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Preface | |
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Introduction to life insurance | |
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Summary | |
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Background | |
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Life insurance and annuity contracts | |
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Introduction | |
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Traditional insurance contracts | |
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Modern insurance contracts | |
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Distribution methods | |
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Underwriting | |
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Premiums | |
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Life annuities | |
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Other insurance contracts | |
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Pension benefits | |
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Defined benefit and defined contribution pensions | |
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Defined benefit pension design | |
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Mutual and proprietary insurers | |
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Typical problems | |
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Notes and further reading | |
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Exercises | |
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Survival models | |
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Summary | |
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The future lifetime random variable | |
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The force of mortality | |
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Actuarial notation | |
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Mean and standard deviation of T<sub>x</sub> | |
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Curtate future lifetime | |
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K<sub>x</sub> and e<sub>x</sub> | |
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The complete and curtate expected future lifetimes, e<sub>x</sub> and e<sub>x</sub> | |
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Notes and further reading | |
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Exercises | |
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Life tables and selection | |
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Summary | |
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Life tables | |
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Fractional age assumptions | |
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Uniform distribution of deaths | |
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Constant force of mortality | |
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National life tables | |
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Survival models for life insurance policyholders | |
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Life insurance underwriting | |
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Select and ultimate survival models | |
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Notation and formulae for select survival models | |
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Select life tables | |
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Notes and further reading | |
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Exercises | |
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Insurance benefits | |
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Summary | |
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Introduction | |
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Assumptions | |
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Valuation of insurance benefits | |
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Whole life insurance: the continuous case, &Abar;<sub>x</sub> | |
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Whole life insurance: the annual case, A<sub>x</sub> | |
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Whole life insurance: the 1 /mthly case, A<sup>(m)</sup><sub>x</sub> | |
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Recursions | |
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Term insurance | |
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Pure endowment | |
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Endowment insurance | |
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Deferred insurance benefits | |
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Relating &Abar;<sub>x</sub>, A<sub>x</sub> and A<sup>(m)</sup><sub>x</sub> | |
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Using the uniform distribution of deaths assumption | |
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Using the claims acceleration approach | |
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Variable insurance benefits | |
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Functions for select lives | |
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Notes and further reading | |
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Exercises | |
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Annuities | |
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Summary | |
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Introduction | |
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Review of annuities-certain | |
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Annual life annuities | |
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Whole life annuity-due | |
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Term annuity-due | |
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Whole life immediate annuity | |
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Term immediate annuity | |
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Annuities payable continuously | |
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Whole life continuous annuity | |
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Term continuous annuity | |
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Annuities payable m times per year | |
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Introduction | |
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Life annuities payable m times a year | |
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Term annuities payable m times a year | |
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Comparison of annuities by payment frequency | |
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Deferred annuities | |
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Guaranteed annuities | |
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Increasing annuities | |
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Arithmetically increasing annuities | |
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Geometrically increasing annuities | |
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Evaluating annuity functions | |
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Recursions | |
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Applying the UDD assumption | |
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Woolhouse's formula | |
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Numerical illustrations | |
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Functions for select lives | |
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Notes and further reading | |
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Exercises | |
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Premium calculation | |
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Summary | |
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Preliminaries | |
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Assumptions | |
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The present value of future loss random variable | |
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The equivalence principle | |
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Net premiums | |
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Gross premium calculation | |
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Profit | |
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The portfolio percentile premium principle | |
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Extra risks | |
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Age rating | |
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Constant addition to �x | |
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Constant multiple of mortality rates | |
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Notes and further reading | |
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Exercises | |
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Policy values | |
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Summary | |
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Assumptions | |
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Policies with annual cash flows | |
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The future loss random variable | |
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Policy values for policies with annual cash flows | |
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Recursive formulae for policy values | |
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Annual profit | |
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Asset shares | |
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Policy values for policies with cash flows at discrete intervals other than annually | |
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Recursions | |
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Valuation between premium dates | |
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Policy values with continuous cash flows | |
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Thiele's differential equation | |
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Numerical solution of Thiele's differential equation | |
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Policy alterations | |
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Retrospective policy value | |
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Negative policy values | |
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Notes and further reading | |
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Exercises | |
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Multiple state models | |
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Summary | |
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Examples of multiple state models | |
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The alive-dead model | |
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Term insurance with increased benefit on accidental death | |
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The permanent disability model | |
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The disability income insurance model | |
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The joint life and last survivor model | |
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Assumptions and notation | |
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Formulae for probabilities | |
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Kolmogorov's forward equations | |
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Numerical evaluation of probabilities | |
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Premiums | |
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Policy values and Thiele's differential equation | |
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The disability income model | |
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Thiele's differential equation - the general case | |
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Multiple decrement models | |
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Joint life and last survivor benefits | |
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The model and assumptions | |
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Joint life and last survivor probabilities | |
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Joint life and last survivor annuity and insurance functions | |
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An important special case: independent survival models | |
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Transitions at specified ages | |
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Notes and further reading | |
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Exercises | |
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Pension mathematics | |
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Summary | |
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Introduction | |
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The salary scale function | |
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Setting the DC contribution | |
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The service table | |
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Valuation of benefits | |
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Final salary plans | |
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Career average earnings plans | |
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Funding plans | |
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Notes and further reading | |
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Exercises | |
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Interest rate risk | |
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Summary | |
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The yield curve | |
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Valuation of insurances and life annuities | |
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Replicating the cash flows of a traditional non-participating product | |
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Diversifiable and non-diversifiable risk | |
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Diversifiable mortality risk | |
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Non-diversifiable risk | |
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Monte Carlo simulation | |
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Notes and further reading | |
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Exercised | |
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Emerging costs for traditional life insurance | |
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Summary | |
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Profit testing for traditional life insurance | |
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The net cash flows for a policy | |
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Reserves | |
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Profit measures | |
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A further example of a profit test | |
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Notes and further reading | |
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Exercises | |
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Emerging costs for equity-linked insurance | |
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Summary | |
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Equity-linked insurance | |
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Deterministic profit testing for equity-linked insurance | |
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Stochastic profit testing | |
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Stochastic pricing | |
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Stochastic reserving | |
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Reserving for policies with non-diversifiable risk | |
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Quantile reserving | |
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CTE reserving | |
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Comments on reserving | |
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Notes and further reading | |
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Exercises | |
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Option pricing | |
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Summary | |
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Introduction | |
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The'no arbitrage�assumption | |
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Options | |
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The binomial option pricing model | |
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Assumptions | |
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Pricing over a single time period | |
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Pricing over two time periods | |
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Summary of the binomial model option pricing technique | |
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The Black-Scholes-Merton model | |
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The model | |
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The Black-Scholes-Merton option pricing formula | |
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Notes and further reading | |
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Exercises | |
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Embedded options | |
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Summary | |
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Introduction | |
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Guaranteed minimum maturity benefit | |
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Pricing | |
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Reserving | |
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Guaranteed minimum death benefit | |
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Pricing | |
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Reserving | |
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Pricing methods for embedded options | |
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Risk management | |
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Emerging costs | |
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Notes and further reading | |
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Exercises | |
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Probability theory | |
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Probability distributions | |
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Binomial distribution | |
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Uniform distribution | |
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Normal distribution | |
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Lognormal distribution | |
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The central limit theorem | |
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Functions of a random variable | |
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Discrete random variables | |
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Continuous random variables | |
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Mixed random variables | |
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Conditional expectation and conditional variance | |
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Notes and further reading | |
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Numerical techniques | |
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Numerical integration | |
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The trapezium rule | |
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Repeated Simpson's rule | |
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Integrals over an infinite interval | |
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Woolhouse's formula | |
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Notes and further reading | |
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Simulation | |
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The inverse transform method | |
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Simulation from a normal distribution | |
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The Box-Muller method | |
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The polar method | |
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Notes and further reading | |
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References | |
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Author index | |
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Index | |