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Unformatted text preview: Chapter 4 Insurance benefits Chapter 4 Insurance benefits ACTSC 232 Introduction to Actuarial Mathematics Tianxiang Shi Department of Statistics and Actuarial Science University of Waterloo Winter 2012 Tianxiang Shi(tim.shi@uwaterloo.ca) Chapter 4 Insurance benefits Outline 1 Introduction and assumptions 2 Valuation of level insurance benefits CM: insurances payable at the moment of death DM: insurances payable at the end of the year of death mthly life insurances 3 Variable insurance benefits 4 Some remarks on selected lives and sums of insureds Tianxiang Shi(tim.shi@uwaterloo.ca) Chapter 4 Insurance benefits Introduction and assumptions Present value of insurance benefits Question Q: In a whole life insurance, a benefit of $ Y is payable at the moment of death, what is the present value (PV) of the insurance benefit? Tianxiang Shi(tim.shi@uwaterloo.ca) Chapter 4 Insurance benefits Introduction and assumptions Present value of insurance benefits Question Q: In a whole life insurance, a benefit of $ Y is payable at the moment of death, what is the present value (PV) of the insurance benefit? A: $ Y v T x or $ Y e T x , which is a r.v. The main topic in Ch.4 is to evaluate the PV of the benefits in various traditional life insurance policies, where the benefits are contingent on the insureds survival. Distribution of the PV of the benefits can be derived from T x The main focus will be analyzing the first two moments. Tianxiang Shi(tim.shi@uwaterloo.ca) Chapter 4 Insurance benefits Introduction and assumptions Interest rate model Throughout this chapter, we assume constant force of interest , unless otherwise stated. accumulate for n periods (1 + i ) n = e n = 1 + i ( m ) m ! mn = (1 d ) n i ( m ) : nominal rate of interest compounded m times per year discount for n periods (1 + i ) n = v n = e n = (1 d ) n = 1 d ( u ) u ! un d ( u ) : nominal rate of discount compounded u times per year Tianxiang Shi(tim.shi@uwaterloo.ca) Chapter 4 Insurance benefits Valuation of level insurance benefits CM: insurances payable at the moment of death Notations for continuous model In the continuous models (CM), the benefit is payable at the moment of death, where both the benefit amount and discount factor can be expressed in terms of T x . b T x : benefits amount at the moment of death. For a level insurance benefit policy, the benefit is constant. In this section, all the actuarial notations are for $ 1 level benefit insurances . v T x : discount factor of the moment of death to present. Z = b T x v T x : PV of the benefits. E [ Z ] : expected present value (EPV), also referred as actuarial present value (APV), or simply actuarial value. Tianxiang Shi(tim.shi@uwaterloo.ca) Chapter 4 Insurance benefits Valuation of level insurance benefits CM: insurances payable at the moment of death CM: Whole life insurance A x Whole life insurance with sum insured $ 1 For a life ( x ) , a benefit of $ 1 is payable at the moment of death....
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