Question

On January 1, 2000 an insurance company has a lump-sum of 450000 which is due to Linden as a life insurance death benefit. He (Linden) chooses to receive the benefit annually over a period of 20 years via equal size payments. The first payment is received on Dec. 31, 2000, and each subsequent payment is made at the end of each of the following years. The yearly benefit that Linden receives is based on valuing the lump-sum (i.e. the 450000) at an effective interest rate of 4% per year. However, the insurance company earns interest at an effective interest rate of 5% per year. Furthermore, every July 1 (the middle of the year) the company pays 500 in expenses and taxes to maintain the policy. At the end of the 20-year period, after the last payment is made to Linden, the company has X remaining. Calculate X. Give your answer rounded to the nearest whole number. Hint: You are basically subtracting out the “time-value” of the annual benefits and the “time value” of the taxes/fees from the “time-value” of the lump-sum.

Answer 1

Principal = 4,50,000

Amount totally paid after 20 years = 4,50,000*(1+0.04)²⁰ = 986005.41

Amount earned if no money was paid = 4,50,000*(1+0.05)²⁰ = 1193983.97

Expenses incurred = 500*[(1+0.04)¹⁹ -1)/0.04]*(1+0.04)*(1+0.02) = 14676.82

Amount X remaining with the company = 1193983.97-986005.41-14676.82 = 193,302