I question the answer - If the first contribution starts one year from now (and not today), and we need to calculate the amount after the end of 20 years, so why shall not we deduct 1 year from the calulation? Because he is not contributing today. So, the first saving earns interest for 19 years only, beginning after the end of year 1 (the time when it was invested).
0----1 year = no contribution
1----2 year = 20000 invested, earns interest of 7%… if you make a timeline as well, only 18 years remain after this till end of year 20. So in all 19 years.
I hope I have not confused anyone and made myself clear enough to get your well formed response. Thanks!
The reason is that this is an ordinary annuity meaning that the annual payments are made at the end of every of the 20 years. If you have your calculator in End-Mode (default setting) than solving vor FV should give you autmatically the correct answer ($ 819,910).
By dividing this result further by 1.07 you assume that the payments are an annuity due (which, however is not the case here). This would only be the case if the problem would state that the couple would start making contributions as of now and at the beginning of every of the next 20 years.
Respectfully, the answer for an annuity due _ is not _ the answer for an ordinary annuity divided by (1 + r); rather, the answer for an annuity due is the answer for an ordinary annuity _ multiplied _ by (1 + r). An annuity due is worth more (present value and future value) than an ordinary annuity.
I’d encourage you to model this situation in Excel: make a column with the times 1 - 20, another with the contributions ($20,000 at each time), and a third with the future value of each contribution (1.07^(20 - t)), then sum the final column.