standard SL contract. 19 In practice, the reinsurance industry usually carries out an isolated valuation of risk based on the payoff distribution of the double-trigger contract (see Table 1). Methods used for this valuation might include the expectation value principle, the standard deviation principle, or the percentile principle, which typically results in different safety loadings on the mean of S DT. 20 However, such a valuation procedure does not take into account both interrelations with the reinsurer s existing portfolio and pricing factors of the capital market. We will study these effects in the following sections. FINANCIAL MODELS OF INSURANCE PRICING In this section, we will determine a fair price for the double-trigger product in a capital market context. We assume an arbitrage-free capital market and also that the product being valued can be duplicated with the financial instruments traded on the capital 18 Unless otherwise indicated, we rounded all figures to whole dollar amounts. 19 The coefficient of variation is the standard deviation divided by the mean. A larger coefficient of variation signifies a higher risk. 20 Bühlmann (1970, pp ); Daykin et al. (1994, pp ).

6 454 THE JOURNAL OF RISK AND INSURANCE TABLE 2 Approximate Double-Trigger CAPM Prices for Different Correlations ((i;r m)) Between the Capital Market Index Level (i) and the Return on the Market Portfolio (r m) (i;r m ) DT (CAPM price) $313,027 $429,793 $547,171 $664,501 market. 21 Moreover, the pricing system is not changed by the double-trigger contract (the so-called competitivity characteristic), and no informational asymmetries exist between the market participants. 22 We first calculate the price of a double-trigger contract that is not threatened by default on the basis of the insurance CAPM. Then we price the contract taking into account a possible default of the contract. Pricing Without Default Risk If r f characterizes the risk-free rate of return, r m is the rate of return of the market portfolio, and is the market price of risk ( = (E(r m ) r f )=var(r m )), then the equilibrium price ( DT ) for the double-trigger contract on the basis of the CAPM is given by 23 DT = 1 1+r f [E (S DT) cov(s DT ;r m)] : (3) Assume that the capital market index level (i) is uncorrelated with the original loss distribution (S ) of the primary insurer (see Table 1, = 0). In addition, assume that the risk-free interest rate stands at 3 percent and that the market portfolio (r m ) yields an expected 8 percent return, with a standard deviation of 4 percent. If the original loss distribution (S ) is assumed to be uncorrelated with the market portfolio (r m ), then the following double-trigger equilibrium prices can be calculated, depending on the correlations (i;r m ) between the index level and the market portfolio. See Table 2. If the capital market index level displays a high correlation with the market portfolio, then the high systematic risk associated with the contract leads to high premiums. Thus, where the correlation coefficient (i;r m ) is 0.6, the premium is more than twice as high as when (i;r m ) = 0, where the premium is given by E(S DT )=(1+r f ). However, the premiums are much lower than those for the SL contract without a second trigger, (E(S SL )=(1 + r f ) $1;955;466), because the double-trigger contract pays much less often. 21 In reality, this premise is frequently breached (Bochicchio et al. (1998, pp. 12, 15)). Møller (2000) discusses financial insurance pricing principles in the presence of an incomplete capital market. 22 Ross (1978); Harrison and Kreps (1979). 23 Fairley (1979); D Arcy and Doherty (1988).

7 PRICING DOUBLE-TRIGGER REINSURANCE CONTRACTS 455 Pricing With Default Risk We now examine pricing a double-trigger contract where a default risk on the reinsurer s payment exists. The literature suggests using a contingent claims approach. 24 The double-trigger contract is assessed within the specific risk situation faced by the reinsurer. The most straightforward situation is one where the suppliers of equity capital and the insurance policyholders come together to invest their money. The loss distribution stipulated in the insurance contract determines the policyholders claims to repayment. We will focus on a one-period model in which ˆ DT characterizes the price of the double-trigger contract in the presence of default risk, r is the risky rate of return on the reinsurer s investment portfolio, and E 0,orE 1, is the reinsurer s equity at time t =0ort = 1, respectively. The following relationship is derived: E 1 = max [(E 0 +ˆ DT)(1+r) S DT ;0] : (4) If the degree of default risk is taken into account, the insurance premium ˆ DT can be found in analogy to the CAPM on the premise that entering into the contract must not worsen the owner s financial position. Using the risk adjustment under the CAPM, the premium implicitly follows from the condition E 0 = 1 1+r f [E (E 1) cov(e 1 ;r m)] : (5) An evaluation of the policyholders deficit max [S DT (E 0 +ˆ DT)(1+r) ;0] or, in other words, an evaluation of the risk of default on the insurance payment, by way of Equation (5), produces the value of the default put option. 25 In line with the put-call parity, the value of the default put option is the difference between the default-free premium of the contract ( DT ) and the premium in the presence of default risk ( ˆ DT ). In a market without arbitrage and without transaction costs, the value of the default put option is exactly the same as the value of any risk management measure (e.g., equity capital increase, retrocession) that ensures fulfillment of the contract. Thus, in principle, proper use of risk policy measures would make it possible to realize any price (between 0 and DT ), with a corresponding safety level, irrespective of the reinsurer s initial situation. In practice, however, the reinsurance company holds a pre-existing insurance portfolio with a specific default put option value. For various reasons, the reinsurance company might wish to remain in the same risk class after writing the double-trigger contract. For example, the reinsurer s risk class can be described by the following ratio: default put option value per dollar premium income. This is the same risk measure and risk policy proposed for an insurance firm in Butsic (1994), Cummins (2000), and Myers and Read (2001). We also use this approach in setting out the problem. In the following, is the default risk-free premium associated with the portfolio of existing insurance policies, ˆ is the premium income actually collected on the portfolio of existing insurance policies, d is the corresponding value of the default put 24 Doherty and Garven (1986); Cummins (1991). 25 Butsic (1994).

8 456 THE JOURNAL OF RISK AND INSURANCE FIGURE 1 Graphic Determination of the Double-Trigger Premium When the Reinsurance Company Wishes the Same Risk Class as It Was Before Issuing the Contract Notes: (a) = possible default put option value/premium combinations associated with existing portfolio; (b) = possible default put option value/premium combinations associated with existing portfolio but now including the double-trigger contract; = default risk-free premium for the existing portfolio; ˆ = premium income obtained from existing portfolio (with default risk); DT = default risk-free double-trigger premium; ˆ DT = double-trigger premium with default risk; d = value of default put option in the existing portfolio; d DT = value of default put option in the double-trigger contract. option, and d DT is the value of the default put option of the double-trigger contract. 26 The reservation price for the double-trigger contract can be determined using the graph in Figure 1. In general, the premium in the presence of default risk is the default risk-free premium less the default put option value. Thus, for the double-trigger contract, the relationship DT = ˆ DT +d DT holds. The ordinate sections in Figure 1 illustrate the default riskfree premiums and + DT. Point A on the opportunity line (a) describes the default put option value/premium combination associated with the reinsurer s existing portfolio. If a double-trigger contract with a default risk-free price ( DT )is added to the portfolio, the double-trigger premium ( ˆ DT ), determined by point B, 26 It is assumed that any positive net present value possessed by the old contracts has already been transferred to the private assets of the owners of the reinsurance firm.

9 PRICING DOUBLE-TRIGGER REINSURANCE CONTRACTS 457 ensures that the previous relationship holding for the default put option value per dollar premium is maintained. We then obtain the double-trigger premium ( ˆ DT )from the following trigonometric relation: ˆ DT = DT ˆ : (6) The risk management measure required to ensure the value of the default put option (d +d DT ) may consist of specific reinsurance arrangements. In this article, however, we look exclusively at steps taken to adjust the equity capital as a risk management measure. We can calculate the new level of equity on the basis of the overall premium and the overall loss distribution of the reinsurer, by way of Equations (4) and (5). Adjustment of the equity capital (or, more generally, the dimensions of the risk management measures) depends heavily on the size and structure of the existing portfolio and its risk interdependencies with the double-trigger contract. We shall demonstrate this by further developing the above example. To separate the price effects of the contract being embedded in the portfolio from the price effects arising from the CAPM risk adjustment, we assume in this example that market participants display risk-neutral behavior (the market price of risk in the pricing Equation (5) is zero). Before the double-trigger contract is signed, assume the following: equity at time t = 0 (E 0 ) is $300,000,000; premium income from the existing portfolio ( ˆ ) is $1,941,712,200; losses arising from the existing portfolio (S ) under a normal distribution with an expected value of $2,000,000,000 and standard deviation of $100,000,000; and rate of return on the investment portfolio (r) under a normal distribution with expected value of 3 percent and standard deviation of 1 percent. The risk-free rate of return must again be 3 percent. This situation fits into Equation (5) and yields a net present value of approximately zero for both groups of stakeholders (i.e., reinsurance shareholders and reinsurance buyers). The default put option value (d ) is $35,373. For the default put option value per $ premium pertaining to the existing portfolio, we obtain a value of $ With respect to the double-trigger contract, we examine where the capital market index level is uncorrelated with the original loss distribution of the primary insurer (see Table 1, = 0). We now analyze three double-trigger contracts that differ only in respect to their correlations to the reinsurer s existing portfolio. The three correlation matrixes are C 0,C 1, and C 2, respectively. See Table 3. The matrixes have been chosen so that the correlations between the reinsurer s existing portfolio E 1 = max[ (E 0 +ˆ )(1 + r) S ;0 ] and the distribution of the double-trigger contracts (S DT )are0(c 0 ), roughly 0:1(C 1 ), and roughly 0:2 (C 2 ), respectively. 27 In the financial approach without default risk, different correlations between the existing portfolio and the double-trigger contract have no influence on the reinsurer s reservation price. Applying Equation (3) to all three double-trigger contracts produces the discounted expected loss (roughly $313,027, see Table 2) as the reservation price without default risk ( DT ). 27 If the figures 0.6 and 0.65 in the correlation matrix C 2 are substituted with the value 1, then, all else being equal, the correlation between the reinsurer s existing portfolio and the distribution of the double-trigger contract is a minimum figure amounting to roughly 0:32.

10 458 THE JOURNAL OF RISK AND INSURANCE TABLE 3 Correlations Between the Capital Market Index Level (i), the Cedent's Claim Distribution (S ), the Rate of Return on the Reinsurer's Investment Portfolio (r), and the Reinsurer's Claim Distribution (S ) C 0 C 1 C 2 i S r S i S r S i S r S i i i S S S r r r S S S In the next step, we determine the lowest price for the double-trigger contract that possesses the default put option value per dollar premium of the existing portfolio. Referring to Equation (6) and Figure 1, the same premium ( ˆ DT ) is obtained for all three double-trigger contracts, in line with the relation ˆ DT = DT ˆ $1;941;712;200 $313;027 $313;021: (7) $1;941;747;573 In our example, the risk management measure needed to ensure the desired safety level is an equity capital increase. The amount of additional equity capital required rises in line with the risk. Thus, in the example, the equity capital increase is roughly $2,649,992 (C 0 ), $3,279,972 (C 1 ), or $5,649,977 (C 2 ). 28 The correlations in C 2 result in more than twice as much additional equity capital being necessary to maintain the desired safety level as in the situation characterized by C 0. The values of the risk management measure (equity capital increase) are $3,204 (C 0 ), $4,078 (C 1 ), and $7,476 (C 2 ). These values are the net present values for the reinsurer s shareholders if the primary insurer pays the premium ˆ DT = $313;021 and if the reinsurer undertakes no further risk management measures. 29 Generally speaking, the costs of this risk management measure will, in the absence of arbitrage, be the same as what would have to be paid for any alternative risk management measures (e.g., retrocession), with the same effect on the default put option value. ACTUARIAL PRICING In the following, we use a portfolio-oriented actuarial valuation model to determine actuarial pricing, which accounts for diversification effects in the reinsurer s portfolio. 28 The necessary adjustment to the equity capital is obtained implicitly by analogously applying Equation (5), i.e., 1 E (E 1) E 0 0; 1+r f where E 1 = max [( E 0 +ˆ +ˆ DT ) (1+r) S S DT ;0 ] : 29 Bearing in mind the risk incentive problems (MacMinn, 1987), we have, however, ruled out such shifts in wealth in the financial approach.

11 PRICING DOUBLE-TRIGGER REINSURANCE CONTRACTS 459 We compare and contrast the premiums we calculate in this way with the premiums obtained under the financial pricing model used above. One prominent actuarial model, presented by Bühlmann and by Straub, 30 has an infinite planning horizon. The object of this model is to calculate the constant premium level that guarantees a certain (small) ruin probability, given a certain level of equity capital and given independent and identically distributed claims. As a result, the required premiums lead to nonnegative expected underwriting results, and the higher the initial equity capital, the smaller the required premiums. By using this approach in our example, we seek a price for the additional contract that neither reduces the expected level of gain at time t = 1 nor increases the existing ruin probability. This approach may be adequate if, for example, the reinsurer is risk-neutral and a regulatory authority has set a required solvency margin. If the ruin restriction is binding, then, in general, substantial safety loading on the expected loss will be generated. Consequently, the premiums paid by the primary insurer will exclusively maintain the original safety level. In addition to the distribution structure of the existing portfolio and the loss distribution of the double-trigger contract, the correlation relationships are particularly influential on the reinsurer s reservation price. Using the same example as above, let E ( [ 1 = max (E0 +ˆ )(1 + r) S ;0 ]) be the equity capital distribution of the existing portfolio before the double-trigger contract is signed. The ruin probability " faced by the reinsurer before signing the double-trigger contract is approximately percent. After signing the double-trigger contract, the following obtains: 31 E 1 = max [( E 0 +ˆ +ˆ DT ) (1+r) S S DT ;0 ] : (8) The reinsurer s ruin probability after the double-trigger contract has been signed is characterized as ". In a formal sense, we apply a stochastic optimizing program with ˆ DT min! subject to (I) E(E 1 ) E(E 1) and (II) " ". Because of restriction (II), the optimization is termed a chance-constrained program, which generally can be solved only approximately by numerical methods. 32 A numerical evaluation of the stochastic optimization problem yields in Table 4 the price limits for the double-trigger contracts C 0,C 1, and C 2, as given by Table 3. As Table 4 shows, the premiums rise sharply, depending on the correlations. In a competitive market, it is doubtful that there will be any demand for products priced in this way. In the following section, we compare the portfolio-oriented actuarial pricing with the financial models of insurance pricing. 30 Bühlmann (1985, pp ); Straub (1997, pp ). See also Bühlmann (1970, pp. 131 ff.); Daykin et al. (1994, pp ). 31 See also footnote Kall and Wallace (1995, S ). For specific distribution assumptions for E 1 (e.g., normal distribution), the above program can be converted into an equivalent deterministic (typically, nonlinear) program.

12 460 THE JOURNAL OF RISK AND INSURANCE TABLE 4 Approximate Values of the Reinsurer's Price Limits for Double- Trigger Contracts With the Correlations C 0,C 1, and C 2 C 0 C 1 C 2 ˆ DT (actuarial price) $446,660 $1,725,009 $4,700,112 ACTUARIAL VERSUS FINANCIAL PRICING Because of different measurements and valuations of risk, actuarial and financial pricing approaches cannot be directly compared. Thus, to gain comparative insights, we adapt each of the two model environments to one another. First, we assume that all random variables are uncorrelated to the market portfolio. In addition, in the two types of approaches, the nature of the insolvency measurement differs (ruin probability versus default put option). Therefore, we substitute the ruin probability restriction (II) from the actuarial model with a default put option restriction. This results in the stipulation that the default put option value per dollar premium of the existing portfolio must not be exceeded after the double-trigger contract has been signed. Again, we analyze the three double-trigger contracts described by C 0,C 1, and C 2 in Table 3, taking first the modified actuarial model. For this case, we use the stochastic optimizing approach to find the lowest price for the double-trigger contract that will neither reduce the expected amount of equity capital at time t = 1 nor exceed the default put option value per dollar premium of the existing portfolio. For the premiums ˆ DT (actuarial price), the following approximate values are obtained: $2,963,013 (C 0 ), $3,592,993 (C 1 ), and $5,962,998 (C 2 ). Table 5 summarizes the results. Table 5 shows that in the actuarial model, the double-trigger contract premiums have to be higher due to the modified insolvency restriction (see, for comparison, Table 4). This is a result of high expected claims from the double-trigger contract in ruin situations. Using the actuarial price for the contract creates from the standpoint of financial theory a wealth transfer from the primary insurer to the reinsurance company. This wealth transfer reaches an extreme level in case C 2 (high double-trigger payoffs coincide with poor returns on the reinsurer s old portfolio), where the value is $5,649,977, which is approximately 18 (!) times the financial price. The discrepancy between the actuarial premium and the financial approach premium is exactly the amount of additional equity capital that would be needed in an arbitrage-free market. The following is a graphic example of what we have established so far. To keep things simple, we look only at a double-trigger contract without a pre-existing portfolio. The basis will again be Equation (5), in which the arbitrage-free double-trigger premium is implicitly determined. The only risk management measure considered is equity capital at time t = 0(E 0 ). For arbitrage reasons, reinsurance buyers would not pay a premium if no equity were invested (and vice versa). A huge amount of equity capital E 0 rules out any default situations, and therefore the double-trigger premium reaches its default free value ( DT ), i.e., the traditional insurance-capm premium. In an equity capital/premium diagram, the required premium is a concave function (see line (a) in Figure 2) Doherty and Garven (1986, p. 1044).

13 PRICING DOUBLE-TRIGGER REINSURANCE CONTRACTS 461 TABLE 5 Comparison of Reinsurer's Approximate Reservation Prices in the Financial and the Actuarial Model C 0 C 1 C 2 Financial approach: DT (price without default risk) $313,027 $313,027 $313,027 ˆ DT (price with default risk) $313,021 $313,021 $313,021 Necessary equity capital increase $2,649,992 $3,279,972 $5,649,977 Value of the risk management measure $3,204 $4,078 $7,476 Modified actuarial approach: ˆ DT (actuarial price) $2,963,013 $3,592,993 $5,962,998 The premium in the modified actuarial model is yielded by ˆ DT (financial model) + (c). Obviously, the actuarial approach dictates a quite particular risk management measure to the customer, who is, in fact, required to put capital into the reinsurance company. From the standpoint of financial theory, this would be fair only if the customer were given a share in the reinsurance company corresponding to the equity capital increase (c). Yet in the actuarial model, the customer provides debt capital without receiving a risk adequate return. This particular type of double-trigger contract, unlike traditional reinsurance contracts, will likely possess substantial interrelations with the reinsurer s preexisting portfolio. In this case, a pricing model such as the shown actuarial model tends to result in extremely high safety loadings. The hereby implied wealth transfer, as measured by the financial model, could end up in a market failure. CONDITIONS LEADING TO A MARKET FOR DOUBLE-TRIGGER REINSURANCE CONTRACTS The price ˆ DT according to the financial model is, under homogeneous expectations, the fair market price for the double-trigger contract as long as the primary insurer uses the same net present value calculation as does the reinsurer. For large primary insurance companies, this assumption is not unrealistic. In a contract signed on these terms, neither party receives advantages or suffers disadvantages. 34 (Re)insurance is irrelevant. However, circumstances could conceivably arise when it would be advantageous for at least one of the parties to close a double-trigger contract. In their well-known article, Mayers and Smith (1982) cite several reasons for the relevance of (re)insurance, e.g.: (Re)insurance can help shift risk away from those stakeholders who are at a relative disadvantage in risk-bearing terms; A (re)insurer may enjoy a comparative advantage in administering claims; A (re)insurer may have experience in monitoring activities that give rise to risk; (Re)insurance may lower expected tax liability. 34 Cummins (1991, p. 291).

14 462 THE JOURNAL OF RISK AND INSURANCE FIGURE 2 Financial Versus Actuarial Pricing of Double-Trigger Reinsurance Contracts Notes: (a) = equity capital/premium combinations yielding a net present value of zero (reservation price in the financial model); (b) = value of the risk management measure (taken to establish the desired safety level); (c) = equity capital increase needed to establish the desired safety level (= E 0 ); DT = default risk-free double-trigger premium (financial model); ˆ DT = double-trigger premium with default risk (financial model); E 0 = equity capital in t =0. Moreover, in an otherwise neoclassical model environment, Doherty and Tinic (1981) derive the relevance of reinsurance from the risk sensitivity of the primary insurer s policyholders. If we insert Doherty and Tinic s (1981) concept into our argument, in principle, we can reach an optimal safety level. In this case, the reservation price of the primary insurer s policy holders deviates from that predicted by the contingent claims approach under Equations (4) and (5). The primary insurer then chooses an optimal risk level via reinsurance to maximize its shareholder value. The Doherty and Tinic (1981) approach is similar to the model presented by Cummins and Sommer (1996), which determines an optimal risk management mix for insurance firms that maximizes expected gains by assuming that the policyholders reservation price is highly sensitive to the primary insurer s risk situation (measured by the value of the default put option). In particular, Cummins and Sommer assume that policyholders ::: do not view insurance solely as a financial asset ::: 35 The ideas of Doherty and Tinic (1981) and Cummins and Sommer (1996) are strongly supported by the 35 Cummins and Sommer (1996, p. 1074).

15 PRICING DOUBLE-TRIGGER REINSURANCE CONTRACTS 463 FIGURE 3 Optimal Default Risk for Double-Trigger Contract When Primary Insurer Is Extremely Risk-Averse Notes: (a) = possible default put option value/premium combinations of the double-trigger contract (reservation price of the reinsurer in the financial model); (b) = reservation price function of the primary insurer; NPV opt DT = maximum net present value; opt DT = optimal double-trigger premium; DT = default risk free double-trigger premium; ˆ DT = double-trigger premium with default risk; d opt DT = optimal value of default put option of the double-trigger contract. empirical analysis of Wakker et al. (1997), who find that the pricing method used by policyholders differs substantially from that given by Equation (5). Where the primary insurer faces no default risk, Wakker et al. (1997) find that insureds are willing to pay premiums that are higher than the expected loss. But such willingness to pay falls sharply as the insolvency risk faced by the primary insurance company rises (i.e., when the firm s ruin probability reaches a certain level). 36 The relevance of insurance, as mentioned above, can be transferred to double-trigger products. This is clarified in Figure 3 and the arguments that follow. 36 The reservation prices of the policyholders fall by one quarter if the probability of ruin faced by the insurance company rises from 0 percent to 1 percent. Similar behavior is documented in Cummins and Sommer (1996) and in Cummins and Danzon (1997).

16 464 THE JOURNAL OF RISK AND INSURANCE Again, we assume that the primary insurer and its policyholders can directly observe the reinsurer s safety level. 37 The reservation price function of the primary insurer (line (b)) depends on the risk situation of the reinsurance company. Line (b) may result directly from the primary insurer s risk attitude, or in line with the view taken by Doherty and Tinic (1981) indirectly from the reservation prices of the primary insurer s policyholders. We reach the optimal safety level of the contract, i.e., the optimal value of the default put option (d opt DT ), when the positive difference between the primary insurer s reservation price and the reservation price of the reinsurer reaches its maximum. This difference represents the maximum net present value (NPV opt DT ) for the reinsurer if it can realize the premium income ( opt DT ) in the market. We can derive another line of reasoning that supports the rationality of reinsurance being purchased and written from articles by Doherty (1991) and Froot and Stein (1998). In the Doherty (1991) framework, shares of publicly held financial services firms are assumed to be embedded into perfectly diversified portfolios held by individual investors. Nevertheless, in contrast to typical neoclassical capital market models, Doherty (1991), as well as Froot and Stein (1998), derive risk-averse behavior on the firm s part and, thus, an incentive for risk management. The reason for this risk-averse behavior lies in the assumption that the cost of capital for a financial services firm is a convex function of the amount of external capital needed for post-loss financing. This generates risk aversion (in the Pratt-Arrow sense) on the part of the financial services firm. 38 If we transfer this issue to our reinsurance problem, the following question arises: How high are the (stochastic) costs of post-loss financing faced by the reinsurer if it signs the double-trigger contract? Knowing these costs will allow the reinsurer to calculate its reservation price. Conversely, by closing the double-trigger contract, the primary insurer can save post-loss financing costs. Insofar as the double-trigger contract can be duplicated by marketable assets (i.e., it consists of tradable risk exposures), perfect risk management is possible and will indeed be carried out. The contract does not create any risk for the reinsurance company that would need to be hedged with the aid of expensive equity capital, which obviates any need to load the premium to reflect the additional cost of such equity capital. Because of the index trigger, this tradable risk component in the double-trigger contract may be quite substantial. However, the greater the nondiversifiable risk component of the indemnity payments of the double-trigger contract, the higher the price of reinsurance. 39 The unit price of this risk component is given by the firm-specific risk aversion. The lower the amount of equity capital initially held by the reinsurer, and thus the higher the cost of post-loss financing, the greater this risk aversion will be. 37 In practice, the primary insurer s information regarding the reinsurer s risk situation (e.g., via ratings) is more delayed and ambiguous. 38 Froot and Stein (1998, p. 65); see also Doherty (1991, p. 234). 39 The nondiversifiable risk component is the risk that the reinsurer still has left in its portfolio after establishing a hedge portfolio that tries to eliminate the nontradable risk. This hedge portfolio (Froot and Stein (1998, p. 66, Equation [12]) was originally introduced by Mayers and Smith (1983).

17 PRICING DOUBLE-TRIGGER REINSURANCE CONTRACTS 465 The position of the primary insurer is the exact opposite: 40 Transfer of the doubletrigger risk to the reinsurer results in a corresponding reduction of risk, especially since extremely large losses can be avoided by the hedge provided by the doubletrigger contract. In the model framework used by Doherty (1991) and Froot and Stein (1998), it is thus possible that by closing the double-trigger contract both parties may benefit from positive net present values. This is the case if the double-trigger cash flows can be hedged more effectively in the reinsurer s portfolio than in the portfolio of the ceding insurer. A further possible reason in favor of such a contract might be if the reinsurer has more equity capital than the primary insurer, which would result in the reinsurer being less risk-averse. If the reinsurer has infinite amounts of equity capital, it will always write the contract if the contract provides a risk premium exceeding the market risk premium (e.g., determined by the CAPM). 41 SUMMARY In this article, we looked at various approaches to pricing a new type of contract that has emerged from the area of ART the double-trigger reinsurance contract. A doubletrigger contract pays off when a condition from the underwriting area is fulfilled and a second condition is given: Typically, a capital market index must lie below a particular attachment point. We investigated the valuation of a double-trigger contract in the original CAPM framework (in other words, without taking default risk into consideration). This type of contract will, as a rule, make necessary large safety loadings on the expected loss payments, which is primarily a consequence of the correlation between the capital market index trigger and the return on the market portfolio. In the presence of a reinsurer s default risk, we determined the reinsurer s reservation price in the event the reinsurer wants to remain in the same risk class after signing the contract. Double-trigger contracts as insurance derivatives typically produce a relatively small probability of high loss while giving rise to relatively low insurance premiums. To maintain the desired risk situation, these contracts must be backed with large amounts of equity capital, if equity capital is the risk management measure to be taken. The amount of additional equity capital induced by the double-trigger contract depends heavily on the correlations between the double-trigger loss distribution and the reinsurer s underwriting and investment portfolio. We contrasted the financial insurance pricing models with an actuarial insurance pricing model that determines the price of a contract where the objective is that the contract should result in neither the reinsurance company s expected profits falling nor its ruin probability rising. Comparing the financial model with the actuarial model dramatically shows the fundamental problems associated with actuarial models. Actuarial pricing can, in certain circumstances, lead the reinsurer into a trap that results in failing to close insurance contracts that would have a positive net present value for the reinsurer. Typical actuarial pricing dictates the type of risk management measure that must be taken, namely, the insertion of additional capital. What is more, it forces 40 See Doherty (1991, p. 237). This argument is in principle compatible with the argument by Mayers and Smith (1982) cited above, which states that some stakeholders might have a comparative advantage in reallocating risk. 41 Froot and Stein (1998, p. 65).

18 466 THE JOURNAL OF RISK AND INSURANCE the reinsurance buyer to provide this safety capital as a debtholder. Taking financial theory into consideration, the reinsurance purchaser would be prepared to provide only a small portion of this capital in the context of the insurance premium; for the largest portion, the purchaser would demand the status of an equity provider. Finally, we investigated conditions leading to a market for double-trigger reinsurance contracts. In particular, we looked at a reinsurance market in which the reinsurance purchasers display extremely risk-averse behavior. We also discussed conditions for the closing of double-trigger reinsurance contracts in the context of the approach proposed by Doherty (1991), as well as Froot and Stein (1998). REFERENCES Baur, E., and K.-U. Schanz, 1999, Alternative Risk Transfer (ART) for Corporations: A Passing Fashion or Risk Management for the 21st Century?, sigma no. 2/1999 (Zurich: Swiss Re). Bochicchio, V., E. Schön, and E. Wolfram, 1998, Integrated Risk Management Solutions. Beyond Traditional Reinsurance and Financial Hedging (Zurich: Swiss Re New Markets). Bühlmann, H., 1970, Mathematical Methods in Risk Theory (Berlin: Springer). Bühlmann, H., 1985, Premium Calculation from Top Down, ASTIN Bulletin, 15: Butsic, R., 1994, Solvency Measurement for Property-Liability Risk-Based Capital Applications, Journal of Risk and Insurance, 61: Canter, M., J. Cole, and R. Sandor, 1996, Insurance Derivatives: A New Asset Class for the Capital Markets and a New Hedging Tool for the Insurance Industry, Journal of Derivatives, 5: Chichilnisky, G., and G. Heal, 1998, Managing Unknown Risks, Journal of Portfolio Management, 25: Cox, S., and R. Schwebach, 1992, Insurance Futures and Hedging Insurance Price Risk, Journal of Risk and Insurance, 59: Cummins, J. D., 1990, Asset Pricing Models and Insurance Ratemaking, ASTIN Bulletin, 20: Cummins, J. D., 1991, Statistical and Financial Models of Insurance Pricing and the Insurance Firm, Journal of Risk and Insurance, 58: Cummins, J. D., 2000, Allocation of Capital in the Insurance Industry, Risk Management & Insurance Review, 3: Cummins, J. D., and P. Danzon, 1997, Price, Financial Quality, and Capital Flows in Insurance Markets, Journal of Financial Intermediation, 6: Cummins, J. D., and H. Geman, 1995, Pricing Catastrophe Insurance Futures and Call Spreads: An Arbitrage Approach, Journal of Fixed Income, March: Cummins, J. D., and D. Sommer, 1996, Capital and Risk in Property-Liability Insurance Markets, Journal of Banking and Finance, 20: D Arcy, S. P., and N. A. Doherty, 1988, The Financial Theory of Pricing Property- Liability Insurance Contracts (Philadelphia, Pa.: Huebner Foundation for Insurance Education). D Arcy, S. P., and V. France, 1992, Catastrophe Futures: A Better Hedge for Insurers, Journal of Risk and Insurance, 59: