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Essays on valuation and risk management for insurers
Plat, H.J.
Publication date
2011
Link to publication
Citation for published version (APA):
Plat, H. J. (2011). Essays on valuation and risk management for insurers.
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Contents
1. INTRODUCTION AND OUTLINE ...1
1.1 VALUATION AND RISK MANAGEMENT FOR INSURERS...1
1.2 OUTLINE...2
1.2.1 Chapter 3: Valuation of Swap Rate Dependent Embedded Options...3
1.2.2 Chapter 4: Valuation of Guaranteed Annuity Options using a Stochastic Volatility Model for Equity Prices 3 1.2.3 Chapter 5: On stochastic mortality modeling...4
1.2.4 Chapter 6: Stochastic portfolio specific mortality and the quantification of mortality basis risk ...4
1.2.5 Chapter 7: Micro-level stochastic loss reserving ...4
2. STOCHASTIC PROCESSES...6
2.1RISK NEUTRAL STOCHASTIC PROCESSES FOR VALUATION...6
2.1.1 Martingales and Measures...6
2.1.2 Affine Jump-Diffusions...8
2.1.3 Gaussian interest rate models...9
2.1.4 Stochastic volatility model for equity prices ...10
2.1.5 Stochastic processes for valuation of unhedgeable insurance risks...10
2.2REAL WORLD STOCHASTIC PROCESSES FOR RISK MANAGEMENT...11
2.2.1 ARIMA Time Series Models ...11
2.2.2 Poisson processes and renewal processes ...12
3. VALUATION OF SWAP RATE DEPENDENT EMBEDDED OPTIONS...14
3.1INTRODUCTION...14
3.2SWAP RATE DEPENDENT EMBEDDED OPTIONS...16
3.3THE UNDERLYING INTEREST RATE MODEL...18
3.3.1 Multi-factor Gaussian models...19
3.3.2 Valuation for other interest rate models ...19
3.4THE SCHRAGER-PELSSER RESULT FOR SWAPTIONS...20
3.5ANALYTICAL APPROXIMATION – DIRECT PAYMENT...21
3.5.1 Determining the expectation of R(Ti) ...22
3.5.2 Determining the variance of R(Ti)...23
3.5.3 Pricing formulas ...23
3.6VALUATION FOR MORE COMPLEX PROFIT SHARING RULES...24
3.6.1 Compounding profit sharing ...25
3.6.2 Profit sharing including the return on an additional asset ...26
3.6.3 Additional management actions or other complex features ...26
3.7NUMERICAL EXAMPLES...28
7.1 Example 1: 10-year average of 7-year swap rate, direct payment...28
3.7.2 Example 2: 10-year average of 7-year swap rate, compounding option...29
3.8CONCLUSIONS...31
APPENDIX 3A: PROOF OF (3.8) ...32
APPENDIX 3B: PROOFS OF (3.11) AND (3.12)...33
4. VALUATION OF GUARANTEED ANNUITY OPTIONS USING A STOCHASTIC VOLATILITY
MODEL FOR EQUITY PRICES...37
4.1INTRODUCTION...37
4.2GUARANTEED ANNUITY CONTRACT...38
4.3THE SCHÖBEL-ZHU-HULL-WHITE MODEL...39
4.4CALIBRATION OF THE SZHW AND BSHW MODEL...41
4.5PRICING THE GUARANTEED ANNUITY OPTION UNDER STOCHASTIC VOLATILITY AND STOCHASTIC INTEREST RATES...44
4.5.1 Taking the equity price as numéraire...44
4.5.2 Explicit formula for the GAO price...46
4.6EXTENSION TO TWO-FACTOR INTEREST RATE MODEL...47
4.7NUMERICAL EXAMPLES...49
4.7.1 Comparison results SZHW model and Black-Scholes Hull-White model...49
4.7.2 Impact of different risk drivers...51
4.7.3 Comparison results of the two-factor model with Chu and Kwok (2007) ...52
4.8CONCLUSIONS...54
APPENDIX 4A:PRICING OF A COUPON BEARING OPTION UNDER A TWO-FACTOR INTEREST RATE MODEL...55
APPENDIX 4B:MOMENTS AND TERMINAL CORRELATION OF THE TWO-FACTOR GAUSSIAN INTEREST RATE MODEL56 APPENDIX 4C:SPECIAL CASE: INDEPENDENT EQUITY PRICE PROCESS OR PURE INTEREST RATE GUARANTEED ANNUITIES...58
C.1 Hull-White model ...58
C.2 Gaussian Two-factor model ...60
APPENDIX 4D:YIELD CURVE SHOCKS...60
APPENDIX 4E:MODEL SETUP OF THE CHU AND KWOK (2007) CASE...61
5. ON STOCHASTIC MORTALITY MODELING...63
5.1INTRODUCTION...63
5.2LITERATURE REVIEW: CRITERIA AND MODELS...64
5.2.1 Criteria for stochastic mortality models ...64
5.2.2 Stochastic mortality models ...64
5.2.3 Problems with modeling cohort effect...66
5.3A NEW STOCHASTIC MORTALITY MODEL...67
5.3.1 The proposed model ...68
5.3.2 Identifiability constraints ...69
5.4FITTING THE MODEL...70
5.4.1 Fitting methodology ...70
5.4.2 Comparison of fit quality with existing models ...71
5.4.3 Fitting the ARIMA processes – U.S. Males ...73
5.5MORTALITY PROJECTIONS –U.S.MALES...74
5.5.1 Simulation results – U.S. Males ...75
5.5.2 Robustness of simulation results ...75
5.5.3 Comparison with other models ...76
5.6RISK NEUTRAL SPECIFICATION OF THE MODEL...76
5.6.1 Risk neutral dynamics ...77
5.6.2 Calibration of the market price of risk...77
5.7PARAMETER UNCERTAINTY...80
5.8CONCLUSIONS...81
APPENDIX 5A:U.S.MALE - ESTIMATES FOR AX AND T-X...83
APPENDIX 5B: SIMULATION RESULTS ENGLAND &WALES AND THE NETHERLANDS...84
APPENDIX 5C: SIMULATION RESULTS ROBUSTNESS TESTS...85
6. STOCHASTIC PORTFOLIO SPECIFIC MORTALITY AND THE QUANTIFICATION OF MORTALITY BASIS RISK ...87
6.1INTRODUCTION...87
6.2.1 The basic model ...90
6.2.2 Fitting the basic model...92
6.2.3 Adding stochastic behavior ...93
6.2.4 Combine the process with the stochastic country population model...94
6.3APPLICATION TO EXAMPLE INSURANCE PORTFOLIOS...95
6.4NUMERICAL EXAMPLE 1:VALUE AT RISK...98
6.4.1 Stochastic country population mortality model...98
6.4.2 Impact on Value at Risk ...99
6.5NUMERICAL EXAMPLE 2: HEDGE EFFECTIVENESS / BASIS RISK...100
6.6CONCLUSIONS...102
APPENDIX 6A: EXAMPLE 2-FACTOR MODEL BASED ON NELSON &SIEGEL...103
APPENDIX 6B: FURTHER RESULTS...104
APPENDIX 6C: HEDGE PORTFOLIOS...105
7. MICRO-LEVEL STOCHASTIC LOSS RESERVING ...106
7.1INTRODUCTION...106
7.2DATA...109
7.3THE STATISTICAL MODEL...115
7.3.1 Position Dependent Marked Poisson Process...115
7.3.2 The Likelihood...116
7.3.3 Distributional assumptions ...118
7.4ESTIMATION RESULTS...120
7.5PREDICTING FUTURE CASH FLOWS...123
7.5.1 Predicting IBNR claims...123
7.5.2 Predicting RBNS claims...124
7.6NUMERICAL RESULTS...127
7.7CONCLUSIONS...135
REFERENCES ...137
