11/07/2007 Profit analysis via Risk-Adjusted Performance Indicators in Life Insurance1 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

11/07/2007 Profit analysis via Risk-Adjusted Performance Indicators in Life Insurance 2 The research questions 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

11/07/2007 Profit analysis via Risk-Adjusted Performance Indicators in Life Insurance 3 The case analyzed: temporary variable annuity  The temporary variable annuity structure Pure Single Premium: U Office Premium: U(1+ ) 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:  (≤ 1) Duration: T years  Insurance portfolio annual OutFlow (portfolio installment):

11/07/2007 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 AssetLiability Call Option Portfolio Zero Coupon Bond Portfolio Index Linked Provision (Portfolio) Capital

11/07/2007 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.

11/07/2007 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

11/07/2007 Profit analysis via Risk-Adjusted Performance Indicators in Life Insurance 7 Interest Rate Uncertainty The single source of uncertainty is the spot rate r t, which is a diffusion process described by the stochastic differential equation (Cox, Ingersoll and Ross, 1985) where Z r (t) is a standard Brownian motion. As known, this specification assumes a mean- reverting drift, with long term rate  speed of adjustment , and a ”square root” diffusion, with volatility parameter . As is well-known, the CIR process implies non-central chi-square transition distribution for r t.

11/07/2007 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  (rate) 4,513%  (force) 4,414%  0,10  5%  0,09

11/07/2007 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 S t ; the diffusion process for the stock index is given by the stochastic differential equation where Z s (t) is a standard Brownian motion with the property Thus we have a geometric Brownian motion, with instantaneous expected return  s and volatility  s, which implies a lognormal transition density for S t

11/07/2007 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.

11/07/2007 Profit analysis via Risk-Adjusted Performance Indicators in Life Insurance 11 The strategic variables Features of the policy  Guaranteed rate of return: g with r f <g≤2%  Participation rate:  The investment alternatives  Perfect matching  Imperfect matching (controlled risk assumption) The return on capital at risk set by top management

11/07/2007 Profit analysis via Risk-Adjusted Performance Indicators in Life Insurance 12 The input data set  The temporary index-linked annuity structure Pure Single Premium: U = 2.325,00€ Loadings = 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:  = [70%; 80%; 90%; 100%] Duration: 10 years Number of policy initially sold = homogeneous contracts Initial Insurer Capital = ,00€ (=4%*U(1+ )*1.000)

11/07/2007 Profit analysis via Risk-Adjusted Performance Indicators in Life Insurance 13 The cash flow mapping: Plain Vanilla TAsset FlowsLiabilities FlowNet 0 NF 0 1 NF 1 2 NF 2 t NF t m NF m

11/07/2007 Profit analysis via Risk-Adjusted Performance Indicators in Life Insurance 14 The cash flow mapping: Path Dependent TAsset FlowsLiabilities FlowNet 0 NF 0 1 NF 1 2 NF 2 t NF t m NF m

11/07/2007 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

11/07/2007 Profit analysis via Risk-Adjusted Performance Indicators in Life Insurance 16 The Expected Profit Plain Vanilla Replication (perfect matching) 0,00%0,50%1,00%1,50%2,00% 70% , , , , ,20 80% , , , , ,40 90% , , , , ,50 100% , , , , ,70

11/07/2007 Profit analysis via Risk-Adjusted Performance Indicators in Life Insurance 17 The Expected Profit Path Dependent Replication (imperfect matching) 0,00%0,50%1,00%1,50%2,00% 70% , , , , ,20 80% , , , , ,20 90% , , , , ,50 100% , , , , ,50

11/07/2007 Profit analysis via Risk-Adjusted Performance Indicators in Life Insurance 18 The CVaR 99% Plain Vanilla Replication (perfect matching) 0,00%0,50%1,00%1,50%2,00% 70% , , , , ,60 80% , , , , ,50 90% , , , , ,40 100% , , , , ,40

11/07/2007 Profit analysis via Risk-Adjusted Performance Indicators in Life Insurance 19 The CVaR 99% Path Dependent Replication (imperfect matching) 0,00%0,50%1,00%1,50%2,00% 70% , , , , ,85 80% , , , , ,79 90% , , , , ,85 100% , , , , ,94

11/07/2007 Profit analysis via Risk-Adjusted Performance Indicators in Life Insurance 20 The investment alternative

11/07/2007 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.

11/07/2007 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.

11/07/2007 Profit analysis via Risk-Adjusted Performance Indicators in Life Insurance 23 The E[RoE] Map: Plain Vanilla Replication

11/07/2007 Profit analysis via Risk-Adjusted Performance Indicators in Life Insurance 24 The E[RoE] Map: Path Dependent Replication

11/07/2007 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.

11/07/2007 Profit analysis via Risk-Adjusted Performance Indicators in Life Insurance 26 The C.R.A.R.o.C. Map: Plain Vanilla Replication

11/07/2007 Profit analysis via Risk-Adjusted Performance Indicators in Life Insurance 27 The C.R.A.R.o.C. Map: Path Dependent Replication

11/07/2007 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 (  ) 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  We selected  = 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.

11/07/2007 Profit analysis via Risk-Adjusted Performance Indicators in Life Insurance 29 The E.V.A. Plain Vanilla Replication (perfect matching) 0,00%0,50%1,00%1,50%2,00% 70%30.801, , , , ,40 80%25.864, , , , ,31 90%20.927, , , , ,20 100%15.989, , , , ,09

11/07/2007 Profit analysis via Risk-Adjusted Performance Indicators in Life Insurance 30 The E.V.A. Path Dependent replication (imperfect matching) 0,00%0,50%1,00%1,50%2,00% 70%53.565, , , , ,18 80%51.781, , , , ,14 90%49.981, , , , ,32 100%48.160, , , , ,99

11/07/2007 Profit analysis via Risk-Adjusted Performance Indicators in Life Insurance 31 Conclusion and future research prospect 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