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**Chapter 8 LIFE ANNUITIES**

Basic Concepts Commutation Functions

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8.1 Basic Concepts We know how to compute present value of contingent payments Life tables are sources of probabilities of surviving We can use data from life tables to compute present values of payments which are contingent on either survival or death

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**Example (pure endowment), p. 155**

Yuanlin is 38 years old. If he reaches age 65, he will receive a single payment of 50,000. If i = .12, find an expression for the value of this payment to Yuanlin today. Use the following entries in the life table: l38 = 8327, l65 = 5411

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Pure Endowment Pure endowment: 1 is paid t years from now to an individual currently aged x if the individual survives Probability of surviving is t px Therefore the present value of this payment is the net single premium for the pure endowment = = t Ex = (t px ) (1 + t) – t = v t t px

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**Example (life annuity), p. 156**

Aretha is 27 years old. Beginning one year from today, she will receive 10,000 annually for as long as she is alive. Find an expression for the present value of this series of payments assuming i = .09 Find numerical value of this expression if px = .95 for each x

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**Series of payments of 1 unit as long as individual is alive**

Life annuity Series of payments of 1 unit as long as individual is alive present value (net single premium) of annuity ax 1 1 1 ….. ….. age x x + 1 x + 2 x + n px 2px npx probability

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**Temporary life annuity**

Series of n payments of 1 unit (contingent on survival) present value ax:n| last payment 1 1 1 ….. age x x + 1 x + 2 x + n px 2px npx probability

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**n - years deferred life annuity**

Series of payments of 1 unit as long as individual is alive in which the first payment is at x + n + 1 present value n|ax first payment 1 1 … … age x x + 1 x + 2 x + n x + n +1 x + n + 2 n+1px n+2px probability Note:

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**Life annuities-due äx ….. … äx:n| ….. n|äx … … 1 1 1 1 px 2px npx 1 1**

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Note but

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**8.2 Commutation Functions**

Recall: present value of a pure endowment of 1 to be paid n years hence to a life currently aged x Denote Dx = vxlx Then nEx = Dx+n / Dx

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**Life annuity and commutation function**

Since nEx = Dx+n / Dx we have Define commutation function Nx as follows: Note: Then:

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**Identities for other types of life annuities**

temporary life annuity n-years delayed life annuity temporary life annuity-due

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**Accumulated values of life annuities**

temporary life annuity since and we have similarly for temporary life annuity-due: and

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Examples (p. 162 – p. 164) (life annuities and commutation functions) Marvin, aged 38, purchases a life annuity of 1000 per year. From tables, we learn that N38 = 5600 and N39 = Find the net single premium Marvin should pay for this annuity if the first 1000 payment occurs in one year if the first 1000 payment occurs now Stay verbally the meaning of (N35 – N55) / D20 (unknown rate of interest) Given Nx = 5000, Nx+1=4900, Nx+2 = 4810 and qx = .005, find i

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Select group Select group of population is a group with the probability of survival different from the probability given in the standard life tables Such groups can have higher than average probability of survival (e.g. due to excellent health) or, conversely, higher mortality rate (e.g. due to dangerous working conditions)

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Notations Suppose that a person aged x is in the first year of being in the select group Then p[x] denotes the probability of survival for 1 year and q[x] = 1 – p[x] denotes the probability of dying during 1 year for such a person If the person stays within this group for subsequent years, the corresponding probabilities of survival for 1 more year are denoted by p[x]+1, p[x]+2, and so on Similar notations are used for life annuities: a[x] denotes the net single premium for a life annuity of 1 (with the first payment in one year) to a person aged x in his first year as a member of the select group A life table which involves a select group is called a select-and-ultimate table

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Examples (p. 165 – p. 166) (select group) Margaret, aged 65, purchases a life annuity which will provide annual payments of 1000 commencing at age 66. For the next year only, Margaret’s probability of survival is higher than that predicted by the life tables and, in fact, is equal to p , where p65 is taken from the standard life table. Based on that standard life table, we have the values D65 = 300, D66 = 260 and N67 = If i = .09, find the net single premium for this annuity (select-and-ultimate table) A select-and-ultimate table has a select period of two years. Select probabilities are related to ultimate probabilities by the relationships p[x] = (11/10) px and p[x]+1 = (21/20) px+1. An ultimate table shows D60 = 1900, D61 = 1500, and ä 60:20| = 11, when i = .08. Find the select temporary life annuity ä[60]:20|

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The following values are based on a unisex life table: N38 = 5600, N39 = 5350, N40 = 5105, N41 = 4865, N42 = It is assumed that this table needs to be set forward one year for males and set back two years for females. If Michael and Brenda are both age 40, find the net single premium that each should pay for a life annuity of 1000 per year, if the first payment occurs immediately.

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