To determine the interest rate that Bank B must pay for Derek to have the same amount in both Bank A and Bank B after 10 years, we will first calculate the future value of each account after the specified period.
Future Value of Bank A's Account
Derek deposits $228.00 every month into Bank A, which pays 13.00% interest compounded monthly. To find the future value (FV) of this annuity after 10 years, we can use the formula for the future value of an ordinary annuity:
\[ FV = P \times \left( \frac{(1 + r)^n - 1}{r} \right) \]
Where:
- \( P \) is the monthly deposit ($228.00)
- \( r \) is the monthly interest rate (annual rate / number of compounding periods)
- \( n \) is the total number of deposits (number of years × number of compounding periods per year)
Given:
- Annual interest rate = 13% → Monthly interest rate \( r = \frac{0.13}{12} \approx 0.01083333 \)
- Number of deposits: \( n = 10 \times 12 = 120 \)
Now substituting the values into the formula:
\[ FV = 228 \times \left( \frac{(1 + 0.01083333)^{120} - 1}{0.01083333} \right) \]
Calculating \( (1 + 0.01083333)^{120} \):
\[ (1 + 0.01083333)^{120} \approx 4.122286 \]
Now, substituting this into the FV formula:
\[ FV = 228 \times \left( \frac{4.122286 - 1}{0.01083333} \right) \] \[ FV \approx 228 \times \left( \frac{3.122286}{0.01083333} \right) \approx 228 \times 288.2831 \approx 65,754.65 \]
Future Value of Bank B's Account
For Bank B, Derek will deposit $2,471.00 annually, and the interest compounds annually. The future value of an annuity due (where deposits are made at the end of the year) is given by similar formulas:
\[ FV = P \times \left( \frac{(1 + r)^n - 1}{r} \right) \]
Where:
- \( P = 2,471.00 \)
- \( r \) = annual interest rate we need to find
- \( n = 10 \) (since he deposits once a year for 10 years)
Setting the future value of Bank B equal to that of Bank A:
\[ 65,754.65 = 2,471 \times \left( \frac{(1 + r)^{10} - 1}{r} \right) \]
We multiply both sides by \( r \):
\[ 65,754.65 \cdot r = 2,471 \cdot ((1 + r)^{10} - 1) \]
Rearranging gives:
\[ (1 + r)^{10} = 1 + \frac{65,754.65 \cdot r}{2,471} \]
Let \( X = (1 + r)^{10} \):
This can be solved using numerical methods or financial calculators. Rearranging into:
\[ X - 65,754.65/2,471 - 1 + r = 0 \]
Solving this numerically yields:
After calculating various values of \( r \), we find that approximately:
\[\text{Annual Rate } r \approx 11.66% \]
Conclusion
Derek would need to invest at a rate of approximately 11.66% in Bank B for the amounts from both Bank A and Bank B to be equal after 10 years.
1 answer