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Preface | |
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Notation index | |
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The Deterministic Model | |
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Introduction and motivation | |
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Risk and insurance | |
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Deterministic versus stochastic models | |
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Finance and investments | |
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Adequacy and equity | |
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Reassessment | |
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Conclusion | |
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The basic deterministic model | |
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Cashflows | |
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An analogy with currencies | |
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Discount functions | |
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Calculating the discount function | |
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Interest and discount rates | |
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Constant interest | |
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Values and actuarial equivalence | |
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Regular pattern cashflows | |
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Balances and reserves | |
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Time shifting and the splitting identity | |
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Change of discount function | |
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Internal rate of return | |
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Forward prices and term structure | |
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Standard notation and terminology | |
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Spreadsheet calculations | |
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Notes and references | |
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Exercises | |
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The life table | |
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Basic definitions | |
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Probabilities | |
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Constructing the life table from the values of q x | |
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Life expectancy | |
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Choice of life tables | |
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Standard notation and terminology | |
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A sample table | |
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Notes and references | |
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Exercises | |
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Life annuities | |
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Introduction | |
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Calculating annuity premiums | |
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The interest and survivorship discount function | |
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Guaranteed payments | |
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Deferred annuities with annual premiums | |
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Some practical considerations | |
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Standard notation and terminology | |
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Spreadsheet calculations | |
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Exercises | |
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Life insurance | |
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Introduction | |
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Calculating life insurance premiums | |
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Types of life insurance | |
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Combined insurance-annuity benefits | |
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Insurances viewed as annuities | |
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Summary of formulas | |
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A general insurance–annuity identity | |
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Standard notation and terminology | |
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Spreadsheet applications | |
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Exercises | |
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Insurance and annuity reserves | |
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Introduction to reserves | |
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The general pattern of reserves | |
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Recursion | |
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Detailed analysis of an insurance or annuity contract | |
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Interest and mortality bases for reserves | |
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Non forfeiture values | |
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Policies involving a ‘return of the reserve' | |
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Universal life and variables annuities | |
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Standard notation and terminology | |
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Spreadsheet applications | |
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Exercises | |
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Fractional durations | |
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Introduction | |
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Cashflows discounted with interest only | |
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Life annuities paid mthly | |
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Immediate annuities | |
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Approximation and computation | |
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Fractional period premiums and reserves | |
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Reserves at fractional durations | |
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Notes and references | |
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Exercises | |
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Continuous payments | |
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Introduction to continuous annuities | |
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The force of discount | |
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The constant interest case | |
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Continuous life annuities | |
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The force of mortality | |
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Insurances payable at the moment of death | |
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Premiums and reserves | |
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The general insurance–annuity identity in the continuous case | |
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Differential equations for reserves | |
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Some examples of exact calculation | |
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Standard notation and terminology | |
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Notes and references | |
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Exercises | |
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Select mortality | |
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Introduction | |
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Select and ultimate tables | |
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Changes in formulas | |
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Projections in annuity tables | |
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Further remarks | |
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Exercises | |
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Multiple-life contracts | |
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Introduction | |
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The joint-life status | |
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Joint-life annuities and insurances | |
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Last-survivor annuities and insurances | |
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Moment of death insurances | |
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The general two-life annuity contract | |
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The general two-life insurance contract | |
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Contingent insurances | |
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Duration problems | |
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Applications to annuity credit risk | |
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Standard notation and terminology | |
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Spreadsheet applications | |
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Notes and references | |
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Exercises | |
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Multiple-decrement theory | |
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Introduction | |
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The basic model | |
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Insurances | |
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Determining the model from the forces of decrement | |
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The analogy with joint-life statuses | |
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A machine analogy | |
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Associated single-decrement tables | |
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Notes and references | |
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Exercises | |
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Expenses | |
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Introduction | |
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Effect on reserves | |
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Realistic reserve and balance calculations | |
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Notes and references | |
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Exercises | |
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The Stochastic Model | |
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Survival distributions and failure times | |
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Introduction to survival distributions | |
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The discrete case | |
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The continuous case | |
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Examples | |
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Shifted distributions | |
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The standard approximation | |
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The stochastic life table | |
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Life expectancy in the stochastic model | |
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Stochastic interest rates | |
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Notes and references | |
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Exercises | |
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The stochastic approach to insurance and annuities | |
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Introduction | |
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The stochastic approach to insurance benefits | |
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The stochastic approach to annuity benefits | |
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Deferred contracts | |
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The stochastic approach to reserves | |
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The stochastic approach to premiums | |
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The variance of r L | |
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Standard notation and terminology | |
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Notes and references | |
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Exercises | |
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Simplifications under level benefit contracts | |
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Introduction | |
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Variance calculations in the continuous case | |
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Variance calculations in the discrete case | |
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Exact distributions | |
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Non-level benefit examples | |
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Exercises | |
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The minimum failure time | |
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Introduction | |
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Joint distributions | |
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The distribution of T | |
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The joint distribution of (T, J ) | |
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Other problems | |
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The common shock model | |
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Copulas | |
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Notes and references | |
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Exercises | |
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Risk Theory | |
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The collective risk model | |
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Introduction | |
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The mean and variance of S | |
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Generating functions | |
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Exact distribution of S | |
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Choosing a frequency distribution | |
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Choosing a severity distribution | |
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Handling the point mass at 0 | |
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Counting claims of a particular type | |
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The sum of two compound Poisson distributions | |
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Deductibles and other modifications | |
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A recursion formula for S | |
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Notes and references | |
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Exercises | |
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Risk Assessment | |
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Introduction | |
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Utility theory | |
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Convex and concave functions: Jensen's inequality | |
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A general comparison method | |
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Risk measures for capital adequacy | |
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Notes and references | |
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Exercises | |
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An introduction to stochastic processes | |
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Introduction | |
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Markov chains | |
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Martingales | |
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Finite-state Markov chains | |
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Notes and references | |
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Exercises | |
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Poisson processes | |
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Introduction | |
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Definition of a Poisson process | |
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Waiting times | |
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Some properties of a Poisson process | |
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Non homogeneous Poisson processes | |
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Compound Poisson processes | |
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Notes and references | |
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Exercises | |
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Ruin models | |
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Introduction | |
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A functional equation approach | |
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The martingale approach to ruin theory | |
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Distribution of the deficit at ruin | |
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Recursion formulas | |
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The compound Poisson surplus process | |
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The maximal aggregate loss | |
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Notes and references | |
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Exercises | |
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Credibility theory | |
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Introductory material | |
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Conditional expectation and variance | |
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General framework for Bayesian credibility | |
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Classical examples | |
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Approximations | |
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Conditions for exactness | |
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Estimation | |
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Notes and references | |
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Exercises | |
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Multi-state models | |
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Introduction | |
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The discrete-time model | |
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The continuous-time model | |
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Notes and references | |
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Exercises | |
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review of probability theory | |
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Introduction | |
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Sample spaces and probability measures | |
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Conditioning and independence | |
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Random variables | |
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Distributions | |
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Expectations and moments | |
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Expectations in terms of the distribution function | |
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The normal distribution | |
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Joint distributions | |
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Conditioning and independence for random variables | |
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Convolution | |
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Moment generating functions | |
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Probability generating functions | |
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Mixtures | |
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Answers to exercises | |
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References | |
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Index | |