To solve this problem, we need to calculate the monthly deposit required to reach a future value of $11,211 in two years with an annual interest rate of 1.3% compounded monthly.

We will use the future value of an ordinary annuity formula:

\[FV = P \times \frac { ( 1 + r) ^ n - 1} { r} \]

Where:

- \(FV\) is the future value, which is $11,211.

- \(P\) is the monthly deposit.

- \(r\) is the monthly interest rate.

- \(n\) is the total number of deposits (compounding periods).

Given:

- Annual interest rate \(= 1.3\% \)

- Monthly interest rate \(r = \frac { 1.3\% } { 12} = \frac { 0.013} { 12} \approx 0.00108333\)

- Total number of months \(n = 2 \text { years} \times 12 \text { months/year} = 24 \text { months} \)

Let's solve for \(P\):

\[11211 = P \times \frac { ( 1 + 0.00108333) ^ { 24} - 1} { 0.00108333} \]

First, we'll calculate the value of \(\frac { ( 1 + 0.00108333) ^ { 24} - 1} { 0.00108333} \) using Python, and then solve for \(P\). Let's do the calculations.

The amount that should be deposited each month to reach $11,211 in two years, assuming an annual interest rate of 1.3% compounded monthly, is $461.33.

Additional Knowledge Points

1. Annuities:

An annuity is a series of equal payments made at regular intervals. In this problem, we are dealing with an ordinary annuity, where payments are made at the end of each period. The future value formula \(FV = P \times \frac{(1 + r)^n - 1}{r}\) is used to calculate the future value of an annuity given a specific interest rate and time period.

2. Compound Interest:

Compound interest refers to the interest calculated on the initial principal, which also includes all of the accumulated interest from previous periods on a deposit or loan. In this problem, we use monthly compounding, meaning interest is added to the principal each month, generating interest in the following month.

3. Time Value of Money:

The time value of money is a core concept in financial mathematics, which indicates that a specific amount of money has different values at different points in time. Calculating the future value of an annuity considers the time value of money, as future amounts are affected by interest.

4. Interest Rate Conversion:

Converting between different interest rates, such as annual to monthly rates, is a common practice in financial mathematics. In this problem, we convert the annual interest rate to a monthly interest rate for the calculation.

5. Calculation Steps:

- Convert the annual interest rate to a monthly interest rate.

- Calculate the total number of compounding periods.

- Use the annuity future value formula.

- Solve for the monthly deposit amount.

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