Question

A 60-year old person buys a 25-year term annuity in arrears contract for a single upfront premium of $1,000,000. Interest earnt by the insurance company is assumed to be 5% over the first 10 years of the contract and then 4% for the remaining term. The amount paid in the first 10 years is two-thirds of the amount paid in the remaining 15 years. Find out the amount paid in year 1 of the contract.

Assume that select mortality applies and there is an annual fee of $20 which is paid at the time of the annuity payment

Answer 1

Input Data
Term of the Annuity	25 Years
Upfront premium	1000000
Interest for first 10 years, r1	5%
Interest for the rest 15 years,r2	4%
Amount paid during first 10 years	2/3 of Amount paid in next 15 years
Select Mortality Fee	$ 20 per annum
	=P*(1-(1+r)^-n)/r
Present Value of Select Mortality Fee	=(20*(1-(1.05)^-5))/.05+(20*(1-(1.04)^-15)/.04
	308.96
Present Value of an Annuity is calculated as
PV = C*(1-((1+r)^-n)) where C is to be worked out
r
where PV is 1000000
PV of Annuity Premium-PV of Annual Select Mortality Fee	=1000000-308.96
	999691.04
Assume 'C' as the annual amount paid in last 15 years,then 2/3 C is the annual amount paid in first 10 years
Applying the formula
999691.04	[2/3C*((1-(1+5%)^-10))/5%]+[C*((1-(1+4%)^-15))/4%]
((1-(1+5%)^-10))/5% =	7.7217
((1-(1+4%)^-15))/4% =	11.1184
999691.04	=2/3C*7.7217+C*11.1184
999691.04	=5.1478C+11.1184C
999691.04	=16.2662C
Hence C	=999691.04/16.2662
	61458.18
2/3C	=61458.18*2/3
	40972.12
Hence $40,972.12 is the Year 1 annuity payment to be received