Question

a 20-years annuity-immediate has annual payments . The first payment is 100 and subsequent payments are increased by 100 until they reach 1000. The remaining payment stay at 1000. the annual effective intersst rate 7.5% . What is the coast of this annuity?

Answer 1

Solution - Calculation of cost of annuity

Part 1 - Calculation of Present value of 10 year deferred annuity

Formula for deferred annuity = 1000 * PVIFA[1/(1+r)ⁿ] * PVIF[1/(1+r)ⁿ]

r = 7.5%

n = 20 Years

Present value of deferred annuity = 1000 * [1/(1.075)²⁰]* [1/(1.075)^{1 to 20}]

Present value of deferred annuity = 1000*0.48519*6.8641

Present value of deferred annuity = 3330.39

Part 2 - Present value of increasing annuity

Formula = (Present value of annuity due - Last year annuity factor*time period)/Interest rate

Formula for present value of annuity due = [[1 - {1/(1+r)ⁿ}] * (1+r)]/r

r = 7.5%, n= 10 years

= [[1-{1/(1.075)¹⁰}] * (1.075)]/0.075

= [(1-0.4852)*(1.075)]/0.075

= 0.5534/0.075= 7.3789

Present value of Incremental annuity=[7.3789 - (0.48519*10)]/0.075

Present value of incremental annuity = (7.3789 - 4.8519)/0.075

Present value of incremental annuity = 3369.33

Part 3 - Total cost of annuity = (3369.33 + 3330.39) = 6700 (rounded off)