Firstly, let's clarify the basic principles of our calculation. We are looking for a formula that would give us the value of equal monthly deposits that will accumulate into a certain amount over a specific period with a constant annual percentage rate (APR).
The formula we use for this purpose is the future value of a series or an ordinary annuity formula. Annuity is a series of equal payments made at fixed intervals for a certain number of periods.
The formula looks like this: P = FV / [((1 + r)^nt - 1) / r]
Explanations:
- P is the monthly deposit,
- FV is the future value of the fund (which is known - $164,739),
- r is the monthly interest rate,
- n is the number of times interest is applied per time period (per year in our case),
- t is the time the money is invested for in years.
Now let's plug values into our formula. The APR is given as 5.6%. We have to convert it to a decimal, so we divide by 100, getting 0.056. However, since interest is compounded monthly, we'll take the monthly rate instead of the annual rate. Doing so requires dividing the decimal rate by 12 (number of months in a year). That would give us 0.056 / 12 = 0.004666667.
Our target value (FV) is $164,739. The total time period is 16 years, but since the compounding is monthly, we should consider the total number of months instead: 16 * 12 = 192 months.
Assuming these figures for r and nt, we can now calculate the monthly deposit amount P using the provided formula:
P = 164739 / ((1 + 0.00466667)^192 - 1) / 0.00466667
Solving this, we find that the monthly deposit that needs to be made is approximately $532.15. It means that, to reach your target of $164,739 in 16 years with a 5.6% APR, you should deposit this amount each month.