Betty received $ 500,000 from a life insurance policy to be distributed to her as an annuity certain in 10 equal annual installments with the first payment made immediately. On the day she receives her third payment, she is offered a monthly perpetuity of X in lieu of the future annual payments. The first payment will be made in exactly one month. The effective annual rate of interest is 8 %. Determine the value of X
I'm struggling to get an accurate figure for this question for the value of X. I'm just doing practise questions for my CT1 exam and if I had the figure I have to work towards I'd figure out my error!
The annual installment is $P$ such that $$ 500,000=P\times \ddot a_{\overline{10}|8\%} $$ that is $P\approx 68,995.13$.
The present value of the future payments after the 3rd payment is $P\times \ddot a_{\overline{7}|8\%}\approx 387,951.31$.
The effective monthly rate is $i=(1.08)^{1/12}-1\approx 0.6434\%$ and then $$ 387,951.31=\frac{X}{i} $$ that is $X\approx 2,496.09$