Chapter 02

Chapter 02 - Introduction The models for life insurances...

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FIN 3220A Actuarial Models I Chapter 2 Life Insurance Page 2 FIN 3220A (2005-2006) Chapter 2 Life Insurance Introduction ± The models for life insurances designed to reduce the financial impact of the random event of untimely death are developed ± Life insurances are characterized by their long-term nature ± The amount of investment earnings, up to the time of payment, provides a significant element of uncertainty ± The uncertainty has two causes: ² The unknown rate of earnings over the investment period ² The unknown length of the investment period ± In this course: ² A deterministic model is used for the unknown investment earnings ² A probability distribution is used to model the uncertainty in regards to the investment period ± Our model will be built in terms of functions of T , the insured’s future- lifetime random variable Page 3 FIN 3220A (2005-2006) Chapter 2 Life Insurance Insurance Payable at the Moment of Death ± We look at a life insurance with the benefit amount and the time of payment depend only on the length of the interval from the issue of the insurance to the death of the insured ± There are two functions in the model: ² A benefit function , b t ² A discount function , v t ± t is the length of the interval from issue to death ± v t is the interest discount factor from the time of payment back to the time of policy issue ² Here, the underlying force of interest is deterministic ² Most of the time, we only look at the case of constant force of interest Page 4 FIN 3220A (2005-2006) Chapter 2 Life Insurance Insurance Payable at the Moment of Death ± The present-value function , z t , is the present value, at policy issue, of the benefit payment ± The elapsed time from policy issue to the death of the insured is the insured’s future-lifetime random variable, T = T ( x ) ² So z T is also a random variable ± The model for the insurance is based on the equation ± To develop a probability model of Z ² The first step will be to define b t and v t ² The next step is to determine some characteristics of the probability distribution of Z that are consequences of an assumed distribution of T tt t z bv = TT Zb v =
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Page 5 FIN 3220A (2005-2006) Chapter 2 Life Insurance Level Benefit Insurance ± An n-year term life insurance provides a payment only if the insured dies within the n -year term of an insurance commencing at issue ± If a unit is payable at the moment of death of ( x ), we have ± Three conventions: ² We define b t , v t and Z only on non-negative values ² For a t value where b t is 0, the value of v t is irrelevant ² The forecast of interest is assumed to be constant, unless stated otherwise ± The expectation of the present-value random variable, Z , is called the actuarial present value , APV , of the insurance ² A more exact term would be expectation of the present value of the payments , 1 , 0, 0, 0.
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Chapter 02 - Introduction The models for life insurances...

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