You receive $12,000 and looking for a bank to deposit the funds. Bank A offers an account with an annual interest rate of 3% compounded semiannually. Bank B offers an account with 2.75% annual interest rate compounded continuously. Calculate the value of the two accounts at the end of the year and recommend.
To calculate the value of the two accounts at the end of the year, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the final amount after interest
P = the principal amount (initial deposit)
r = the annual interest rate (expressed as a decimal)
n = the number of times the interest is compounded per year
t = the number of years
For Bank A:
Principal amount (P) = $12,000
Annual interest rate (r) = 3% = 0.03
Compounded semiannually, so n = 2 (twice per year)
Time (t) = 1 year
Using the formula:
A = $12,000(1 + 0.03/2)^(2*1)
A = $12,000(1 + 0.015)^2
A = $12,000(1.015)^2
A ≈ $12,000(1.030225)
A ≈ $12,362.70
For Bank B:
Principal amount (P) = $12,000
Annual interest rate (r) = 2.75% = 0.0275
Compounded continuously, so n → ∞ (infinitely)
Time (t) = 1 year
Using the formula:
A = P * e^(rt)
Where e ≈ 2.71828 (the base of natural logarithm)
A = $12,000 * e^(0.0275*1)
A = $12,000 * e^(0.0275)
A ≈ $12,000 * 1.028143
A ≈ $12,337.72
Therefore, the value of the account at Bank A after 1 year would be approximately $12,362.70, and the value of the account at Bank B would be approximately $12,337.72.
Based on these calculations, I would recommend choosing Bank A as the value of the account is slightly higher. However, keep in mind that this decision should also consider other factors such as the bank's reputation, fees, customer service, and any additional benefits offered.