Manual for SOA Exam MLC.

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1 Chapter 4. Life Insurance. c 29. Miguel A. Arcones. All rights reserved. Extract from: Arcones Manual for the SOA Exam MLC. Fall 29 Edition. available at c 29. Miguel A. Arcones. All rights reserved. 1/12

2 Non level payments paid at the time of death Suppose that if failure happens at time t, then the benefit payment is b t. The present value of the benefit payment is denoted by B x = b Tx ν Tx. The actuarial present value of this benefit is E[B x ] = b t ν t f Tx (t) dt = b t ν t tp x µ x+t dt. c 29. Miguel A. Arcones. All rights reserved. 2/12

The present value of the benefit payment is denoted by B x = b Tx ν Tx.

3 Example 1 For a whole life insurance on (6), you are given: (i) Death benefits are paid at the moment of death. (ii) Mortality follows a de Moivre model with terminal age 1. (iii) i = 7%. (iv) b t = (2)(1.4) t, t. Calculate the mean and the standard deviation of the present value random variable for this insurance. Solution: The present value random variable is Z = b T6 ν T 6 = (2)(1.4) T 6 (1.7) T 6 = (2) The density of T 6 is t p 6 µ 6 (t) = 1 4, t 4. ( ) 1.4 T c 29. Miguel A. Arcones. All rights reserved. 3/12

Calculate the mean and the standard deviation of the present value random variable for this insurance.

4 Solution: The present value random variable is Z = b T6 ν T 6 = (2)(1.4) T 6 (1.7) T 6 = (2) The density of T 6 is ( ) 1.4 T Hence, E[Z] = 4 = , E[Z 2 ] = 4 = , (2) tp 6 µ 6 (t) = 1 4, t 4. ( ) 1.4 t ( (2)( dt = 1.7) 1) 4 ln(1.4/1.7) ( 1.4 ( ) 1.4 2t 8 (2) dt = (2)2 ( 1.7) 1) 8 ln(1.4/1.7) Var(Z) = ( ) 2 = , Var(Z) = = c 29. Miguel A. Arcones. All rights reserved. 4/12

7, (2) tp 6 µ 6 (t) = 1 4, t 4. ( ) 1.4 t 1 1.7 4 ( (2)( 1.4 4 dt = 1.7) 1) 4 ln(1.4/1.7) ( 1.4 ( ) 1.4 2t 8 (2) 2 1 1.

5 If b t = t, t, we say that we have an increasing life insurance in the continuous case. Its actuarial present value is denoted by ( )x = tν t tp x µ x+t dt. c 29. Miguel A. Arcones. All rights reserved. 5/12

( )x = tv t tp x µ x+t dt = te (δ+µ)t dt = µ (µ +

8 Let t be the least integer greater than or equal to t. t is called the ceiling of t. We have that t = k if k is an integer such that k 1 < t k. Notice that 1 if < t 1, 2 if 1 < t 2, t = 3 if 2 < t 3, If b t = t, t, we say that we have an increasing life insurance in the piecewise continuous case. Its actuarial present value is denoted by ( )x = t ν t tp x µ x+t dt = k=1 (k 1,k] kν t tp x µ x+t dt. c 29. Miguel A. Arcones. All rights reserved. 8/12

Notice that 1 if < t 1, 2 if 1 < t 2, t = 3 if 2 < t 3, If b t = t, t, we say that we have an increasing life

9 If b t = t, t n, zero otherwise, we say that we have an n year term increasing life insurance in the continuous case. Its actuarial present value is denoted by ( ) n 1 x:n = tν t tp x µ x+t dt. c 29. Miguel A. Arcones. All rights reserved. 9/12

Its actuarial present value is denoted by ( ) n 1 x:n = tν t

10 If b t = t, t n, zero otherwise, we say that we have an n year term increasing life insurance in the piecewise continuous case. Its actuarial present value is denoted by ( ) n 1 x:n = t ν t tp x µ x+t dt. c 29. Miguel A. Arcones. All rights reserved. 1/12

11 If b t = n t, t n, zero otherwise, we say that we have an n year term decreasing life insurance in the continuous case. Its actuarial present value is denoted by ( ) n 1 D A x:n = (n t)ν t tp x µ x+t dt. c 29. Miguel A. Arcones. All rights reserved. 11/12

Its actuarial present value is denoted by ( ) n 1 D A x:n = (n

12 If b t = n t, t n, zero otherwise, we say that we have an n year term decreasing life insurance in the piecewise continuous case. Its actuarial present value is denoted by ( ) n 1 D A x:n = n t ν t tp x µ x+t dt. c 29. Miguel A. Arcones. All rights reserved. 12/12

Its actuarial present value is denoted by ( ) n 1 D A x:n = n t ν

Manual for SOA Exam MLC.

Manual for SOA Exam MLC. Chapter 4. Life Insurance. Extract from: Arcones Manual for the SOA Exam MLC. Fall 2009 Edition. available at http://www.actexmadriver.com/ 1/14 Level benefit insurance in the continuous case In this chapter,