Assume that the population living at the age of 50 is 750,000 and the partial mortality rate given below as follow.
Age............................... Number of dying
50.................................. 3,842
51.................................. 2,067
52.................................. 2,172
53.................................. 3,014
54.................................. 3,200
Suppose further that all people living at the age of 50 purchased 5 year term insurance for birr 8,000. Annual premium are paid at the beginning of the year and death claims are to be paid at the end of the year in which death occurred. Interest rate is 10%. DETERMINE
A. The Net Single Premium
B. The Net Level Premium
C. The Gross Premium, given the loading factor to be 5% of the Net Single Premium computed above.
A. The Net Single Premium can be determined using the Net Single Premium formula:
Net Single Premium = Sum of (Number of dying at each age * Present value of 5-year term insurance at each age)
To calculate the present value of 5-year term insurance, we need to discount the future death claims at an interest rate of 10%.
Present Value of 5-year term insurance at age 50 = (Death claim at age 50)/(1 + 0.1)^5
= 8,000/(1.1^5)
= 5,018.93 birr
Present Value of 5-year term insurance at age 51 = (Death claim at age 51)/(1 + 0.1)^4
= 8,000/(1.1^4)
= 6,389.27 birr
Present Value of 5-year term insurance at age 52 = (Death claim at age 52)/(1 + 0.1)^3
= 8,000/(1.1^3)
= 6,880.79 birr
Present Value of 5-year term insurance at age 53 = (Death claim at age 53)/(1 + 0.1)^2
= 8,000/(1.1^2)
= 7,447.77 birr
Present Value of 5-year term insurance at age 54 = (Death claim at age 54)/(1 + 0.1)^1
= 8,000/(1.1^1)
= 7,738.64 birr
Now we can calculate the Net Single Premium:
Net Single Premium = (3,842 * 5,018.93) + (2,067 * 6,389.27) + (2,172 * 6,880.79) + (3,014 * 7,447.77) + (3,200 * 7,738.64)
= 19,299,518.30 birr
Therefore, the Net Single Premium is 19,299,518.30 birr.
B. The Net Level Premium can be determined using the Net Level Premium formula:
Net Level Premium = Net Single Premium / Present value of a 5-year term insurance for a 50-year-old
To calculate the present value of a 5-year term insurance for a 50-year-old, we need to discount the annual premium payments at an interest rate of 10%.
Present value of a 5-year term insurance for a 50-year-old = 8,000/[(1 + 0.1) + (1 + 0.1)^2 + (1 + 0.1)^3 + (1 + 0.1)^4 + (1 + 0.1)^5]
= 8,000/[1.1 + 1.21 + 1.331 + 1.4641 + 1.61051]
= 8,000/6.61661
= 1,209.111 birr
Now we can calculate the Net Level Premium:
Net Level Premium = 19,299,518.30 birr / 1,209.111 birr
= 15,961.88 birr
Therefore, the Net Level Premium is 15,961.88 birr.
C. The Gross Premium can be determined using the Gross Premium formula:
Gross Premium = Net Single Premium + (Loading factor * Net Single Premium)
Since the loading factor is given as 5% of the Net Single Premium, we can calculate it as:
Loading factor = 0.05 * 19,299,518.30 birr
= 964,975.92 birr
Now we can calculate the Gross Premium:
Gross Premium = 19,299,518.30 birr + 964,975.92 birr
= 20,264,494.22 birr
Therefore, the Gross Premium is 20,264,494.22 birr.