Great question! This is a future value of an ordinary annuity problem because you are making equal monthly deposits into a savings account to reach a specific amount in the future.
Let's work through it step by step!
Step 1: Identify the information given.
We know:
- Savings goal = $2,000
- Annual interest rate = 2%
- Compounding frequency = Monthly
- Time = 2 years
- Monthly deposit = Unknown
Our goal is to determine how much money must be deposited each month to accumulate $2,000 after two years.
Step 2: Convert the annual interest rate to a monthly rate.
The annual interest rate is 2%, or 0.02 as a decimal.
Since interest is compounded monthly, divide by 12:
Monthly Interest Rate = 0.02 / 12
r = 0.0016666667
This means the savings account earns approximately 0.1667% interest per month.
Step 3: Calculate the number of monthly deposits.
We are saving for 2 years.
There are 12 months in one year, so:
n = 2 × 12
n = 24 months
Therefore, we will make 24 monthly deposits.
Step 4: Use the Future Value of an Ordinary Annuity Formula.
The formula is:
FV = PMT × [((1 + r)^n - 1) / r]
Where:
FV = Future value, or the amount we want to save
PMT = Monthly deposit
r = Monthly interest rate
n = Total number of deposits
We know everything except PMT.
Step 5: Rearrange the Formula to Solve for PMT.
Start with:
FV = PMT × [((1 + r)^n - 1) / r]
Divide both sides by the annuity factor:
PMT = FV × r / [(1 + r)^n - 1]
Now substitute the values:
PMT = [2,000 × (0.02/12)] / [(1 + 0.02/12)^24 - 1]
Step 6: Calculate the Monthly Deposit.
First, calculate the monthly growth factor:
1 + 0.02/12 = 1.0016666667
Raise it to the 24th power:
(1.0016666667)^24 ≈ 1.040776
Subtract 1:
1.040776 - 1 = 0.040776
Calculate the numerator:
2,000 × (0.02/12) = 3.333333
Now divide:
PMT ≈ 3.333333 / 0.040776
PMT ≈ $81.7472
Round to the nearest cent:
Monthly Deposit = $81.75
Step 7: Check Our Answer.
Let's see how much money you will personally contribute over 24 months.
Total Deposits = Monthly Deposit × Number of Months
= $81.75 × 24
Total Deposits = $1,962.00
Your savings goal is $2,000.
The remaining amount comes from interest earned on your deposits.
Interest Earned ≈ $2,000 - $1,962.00
Interest Earned ≈ $38.00
Because the monthly payment was rounded upward to $81.75, the actual ending balance will be slightly more than $2,000.
Final Answer:
You should deposit $81.75 at the end of each month for 24 months into a savings account earning 2% annual interest compounded monthly.
This will allow you to accumulate at least $2,000 for your semester abroad.
Important Concept to Remember
Why can't we simply divide $2,000 by 24 months?
Without interest:
$2,000 / 24 = $83.33 per month
However, because the savings account earns interest, you do not need to contribute the entire $2,000 yourself.
Your deposits earn additional money over time.
That is why the required monthly deposit decreases from approximately $83.33 to $81.75.
One Final Note:
This solution assumes deposits are made at the end of each month, which is called an ordinary annuity.
If deposits are made at the beginning of each month, it is called an annuity due. In that case, each deposit earns one additional month of interest, so the required monthly deposit would be slightly smaller.
For a standard future value of an ordinary annuity problem, $81.75 per month is the correct answer.
