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?
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
To calculate the monthly investment needed to reach a savings goal, we can use the formula for future value of an ordinary annuity:
Future Value (FV) = Monthly Investment × [(1 + Monthly Interest Rate)^ (Number of Months) - 1] / Monthly Interest Rate
Given:
Desired Future Value (FV) = $800,000
Return Rate = 5.3% = 0.053 (monthly interest rate)
Number of Years (t) = 40
Number of Months (n) = t × 12 = 40 × 12 = 480
Substituting these values in the future value formula:
$800,000 = Monthly Investment × [(1 + 0.053)^(480) - 1] / 0.053
Now, we need to solve this equation to find the monthly investment required.
Here, you can use a financial calculator or spreadsheet software to quickly evaluate the equation. However, let's break down the steps:
1. Simplify the equation:
800,000 = Monthly Investment × [(1.053^480) - 1] / 0.053
2. Solve the term [(1.053^480) - 1] / 0.053 using a calculator:
[(1.053^480) - 1] / 0.053 ≈ 6340.99
3. Rearrange the equation to solve for Monthly Investment:
Monthly Investment ≈ 800,000 / 6340.99 ≈ $126.05 (rounded to the nearest cent)
Therefore, to accumulate $800,000 in 40 years with a 5.3% compounded monthly rate of return, you would need to invest approximately $126.05 per month.
Now, let's move on to the second part of your question:
If you deposited the amount you needed to achieve your goal in 20 years, the value of your savings after 10 years can be calculated using the future value formula:
Future Value (FV) = Present Value (PV) × (1 + Monthly Interest Rate)^ (Number of Months)
Given:
PV (Present Value) = $800,000
Return Rate = 5.3% = 0.053 (monthly interest rate)
Number of Years (t) = 10
Number of Months (n) = t × 12 = 10 × 12 = 120
Substituting these values in the future value formula:
FV = 800,000 × (1 + 0.053)^120
Again, you can use a calculator or spreadsheet software to evaluate this equation:
FV ≈ 800,000 × (1.053)^120 ≈ $2,038,424.86 (rounded to the nearest cent)
Therefore, if you deposited $800,000 and allowed it to grow for 10 years with a 5.3% compounded monthly rate of return, your savings would be worth approximately $2,038,424.86.