Market-Consistent Actuarial ValuationSeries: EAA Series

Wüthrich, Mario V., Bühlmann, Hans, Furrer, Hansjörg

2nd Edition., 2010, XI, 157 p., Softcover
ISBN: 978-3-642-14851-4



It is a challenging task to read the balance sheet of an insurance company. This derives from the fact that different positions are often measured by different yardsticks. Assets, for example, are mostly valued at market prices whereas liabilities are often measured by established actuarial methods. However, there is a general agreement that the balance sheet of an insurance company should be measured in a consistent way.

Market-Consistent Actuarial Valuation presents powerful methods to measure liabilities and assets in a consistent way. The mathematical framework that leads to market-consistent values for insurance liabilities is explained in detail by the authors. Topics covered are stochastic discounting with deflators, valuation portfolio in life and non-life insurance, probability distortions, asset and liability management, financial risks, insurance technical risks, and solvency.


Content Level » Graduate
Keywords » Life-insurance - Market-consistent actuarial value - Non-Life Insurance - Risk theory
Related subjects » Finance & Banking - Quantitative Finance


Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Three pillar approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Solvency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 From the past to the future . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Full balance sheet approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.5 Recent financial failures and difficulties . . . . . . . . . . . . . . . . . . . . . 6
2 Stochastic discounting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1 Basic discrete time model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Market-consistent valuation in the basic discrete time model . . 12
2.2.1 Task of modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.2 Understanding deflators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.3 Toy example for deflators . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 Valuation at time t > 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.4 The meaning of basic reserves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.5 Equivalent martingale measures . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.6 Insurance technical and financial variables . . . . . . . . . . . . . . . . . . 34
2.6.1 Choice of numeraire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.6.2 Probability distortion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.7 Conclusions on Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3 Valuation portfolio in life insurance . . . . . . . . . . . . . . . . . . . . . . . . 43
3.1 Deterministic life insurance model . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2 Valuation portfolio for the deterministic life insurance model . . 44
3.3 General valuation procedure for deterministic insurance
technical risks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.4 Self-financing property of the VaPo (deterministic insurance
technical risk) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
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3.5 VaPo protected against insurance technical risks . . . . . . . . . . . . . 49
3.5.1 Construction of the VaPo protected against insurance
technical risks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.5.2 Probability distortion of life tables . . . . . . . . . . . . . . . . . . . 54
3.6 Back to the basic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.7 Approximate valuation portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.8 Conclusions on Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.9 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4 Financial risks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.1 Asset and liability management . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.2 Procedure to control financial risks . . . . . . . . . . . . . . . . . . . . . . . . 72
4.3 Financial modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.3.1 Stochastic discounting of financial variables . . . . . . . . . . . 73
4.3.2 Modelling Margrabe options . . . . . . . . . . . . . . . . . . . . . . . . 75
4.3.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.4 Pricing Margrabe options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.4.1 Pricing using Esscher transforms . . . . . . . . . . . . . . . . . . . . 78
4.4.2 Application of the Esscher transform to the
multi-dimensional Wiener process . . . . . . . . . . . . . . . . . . . 81
4.4.3 Hedging Margrabe options . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.4.4 Target capital . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5 Valuation portfolio in non-life insurance . . . . . . . . . . . . . . . . . . . 89
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.2 Construction of the VaPo in non-life insurance . . . . . . . . . . . . . . 93
5.3 VaPo protected against insurance technical risks, pragmatic
approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.4 VaPo protected against insurance technical risks, theoretical
considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.5 Loss development triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.5.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.5.2 Chain-ladder method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.5.3 Estimation of insurance technical risks in the
chain-ladder model, single accident years . . . . . . . . . . . . . 112
5.5.4 Aggregation of parameter estimation errors across
different accident years . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.6 Unallocated loss adjustment expenses . . . . . . . . . . . . . . . . . . . . . . 129
5.6.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.6.2 Pure claims payments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.6.3 ULAE charges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
5.6.4 New York-method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
5.6.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
5.7 Conclusions on the non-life VaPo . . . . . . . . . . . . . . . . . . . . . . . . . . 136
Contents XI
6 Selected Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
6.1 Sources of losses and profits, profit sharing. . . . . . . . . . . . . . . . . . 139
6.2 Remarks on the self-financing property of the insurance
technical liabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
6.3 Claims development result in non-life insurance . . . . . . . . . . . . . 144
6.4 Legal quote in life insurance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155