Optimal reinsurance strategies

Transcription

1 Optimal reinsurance strategies Maria de Lourdes Centeno CEMAPRE and ISEG, Universidade de Lisboa July 2016 The author is partially supported by the project CEMAPRE MULTI/00491 financed by FCT/MEC through national funds and when applicable co-financed by FEDER, under the Partnership Agreement PT2020. M.L. Centeno (ISEG) Optimal reinsurance strategies July / 13

2 What is Actuarial Science? Actuarial science is the discipline that applies mathematical and statistical methods to assess risk in insurance and finance Longevity risk Mortality risk Credit risk Catastrophic risk Market risk With the aim of creating value from pooling and managing risk Solvency II uses Value-at-Risk to access risk. We would defend the use of other Risk Measures. M.L. Centeno (ISEG) Optimal reinsurance strategies July / 13

3 General insurance questions Estimate the distribution of the frequency and severity of claims (Statistics) Assess the risk and determine the price (GLM) Experience Rating - Credibility Theory and Bonus Malus (Bayesian Statistics, Markov Chains) Calculate the probability of ruin in a finite or infinite horizon (Stochastic Processes) Determine the reserves that should be set aside to pay claims (GLM, Bayesian Models, Time Series) Determine how to invest the premium received (Mathematical Finance) How and how much should be transferred through reinsurance (Functional Analysis, Probability Theory, Stochastic Processes) M.L. Centeno (ISEG) Optimal reinsurance strategies July / 13

4 Life insurance questions in Actuarial Science Product design: protection and investment Pricing Determine the reserves to pay future claims Risks for the insurer which actuaries need to consider: mortality, anti-selection, withdrawals, expenses, investment, profitability M.L. Centeno (ISEG) Optimal reinsurance strategies July / 13

5 What is Reinsurance Reinsurance is insurance that is purchased by an insurance company (the "ceding company" or "cedant" under the arrangement) from one or more other insurance companies (the "reinsurer") directly or through a broker as a means of risk management. The ceding company and the reinsurer enter into a reinsurance agreement which details the conditions upon which the reinsurer would pay a share of the claims incurred by the ceding company. The reinsurer is paid a "reinsurance premium" by the ceding company, which issues insurance policies to its own policyholders. M.L. Centeno (ISEG) Optimal reinsurance strategies July / 13

6 Traditional Forms of Reinsurance A quota-share treaty reinsures a fixed percentage of each subject policy A surplus treaty is a type of proportional or pro-rata reinsurance treaty where the ceding company determines the maximum loss that it can retain for each risk in the portfolio. An excess of loss treaty reinsures, up to a limit, the part of each claim (or a share) that is in excess of some specified attachment point (cedant s retention). When reinsuring the layer L xs M, the insurer is ceding to the reinsurer, for each loss X, the amount Z (M, L) = min(l, (X M) + ) = with y + = max{0, y}. 0 if X M X M if M < X M + L L if X > M + L, A stop loss treaty is similar to a XL treaty, but on the aggregate claims. M.L. Centeno (ISEG) Optimal reinsurance strategies July / 13

7 Optimal Forms of Reinsurance Consider a portfolio where the aggregate claims are N j=0 Y j, where N is the number of claims in a period of time and Y j is the severity of the j th claim. The usual assumption is that {Y j } j=1,2,... are i.i.d. random variables with the same distribution as Y. The insurer may acquire reinsurance on a per claim" basis. This means that a reinsurance treaty is a function Z : [0, + ) [0, + ) mapping each possible claim size Y to the value refunded under the reinsurance contract. In return the insurer must pay a premium P(Z ) per time period. We assume that the pricing rule Z P(Z ) is fixed by the reinsurer and is known by the direct insurer who seeks to choose the best possible" treaty taking into account that pricing rule. M.L. Centeno (ISEG) Optimal reinsurance strategies July / 13

8 Optimal Forms of Reinsurance The retained aggregate claims will be N j=0(y j Z (Y j )), the risk net of reinsurance premiums, in each period is the random variable T Z = P(Z ) + N j=0 (Y j Z (Y j )), and the net profit is L Z = c T Z, where c is the gross premium. If Pr{N 1} = 1, and X represents the aggregate claims we are dealing with aggregate reinsurance. M.L. Centeno (ISEG) Optimal reinsurance strategies July / 13

9 Optimal Forms of Reinsurance Objective: Find Ẑ Z = {Z : [0, + ) R Z is measurable and 0 Z (y) y, y 0} such that ρ(tẑ ) = min {ρ(t Z ) : Z Z}, for some suitable functions ρ subject or not to some constraints. Alternatively, such that γ(lẑ ) = max {γ(l Z ) : Z Z}, for some suitable functions γ (the problems will be equivalent if the risk measure is cash invariant). M.L. Centeno (ISEG) Optimal reinsurance strategies July / 13

10 Optimal Forms of Reinsurance Examples of functions ρ or γ : Value-at Risk: ρ(x ) = VaR X (α) = min{v : Pr{X > v} α} Conditional Tail Expectation: CTE X (α) = 1 α α 0 VaR X (s)ds Coherent comonotonic risk measures: ρ(x ) = 0 w(1 F X (x))dx,for any X, such that Pr{X < 0) = 0, where w : [0, 1] [0, 1] is a continuous nondecreasing concave function, with w(0) = 0 and w(1) = 1. CTE X (α) is a special case, considering w(t) = min(t/α, 1) Expected utility of wealth, for an exponential utility function, with coeffi cient of risk aversion R > 0: γ(x ) = E [ e RL ] Z M.L. Centeno (ISEG) Optimal reinsurance strategies July / 13

11 Optimal Forms of Reinsurance Problem Find ( ˆR, Ẑ ) ]0, + [ Z + such that ˆR = RẐ = max { R Z : Z Z +}. R Z - adjustment coeffi cient of the retained risk for a particular reinsurance policy, Z Z, i.e. R Z is defined as the strictly positive value of R which solves the equation [ ] G (R, Z ) = E e RL Z = 1. (1) M.L. Centeno (ISEG) Optimal reinsurance strategies July / 13

12 Optimal Forms of Reinsurance The solution of each problem depends on the reinsurance premium calculation principle P(Z ) We have solved some of these problems. The most important result that we achieved was that the solution to the adjustment coeffi cient problem, when P(Z ) is a variance related premium calculation principle is such that y = Z (y) + 1 R Z (y) + α ln. α M.L. Centeno (ISEG) Optimal reinsurance strategies July / 13

13 Guerra, M. and Centeno, M.L. (2008), Optimal reinsurance policy: the adjustment coeffi cient and the expected utility criteria. Insurance Mathematics and Economics, 42, M.L. Centeno (ISEG) Optimal reinsurance strategies July / 13 References on our research on reinsurance Guerra, M. and Centeno M. L. (2012), Are quantile risk measures suitable for risk-transfer decisions? Insurance: Mathematics and Economics, Volume 50/3, Centeno, M.L. and Guerra, M. (2010), The optimal reinsurance strategy - The individual claim case. Insurance: Mathematics & Economics, 46/3, Guerra, M. and Centeno, M. L. (2010), Optimal Reinsurance for variance Related Premium Calculation Principles. ASTIN Bulletin, 40/1, Guerra, M. and Centeno, M. L. (2010), Optimal per claim reinsurance for dependent risks, CEMAPRE working papers. M. Guerra and M. L. Centeno (2010), Optimal trading under coherent comonotonic risk measures, CEMAPRE working papers.