Pricing Double-Trigger. Reinsurance Contracts: Financial versus Actuarial Approach

Transcription

1 Pricing Double-Trigger Reinsurance Contracts: Financial versus Actuarial Approach Helmut Gründl Hato Schmeiser Humboldt-Universität zu Berlin Humboldt-Universität zu Berlin, Germany 1

2 I. Introduction Double-Trigger () reinsurance contracts => potential coverage depends on both underwriting and financial risk Our Focus: Specific -form => traditional stop loss reinsurance contract that makes the reinsurer liable for payments only when a capital market index of a level i is below a certain threshold value Y We compare and contrast the pricing of -contracts in the context of different types of insurance pricing models Perspective of a reinsurance company Humboldt-Universität zu Berlin, Germany 2

3 II. Individual Contract Level Focus: Loss distribution S of a -contract (t = 0,1) S 1 S i Y SL with S SL min MA, max(s* A,0) i Capital market index level at t = 0,1 Y Threshold S SL S * Payoff of a stop loss contract with layer M excess of A (Whole account) loss distribution of the primary insurer Humboldt-Universität zu Berlin, Germany 3

4 Simulation example (based on a Tempest RE/CSAA deal) Data: S * A M i Y Log-normally distributed, mean $75 Mio., std $10 Mio. $80 Mio. $100 Mio. (=> M _ A = $20 Mio.) Truncated normally distributed (i 0), mean 5,000 points, std. 1,000 points 4,000 points Result: Approximate payments of the -contract with different correlations () between i and S * Humboldt-Universität zu Berlin, Germany 4

5 Table 1: E(S ) $824,498 $322,418 $46,718 σ(s ) $3,162,777 $1,881,537 $599,137 prob S 0 9.4% 4.6% 1.1% In reinsurance practice: isolated valuation of risk based on the payoff distribution (e.g. the standard deviation principle or the percentile principle) Humboldt-Universität zu Berlin, Germany 5

6 III. Financial Models of Insurance Pricing a) Pricing Without Default Risk E.g.: CAPM equilibrium price for the -contract (t = 0,1) 1 1 r f covs, r E S m r f r m Riskless rate of return Market price of risk Rate of return of the market portfolio Humboldt-Universität zu Berlin, Germany 6

7 Example: Capital market index i uncorrelated with the loss distribution S * of the primary insurer (see table 1 with = 0) Riskless rate of return (r f ) = 3% Rate of return of the market portfolio (r m ): mean = 8% ; std = 4% Loss distribution S * of the primary insurer is uncorrelated with the market portfolio Table 2: i, r ) ( m (CAPM price) $313,027 $429,793 $547,171 $664,501 Humboldt-Universität zu Berlin, Germany 7

8 b) Pricing With Default Risk Contingent claims approach (t = 0,1), cf. Doherty/Garven (JoF 1986) E E maxe ˆ 1 r S, 0 E ˆ r Reinsurer s equity capital Linear pricing function (e.g. CAPM) Price of the -contract in the presence of default risk Risky rate of return on the reinsurer s investment portfolio Value of the default put option: d max S E ˆ 1 r 0, 0 Humboldt-Universität zu Berlin, Germany 8

9 Hence: ˆ d Arbitrage free market without transaction costs: d is the value of any risk management measure that ensures fulfilment of the contract => Possibility to realise any price (between 0 and ) In practice: reinsurer holds a pre-existing portfolio with a specific default put option value d * Reinsurer might for various reasons wish to remain in the same risk class (default put option value/premium income) after writing the -contract (cf. Myers/Read (JRI 2001)) Humboldt-Universität zu Berlin, Germany 9

10 Figure 1: insurance premiums (financial model) * * ˆ ˆ * * * ˆ ˆ * d* d default put option l Humboldt-Universität zu Berlin, Germany 10

11 Example: The only risk management measure to ensure the value of the default put option (d * + d ) be an adjustment of equity capital Market participants are risk-neutral Reinsurer s pre-existing portfolio: E 0 Equity Capital: $300,000,000 * ˆ Premium income existing portfolio: $1,941,712,200 S * Normally distributed losses arising from the existing portfolio: mean $2,000,000,000, std = $100,000,000 r Risky rate of return (normal distribution): mean 3%, std 1% r f Riskless rate of return: 3% Humboldt-Universität zu Berlin, Germany 11

12 => Default put option value/premium income: $ contract: capital market index i uncorrelated with the loss distribution S * of the primary insurer (see table 1 with = 0) Three -contracts (C 0, C 1, C 2 ) that differ only with respect to their correlations to the reinsurer s existing portfolio Table 3: 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* Humboldt-Universität zu Berlin, Germany 12

13 => Correlation * between the existing portfolio and the payoff distribution of the -contract: Table 4: C 0 : * = 0 C 1 : * -0.1 C 2 : * -0.2 financial approach: C 0 C 1 C 2 (price without default risk) $313,027 $313,027 $313,027 ˆ (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 Humboldt-Universität zu Berlin, Germany 13

14 IV. Actuarial Pricing E.g.: Prominent portfolio-oriented actuarial valuation model, cf. Bühlmann (1970, 1985); Straub (1997) Determine the lowest price for the additional contract (here: the contract) that (I) does not reduce the expected gain at time t = 1 and (II) does not deteriorate the present safety level (e.g. ruin probability) of the reinsurer Chance-constrained program Humboldt-Universität zu Berlin, Germany 14

15 V. Actuarial versus Financial Pricing In general: actuarial model and financial model cannot be compared directly But: the two model environments can be adapted to one another by using the following two assumptions: All random variables are uncorrelated to the market portfolio Restriction (II) in the actuarial model: Default put option value/premium must not be exceeded after the -contract has been signed Humboldt-Universität zu Berlin, Germany 15

16 Table 5: C 0 C 1 C 2 financial approach: (price without default risk) $313,027 $313,027 $313,027 ˆ (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 (actuarial price) $2,963,013 $3,592,993 $5,962,998 Humboldt-Universität zu Berlin, Germany 16

17 Figure 2: reinsurance premiums (financial model) Actuarial -price = (c) ˆ + (c) -premium with default risk (financial model) ˆ (b) (a) E 0 E 0 E 0 equity capital Humboldt-Universität zu Berlin, Germany 17

18 VI. Conditions Leading to a Market for -contracts Neo-classical model environment => (re)insurance is irrelevant Several reasons for the relevance of (re)insurance can be find in Mayers/Smith (JPuE 1982) Doherty/Tinic (JoF 1981): - Primary insurer s policyholders reservation price deviates from that predicted by the contingent claims approach - Primary insurer then chooses an optimal risk level via reinsurance to maximise ist shareholder value - Ideas of Doherty/Tinic are supported by Cummins/Sommer (JBF 1996) and Wakker/Thaler/Tversky (JRU 1997) Humboldt-Universität zu Berlin, Germany 18

19 Figure 3: insurance premiums opt NPV, opt ˆ reservation price function of the primary insurer reservation price function of the reinsurer d,opt default put option l Humboldt-Universität zu Berlin, Germany 19

20 Doherty (JRI 1991), Froot/Stein (JFE 1998) => central assumption: cost of capital is a convex function of the amount of external capital needed for post-loss financing => risk-averse behaviour Position of the reinsurer: diversifiable risk component of the -contact => perfect risk management will be carried out non-diversifiable risk component: unit price of this risk component is given by the firm-specific risk aversion (low amount of equity capital => high costs of post-loss financing => generates a high degree of risk-aversion (in the Pratt-Arrow sense)) Position of the primary insurer is the exact converse Humboldt-Universität zu Berlin, Germany 20

21 VII. Conclusion CAPM-Framework: -contract will in general cause large safety loadings on the expected loss payments Contingent claims approach: -premium may be low; contract must be backed with large amounts of equity capital Comparing the financial with the actuarial model shows some fundamental problems associated with actuarial models Humboldt-Universität zu Berlin, Germany 21