Profit analysis via risk-adjusted performance indicators in life insurance

1. Profit analysis via risk-adjusted performance indicators in life insurance Rosa Cocozza, Università di Napoli Federico II Emilia Di Lorenzo, Università di Napoli Federico II Albina Orlando, Consiglio Nazionale delle Ricerche Marilena Sibillo, Università di Salerno Profit analysis via Risk-Adjusted Performance Indicators in Life Insurance

2. Theresearchquestions • The optimization of a structured insurance policy • Which is the “optimal” price? • In a pure “state price” approach the “optimal” price is the arbitrage free since this is the equilibrium price • In a practical perspective the “optimal” price is one • that can be appraised by policyholders • that is “adequately profitable” to the insurer • Which factor influence the “optimal” price? • The features of the policy • The investment portfolio (replication alternatives) • The return on capital at risk set by top management Profit analysis via Risk-Adjusted Performance Indicators in Life Insurance

3. Thecase analyzed: temporary variable annuity • The temporary variable annuity structure • Pure Single Premium: U • Office Premium: U(1+l) • Guaranteed annual rate: g (<risk-free rate) • Guaranteed minimal payment: R • Additional Benefit indexed to the periodical performance of a reference Stock Exchange Index • Participation rate: h (≤ 1) • Duration: T years • Insurance portfolio annual OutFlow (portfolio installment): Profit analysis via Risk-Adjusted Performance Indicators in Life Insurance

4. The investment coverage • As shown by Brennan and Schwartz (1976;1979) and Boyle and Schwartz (1977), the equity (index) linked policy with a minimum guaranteed rate of return is equivalent • either to a plan providing a fixed benefit plus a call option, • or to a plan providing a benefit of the value of the reference portfolio plus a put option Profit analysis via Risk-Adjusted Performance Indicators in Life Insurance

5. The valuation approach • As shown by Cocozza (2005), the insurance business evolution can be described by means of • the intermediation portfolio (economic value approach) • in order to gain an insight into the value of the business, since it accounts for the difference between asset and liability at any time of the portfolio cycle • the income flows (current earning approach) • in order to gain an insight into the dynamics of the business, since it accounts for the difference between the profit components periodically accrued • In order to compute risk-adjusted performance measures, we used both. Profit analysis via Risk-Adjusted Performance Indicators in Life Insurance

6. Specification of the valuation model • The reference funds backing life insurance liabilities contain both bond and call option • In order to model properly the yearly returns, which determine the readjustment of the contractual benefits, we have to model both • interest rate risk • and stock (index) market risk. • We adopt a two-factor diffusion model obtained by joining a one-factor CIR model for the interest rate risk and a Black-Scholes model for the stock index market risk Profit analysis via Risk-Adjusted Performance Indicators in Life Insurance

7. Interest Rate Uncertainty • The single source of uncertainty is the spot rate rt, which is a diffusion process described by the stochastic differential equation (Cox, Ingersoll and Ross, 1985) where Zr(t) is a standard Brownian motion. • As known, this specification assumes a mean-reverting drift, with long term rate g, speed of adjustment a, and a ”square root” diffusion, with volatility parameter r. As is well-known, the CIR process implies non-central chi-square transition distribution for rt . Profit analysis via Risk-Adjusted Performance Indicators in Life Insurance

8. The term structure of the interest rate • Interest rate have been calibrated to Eonia (Euro OverNight Index Average) and Euribor (Euro Interbank Offered Rate) rates on 11/04/2007 Profit analysis via Risk-Adjusted Performance Indicators in Life Insurance

9. Stock Price Uncertainty • Also for the stock market we assume a single source of uncertainty, expressed by the stock index St; the diffusion process for the stock index is given by the stochastic differential equation where Zs(t) is a standard Brownian motion with the property • Thus we have a geometric Brownian motion, with instantaneous expected return msand volatility ss, which implies a lognormal transition density for St Profit analysis via Risk-Adjusted Performance Indicators in Life Insurance

10. The valuation context • In the CIR model the preferences prevailing on the market (the market price of interest rate risk) are specified by the function: • under the CIR approach – which is a general equilibrium approach – it is shown that this form of the preference function avoids riskless arbitrage. • The market price of risk for the stock market has the classical form: • thus no additional parameter is needed in order to specify the preferences in this case. Profit analysis via Risk-Adjusted Performance Indicators in Life Insurance

11. The strategic variables • Features of the policy • Guaranteed rate of return: g with rf<g≤2% • Participation rate: h • The investment alternatives • Perfect matching • Imperfect matching (controlled risk assumption) • The return on capital at risk set by top management Profit analysis via Risk-Adjusted Performance Indicators in Life Insurance

12. Theinput data set • The temporary index-linked annuity structure • Pure Single Premium: U = 2.325,00€ • Loadings l = 7% • Office premium = 2.500,00€ • Guaranteed annual rate: g = [0%; 0,5%; 1%; 1,5%; 2%] • Guaranteed minimal payment: R calculated with actuarial methodology on the basis of • the different g-rates • the mortality table IPS55 • entry age of the insured = 40 years • Additional Benefit indexed to the performance of the SPMIB as quoted on the Italian Stock Exchange (Fixing Value Bloomberg) • Participation rate: h = [70%; 80%; 90%; 100%] • Duration: 10 years • Number of policy initially sold = 1.000 homogeneous contracts • Initial Insurer Capital = 100.000,00€ (=4%*U(1+l)*1.000) Profit analysis via Risk-Adjusted Performance Indicators in Life Insurance

13. The cash flow mapping: Plain Vanilla Profit analysis via Risk-Adjusted Performance Indicators in Life Insurance

14. The cash flow mapping: Path Dependent Profit analysis via Risk-Adjusted Performance Indicators in Life Insurance

15. The simulation of the final result • Computation of the final result has been derived by numerical methods, using Monte Carlo simulations for the bivariate process (path = 10000) • The risk sources are the interest rate (modeled on the basis of the CIR Process) and the price of the underlying index (modeled on the basis of a lognormal distribution) • The intermediate net flows are reinvested at the corresponding (expected) rates up to the expiration date • Computation provide us with the Expected Profit E[P] at the end of policy duration both for • the plain vanilla replication (perfect matching) and • the path dependent replication (imperfect matching) • The error term sets for the simulation procedure have been selected constant across different alternatives in order to maximize the comparison value Profit analysis via Risk-Adjusted Performance Indicators in Life Insurance

16. The Expected ProfitPlain Vanilla Replication (perfect matching) Profit analysis via Risk-Adjusted Performance Indicators in Life Insurance

17. The Expected ProfitPath Dependent Replication (imperfect matching) Profit analysis via Risk-Adjusted Performance Indicators in Life Insurance

18. The CVaR 99%Plain Vanilla Replication (perfect matching) Profit analysis via Risk-Adjusted Performance Indicators in Life Insurance

19. The CVaR 99%Path Dependent Replication (imperfect matching) Profit analysis via Risk-Adjusted Performance Indicators in Life Insurance

20. The investment alternative Profit analysis via Risk-Adjusted Performance Indicators in Life Insurance

21. Decision Making Criteria • The economic sustainability and/or profitability can be therefore appraised through the following: • Expcted Return on Capital = E[RoE] • Risk-Adjusted Return on Risk-Adjusted Capital = C.R.A.R.o.C. • Economic Value Added = E.V.A. Profit analysis via Risk-Adjusted Performance Indicators in Life Insurance

22. The E[RoE] • The Expected Return on Capital can be evaluated by means of: • We concentrated on an average intertemporal measure, since we are concerned with the global profitability of the issue. Profit analysis via Risk-Adjusted Performance Indicators in Life Insurance

23. The E[RoE] Map:Plain Vanilla Replication Profit analysis via Risk-Adjusted Performance Indicators in Life Insurance

24. The E[RoE] Map:Path Dependent Replication Profit analysis via Risk-Adjusted Performance Indicators in Life Insurance

25. The C.R.A.R.o.C. • The Coherent Risk Adjusted Return on Risk Adjusted Capital can be evaluated by means of: • We concentrated on an average intertemporal measure, since we are concerned with the global profitability of the issue. Profit analysis via Risk-Adjusted Performance Indicators in Life Insurance

26. The C.R.A.R.o.C. Map:Plain Vanilla Replication Profit analysis via Risk-Adjusted Performance Indicators in Life Insurance

27. The C.R.A.R.o.C. Map:Path Dependent Replication Profit analysis via Risk-Adjusted Performance Indicators in Life Insurance

28. The profitability decision • In order to evaluate the profitability condition of the issue it is possible to compare the expected profit the return on the capital at risk (g) set by top management, that is the E.V.A. • Therefore any guaranteed rate g which is able to create a positive E.V.A. can be take into consideration and, among different alternatives, the one that offer the higher E.V.A. is preferable • Given the VA indicator, naturally those issues which are able to maximize the profit and reduce the CVaR are the best, but they have to respect the limit of the annual rate g • We selected g = 11% as a proxy of the threshold value. This is the average gross annual rate of return of the insurance companies listed on the Italian Stock Exchange over the last three years. Profit analysis via Risk-Adjusted Performance Indicators in Life Insurance

29. The E.V.A.Plain Vanilla Replication (perfect matching) Profit analysis via Risk-Adjusted Performance Indicators in Life Insurance

30. The E.V.A.Path Dependent replication (imperfect matching) Profit analysis via Risk-Adjusted Performance Indicators in Life Insurance

31. The computation of Risk-adjusted performance indicators can serve for Pricing decision Portfolio decision “Production” decision The computation through simulative models is prone to model risk Other interest rate model, mainly multifactor models Volatility surface to price the inner smile GARCH model in option pricing Stochastic mortality Conclusion and future research prospect Profit analysis via Risk-Adjusted Performance Indicators in Life Insurance