the first is 59,000 invested up front at 6.3% per year compounded monthly.
the second is 200 invested at the end of each month at 6.3% per year compounded monthly.
you can put this in a financial calculator and the calculator will do the computations for you.
your inputs would be:
present value = -59000 (negative because it's money going out from you)
future value = what you want to find
payments at the end of each month are -200 (negative because it's money going out from you.
interest rate is 6.3/12 = .525% per month.
the 59000 is invested at the beginning of the investment period.
the 200 is invested at the end of each month.
calculator says that the future value is 1,238,479.26
if you do the calculations separately, then the first calculation (the 59000 invested up front) will have a future value of 775,7111.5267.
the second calculation (the 200 at the end of each month) will have a future value of 462,767.7332.
add these together (without rounding to the nearest penny) and you get a combined future value of 1,238,479.2599 rounded to 4 decimal places which becomes 1,238,479.26 rounded to 2 decimal places.
if you rounded each separate calculation to the nearest penny, then the combined total would be 1,238,479.26.
there is no discrepancy because the first figure is rounded up and the second figure is rounded down.
algebraically, you should not found until the very end, which is when the final figures are added together.
financially, it's probably up to the policies of the financial institution whether they round after combining the figures or round after combining the figures.
if the investments are in the same account then i would assume the rounding is done once at the end after the figures are tallied together.
however, i don't know what the individual financial institution policies are, so i can't say definitively, one way or the other.
i used a financial calculator.
you could also use formulas and calculate from them.
future value of 59000 at 6.3% compounded annually would give you the following equation:
f = 59000 * (1 + .063/12) ^ (41 * 12) = 775,711.5267
future value of 200 at the end of each month at 6.3% cmpounded annually would give you the following equation:
f = (200 * ((1 + .063/12) ^ (41*12) -1)) / (.063/12) = 462,767.7332
these are the same values i got using the financial calculator, as they should be.
here are the results from using the online financial calculator at https://arachnoid.com/finance/index.html
the investments are shown as negative because it's money going out.
the future value is shown as positive because it's money coming in.