[EB] Life Insurance Part8 PDF

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Chapter Five: Annuities Net Single Premiums

CHAPTER FIVE ANNUITIES NET SINGLE PREMIUMS

Learning Objectives After studying this chapter, the students will be able to:  Know the types of net premiums.  Recognize Steps in calculating the net single premium.  Calculate the net single premium for pure endowment.  Know the different types of annuities and the difference between each of them.  Calculate the net single premium required for the different annuities.  Calculate the payment of annuity that can be purchased with a cash amount.

5.1 INTRODUCTION The base to determine life insurance and annuities premiums is the preceding mortality and interest assumption, because the net premium takes into account mortality and interest factors only. To complete life insurance premiums two assumptions are made: 85

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a) Premiums are paid at the beginning of the year. b) Claims are paid at the end of the year. The first assumption is realistic because the life policy cannot be force unless the first premium paid in advance. The assumption that claims are paid at the end of the year is not in accordance, with the facts, since claims are paid immediately after the necessary forms are completed. Nevertheless, the actuary to simplify the calculation of the premiums uses this assumption.

Net Premiums may be classified into two types: a) Net single premium. b) Net annual premium. Since the calculation of the annual premium is based upon the net single premium for the same kind and amount of insurance, we shell explain first determining the net single premium.

5.2 STEPS IN CALCULATING THE NET SINGLE PREMIUM There are some information should to be known by the actuary to calculate a net single premium for both life insurance and annuities. Actuary must know: a) The plan of insurance. b) The mortality table to be used. 86

Chapter Five: Annuities Net Single Premiums

c) The interest rate to be used. With this information, six steps are involving in computing a net single premium.

1. Determine what constitutes a claim: Life insurance policies are bought for two main reasons: i. Death benefits: protect dependents in case of Insured’s death. ii. Living benefits: protect the insured in case of the insured live long, he needs a lump sum amount paid on certain future date or life annuities. iii. Death and living benefits.

Step 2. Determine when claims are paid. Claims may be paid any time through the year. Life insurance policies, in which, claim paid as incurred, but the actuary made an assumption that claims are paid at the end of the year. Annuity claims are paid at the beginning, or at the end of each period depends on the annuity type.

Step 3. Determine the probability of payment each year. In computing the premium, the number insured is the number living as shown on the mortality table for the issued age. The number of death claims each year comes out from the death 87

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per year column for that year and for each succeeding year within the policy period in the mortality table. Then calculate the probability of payment each year.

Step 4. Determine the expected value of payment each year. The expected payment value is the product of the amount to be paid and the probability of payment for each year.

Step 5. Determine the present values of all future claims Calculate the present values of all expected payments by choosing an appropriate interest rate.

Step 6. Determine the net single premium: The net single premium is the sum of the present value of all payment expectation. In all cases the present value of the net premiums will equal the present value of all future benefits. Regardless of the type of policy have the following formula for net premiums from the basic equation of value:  The present value at the date   The present value at date      of issue of net premiums   of issue of benefits     

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Chapter Five: Annuities Net Single Premiums

I- Net Single premiums For Living Benefits: 1- Pure Endowment Pure endowment is a single payment that will be paid to a person of at a certain time of the person is still living at that time. To get a formula for a pure endowment we assume that lx persons aged x establish a fund by equal contributions at the present time to pay L.E. 1 to each member that survive at age x+n, the amount of money needed in "n" years will equal lx+n survivors in the mortality table. The present value of this fund is vn × lx+n, since the money is not needed until n years from now. Let us suppose that the symbol

n

Ex or, Ax:

1

n

is used to

represent each person’s contribution to the fund, we can equate the total original premiums paid. At issue date of the policy by lx persons who entering into this policy will equal lx. nEx. The total present value of the benefits to be paid to lx+n survivors at the end of n years, using the mutual fund method, (equation of value) we have lx × nEx = (1 + i)-n × lx+n We substitute the letter v for (1 + i)-1, then vn = (1 + i)-n 89

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Substituting in the formula and solving for nEx, we have, n Ex 

v n l x n v n. n px lx

The symbol nEx is used for the net single premium for L.E. 1 pure endowment policy issued to a person aged x for n years. The net single premium nEx depends for its value upon the interest rate and the probability for the final payment of L.E. 1 will be received, and may be considered as the present value of the pure endowment. We now introduce the commutation functions that are used by life insurance actuaries to reduce the numerical calculation to solve life insurance and annuity problems. All of commutation functions combine present value of I factors, and values from the mortality table. Thus we have different tables of commutation required for each combination of interest rate and mortality table these functions defined below.

1- Survival Function a) vx × lx = Dx b) Nx = Dx + Dx+1 + ………. + D- 1

2- Dying functions a)vx+1× dx = Cx b)Mx = Cx + Cx+1 +………….. + C- 1 90

Chapter Five: Annuities Net Single Premiums

If we multiply both the numerator and denominator of the equation of the net single premium of a pure endowment by vx, we have 

v x n l x n  E n x vx lx

Substituting commutation functions, this expression can be written as follows: n Ex 

D x n Dx

EXAMPLE (5.1): Find the net single premium of a L.E. 50,000 pure endowment payable in 25 year to a person now aged 30. SOLUTION: x = 30

n = 25

R = 50,000

Pure endowment the net single premium NSP = R× nEx  R

D x n Dx

Then NSP = 50,000× 25E30

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 50,000

D30 25 D30

 50,000

D55 D30

 50,000 

1,639,330  L. E.20,985.94 3,905,782

EXAMPLE (5.2): A person now aged 20 purchases a pure endowment payable at age 45. The purchase price is L.E.20,000. Calculate the face value of this policy. SOLUTION: x = 20

n = 25

R=?

Pure endowment policy the net single premium = present value = the purchase price = 20,000 Where NSP = R× nEx R 

Dx n Dx

Then NSP = R ×25E20

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Chapter Five: Annuities Net Single Premiums

 R

D20 25 D20

R 

D45 D20

Substituting for N.S.P = 20,000. we have 20,000  R 

2,392,905 5,351,273

20,000 = R × 0.447166 R = L.E. 44,726.12 EXAMPLE (5.3): A man now aged 30 purchases a life insurance policy give him L.E.10,000 if and only if he still alive at age 55. Calculate the purchase price of this policy. SOLUTION: x = 30

n = 25

R = 10,000

If he still alive at age 55= pure endowment purchase price = present value the net single premium =? n

E x  Ax:

1

n

Since P.V  R  n Ex  R

D x n Dx

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1

Note 25E30 is the same A30: 25 Then P.V  R  n E x  10, 000  10,000 

D3025 D 10,000  55 D30 D30

1,639,330 3,905,782

P.V = L.E.4,197.19 EXAMPLE (5.4): A man now aged 40 has L.E. 60,000 cash if he deposits this with an insurance company what sum should he receive at age 60 if he agrees to forfeit all rights in event of death before age 60. SOLUTION: x = 40

n = 20

R=?

Pure endowment the present value = 60,000

P.V = R× nEx 60,000 = R× 20E40 60,000 R 

D60 D40

60,000  R 

1,306,724  R  0.461251 2,833,002

R =L.E.130,081.02 94

Chapter Five: Annuities Net Single Premiums

EXAMPLE (5.5): What pure endowment payable at age 55 could a man aged 25 purchase with L.E. 5,000 cash? SOLUTION: x = 25

n = 30

P.V = 5,000

pure endowment present value ?

P.V = R × nEx P.V  R

D55 D25

5,000  R 

1,639,330  R  0.358451 4,573,377

R = L.E.13,948.91 EXAMPLE (5.6) A man now aged 30 is promised L.E. 20,000 if he still alive at age 65 .Find the present value. SOLUTION x = 30

n = 35

R = 20,000

pure endowment present value ? P.V = R × nEx P .V  

D65  20,000 D30 995,688  20,000  L.E .5,098.53 3,905,782

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EXAMPLE (5.7): A man now aged 20 has L.E. 6,000 cash if he deposits this with an insurance company what sum should he receive at age 55 if he survival at this age. SOLUTION: x = 20 n = 35 R=? P.V = pure endowment the net single premium = L.E.6,000

N.S.P = R × nEx 6,000 R 

D55 D20

1,639,330 5,351,273 R  L.E.19,585.83 6,000  R 

2-Life Annuities As pointed out earlier in the previous chapter that life annuities being assumed that each payment contingent on a particular individual being alive (the annuitant). Annuities are classified according to the number of payments, we have l) Whole life Annuity, in which, the numbers of payments are to continue as long as the annuitant lives. 2) Temporary life annuity, in which, the number of payments cease after a certain number of years, even though the annuitant be then still living or at the death of the annuitant which comes first. 96...