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1 | (10) |
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1.1 Full Balance Sheet Approach |
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3 | (1) |
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1.2 Solvency Considerations |
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4 | (1) |
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1.3 Further Modeling Issues |
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5 | (1) |
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6 | (5) |
Part I Financial Valuation Principles |
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2 State Price Deflators and Stochastic Discounting |
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11 | (24) |
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2.1 Zero Coupon Bonds and Term Structure of Interest Rates |
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11 | (7) |
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2.1.1 Motivation for Discounting |
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11 | (1) |
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2.1.2 Spot Rates and Term Structure of Interest Rates |
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12 | (3) |
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2.1.3 Estimating the Yield Curve |
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15 | (3) |
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2.2 Basic Discrete Time Stochastic Model |
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18 | (8) |
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2.2.1 Valuation at Time 0 |
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19 | (3) |
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2.2.2 Interpretation of State Price Deflators |
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22 | (1) |
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2.2.3 Valuation at Time t > 0 |
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23 | (3) |
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2.3 Equivalent Martingale Measure |
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26 | (5) |
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2.3.1 Bank Account Numeraire |
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26 | (1) |
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2.3.2 Martingale Measure and the FTAP |
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27 | (4) |
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31 | (4) |
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35 | (62) |
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3.1 General Gaussian Spot Rate Models |
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35 | (3) |
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3.2 One-Factor Gaussian Affine Term Structure Models |
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38 | (3) |
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3.3 Discrete Time One-Factor Vasicek Model |
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41 | (15) |
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3.3.1 Spot Rate Dynamics on a Yearly Grid |
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42 | (3) |
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3.3.2 Spot Rate Dynamics on a Monthly Grid |
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45 | (2) |
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3.3.3 Parameter Calibration in the One-Factor Vasicek Model |
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47 | (9) |
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3.4 Conditionally Heteroscedastic Spot Rate Models |
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56 | (4) |
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3.5 Auto-Regressive Moving Average (ARMA) Spot Rate Models |
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60 | (5) |
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3.5.1 AR(1) Spot Rate Model |
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61 | (1) |
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3.5.2 AR(p) Spot Rate Model |
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62 | (1) |
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3.5.3 General ARMA Spot Rate Models |
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63 | (1) |
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3.5.4 Parameter Calibration in ARMA Models |
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64 | (1) |
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3.6 Discrete Time Multifactor Vasicek Model |
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65 | (22) |
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3.6.1 Motivation for Multifactor Spot Rate Models |
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65 | (2) |
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3.6.2 Multifactor Vasicek Model (with Independent Factors) |
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67 | (5) |
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3.6.3 Parameter Estimation and the Kalman Filter |
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72 | (15) |
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3.7 One-Factor Gamma Spot Rate Model |
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87 | (5) |
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3.7.1 Gamma Affine Term Structure Model |
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87 | (3) |
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3.7.2 Parameter Calibration in the Gamma Spot Rate Model |
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90 | (2) |
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3.8 Discrete Time Black-Karasinski Model |
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92 | (5) |
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3.8.1 Log-Normal Spot Rate Dynamics |
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92 | (1) |
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3.8.2 Parameter Calibration in the Black-Karasinski Model |
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93 | (2) |
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3.8.3 ARMA Extended Black-Karasinski Model |
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95 | (2) |
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4 Stochastic Forward Rate and Yield Curve Modeling |
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97 | (34) |
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4.1 General Discrete Time HJM Framework |
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98 | (2) |
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4.2 Gaussian Discrete Time HJM Framework |
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100 | (6) |
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4.2.1 General Gaussian Discrete Time HJM Framework |
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100 | (2) |
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4.2.2 Two-Factor Gaussian HJM Model |
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102 | (3) |
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4.2.3 Nelson-Siegel and Svensson HJM Framework |
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105 | (1) |
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106 | (19) |
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4.3.1 Derivations from the Forward Rate Framework |
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106 | (3) |
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4.3.2 Stochastic Yield Curve Modeling |
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109 | (16) |
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Appendix Proofs of Chap. 4 |
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125 | (6) |
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5 Pricing of Financial Assets |
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131 | (24) |
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5.1 Pricing of Cash Flows |
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132 | (5) |
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5.1.1 General Cash Flow Valuation in the Vasicek Model |
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132 | (3) |
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5.1.2 Defaultable Coupon Bonds |
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135 | (2) |
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137 | (9) |
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5.2.1 A Log-Normal Example in the Vasicek Model |
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139 | (4) |
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5.2.2 A First Asset-and-Liability Management Problem |
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143 | (3) |
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5.3 Pricing of Derivative Instruments |
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146 | (3) |
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Appendix Proofs of Chap. 5 |
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149 | (6) |
Part II Actuarial Valuation and Solvency |
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6 Actuarial and Financial Modeling |
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155 | (14) |
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6.1 Financial Market and Financial Filtration |
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155 | (2) |
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6.2 Basic Actuarial Model |
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157 | (7) |
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6.3 Improved Actuarial Model |
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164 | (5) |
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169 | (36) |
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7.1 Construction of the Valuation Portfolio |
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170 | (7) |
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7.1.1 Financial Portfolios and Cash Flows |
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171 | (1) |
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7.1.2 Construction of the VaPo |
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171 | (3) |
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7.1.3 Best-Estimate Reserves |
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174 | (3) |
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177 | (10) |
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7.2.1 Examples in Life Insurance |
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177 | (4) |
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7.2.2 Example in Non-life Insurance |
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181 | (6) |
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7.3 Claims Development Result and ALM |
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187 | (10) |
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7.3.1 Claims Development Result |
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187 | (1) |
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7.3.2 Hedgeable Filtration and ALM |
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188 | (4) |
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192 | (5) |
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7.4 Approximate Valuation Portfolio |
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197 | (8) |
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8 Protected Valuation Portfolio |
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205 | (56) |
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8.1 Construction of the Protected Valuation Portfolio |
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205 | (2) |
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207 | (27) |
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8.2.1 Risk-Adjusted Reserves |
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207 | (2) |
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8.2.2 Claims Development Result of Risk-Adjusted Reserves |
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209 | (2) |
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8.2.3 Fortuin-Kasteleyn-Ginibre (FKG) Inequality |
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211 | (2) |
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8.2.4 Examples in Life Insurance |
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213 | (10) |
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8.2.5 Example in Non-life Insurance |
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223 | (7) |
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8.2.6 Further Probability Distortion Examples |
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230 | (4) |
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234 | (27) |
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8.3.1 Non-life Insurance Run-Off |
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234 | (10) |
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8.3.2 Life Insurance Examples |
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244 | (17) |
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261 | (76) |
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261 | (7) |
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9.1.1 Definition of (Conditional) Risk Measures |
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261 | (4) |
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9.1.2 Examples of Risk Measures |
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265 | (3) |
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9.2 Solvency and Acceptability |
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268 | (10) |
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9.2.1 Definition of Solvency and Acceptability |
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268 | (6) |
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9.2.2 Free Capital and Solvency Terminology |
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274 | (3) |
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277 | (1) |
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9.3 No Insurance Technical Risk |
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278 | (21) |
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9.3.1 Theoretical ALM Solution and Free Capital |
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278 | (5) |
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9.3.2 General Asset Allocations |
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283 | (3) |
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9.3.3 Limited Liability Option |
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286 | (5) |
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291 | (5) |
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9.3.5 Hedging Margrabe Options |
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296 | (3) |
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9.4 Inclusion of Insurance Technical Risk |
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299 | (27) |
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9.4.1 Insurance Technical and Financial Result |
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300 | (2) |
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9.4.2 Theoretical ALM Solution and Solvency |
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302 | (7) |
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9.4.3 General ALM Problem and Insurance Technical Risk |
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309 | (4) |
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9.4.4 Cost-of-Capital Loading and Dividend Payments |
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313 | (8) |
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9.4.5 Risk Spreading and Law of Large Numbers |
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321 | (4) |
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9.4.6 Limitations of the Vasicek Financial Model |
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325 | (1) |
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9.5 Portfolio Optimization |
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326 | (11) |
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9.5.1 Standard Deviation Based Risk Measure |
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327 | (6) |
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9.5.2 Estimation of the Covariance Matrix |
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333 | (4) |
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10 Selected Topics and Examples |
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337 | (70) |
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10.1 Extreme Value Distributions and Copulas |
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337 | (2) |
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10.2 Parameter Uncertainty |
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339 | (17) |
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10.2.1 Parameter Uncertainty for a Non-life Run-Off |
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339 | (13) |
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10.2.2 Modeling of Longevity Risk |
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352 | (4) |
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10.3 Cost-of-Capital Loading in Practice |
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356 | (10) |
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10.3.1 General Considerations |
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356 | (2) |
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10.3.2 Cost-of-Capital Loading Example |
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358 | (8) |
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10.4 Accounting Year Factors in Run-Off Triangles |
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366 | (3) |
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366 | (2) |
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10.4.2 Predictive Distribution |
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368 | (1) |
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10.5 Premium Liability Modeling |
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369 | (12) |
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10.5.1 Modeling Attritional Claims |
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371 | (4) |
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10.5.2 Modeling Large Claims |
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375 | (1) |
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376 | (5) |
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10.6 Risk Measurement and Solvency Modeling |
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381 | (21) |
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10.6.1 Insurance Liabilities |
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381 | (4) |
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10.6.2 Asset Portfolio and Premium Income |
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385 | (2) |
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10.6.3 Cost Process and Other Risk Factors |
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387 | (1) |
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10.6.4 Accounting Condition and Acceptability |
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388 | (2) |
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10.6.5 Solvency Toy Model in Action |
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390 | (12) |
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402 | (5) |
Part III Appendix |
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11 Auxiliary Considerations |
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407 | (12) |
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11.1 Helpful Results with Gaussian Distributions |
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407 | (1) |
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11.2 Change of Numeraire Technique |
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408 | (11) |
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11.2.1 General Changes of Numeraire |
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408 | (2) |
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11.2.2 Forward Measures and European Options on ZCBs |
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410 | (5) |
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11.2.3 European Options with Log-Normal Asset Prices |
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415 | (4) |
References |
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419 | (8) |
Index |
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427 | |