Question

How much should you invest each month in order to have $800,000 if your rate of return is 5.3% compounded monthly and you want to achieve your goal in 40 years?

If you deposit the amount you need to achieve your goal in 20 years, how much will your savings be worth after 10 years?

Answer 1

Part 1:
Use the amortization formula:
P=(L)R^n(R-1)/(R^n-1)
where
P=Payment per period
L=Loan amount
R=1+interest rate per period
n=number of periods

For example, if monthly payment is required for a 100,000 loan over 30 years at 5% p.a., then
L=100,000
R=(1+0.05/12)
n=30*12=360
(Do NOT truncate the value of R to 1.004167, it will reduce accuracy considerably)
so monthly payment,
P=100000*(1+0.05/12)^360*(0.05/12)/((1+0.05/12)^360-1)
=536.82(16)

If R were truncated to 1.004167
the answer would be
536.84(61).

The second part is not very clear as to how the amount was achieved. Would it be using the same monthly payment, or the same future value (namely 100,000)?
In either case, calculate the value after 20 years using the formula for amortization, then apply the compound interest formula to find the value after 10 more years.
FV=PV(R^120)
FV=value after 30 years
PV=value after 20 years
R=interest rate