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 (value of the account)
P = the principal amount (initial deposit)
r = the annual interest rate (as a decimal)
n = the number of times interest is compounded per year
t = the number of years
For Bank A:
Principal amount (P) = $12,000
Annual interest rate (r) = 3% or 0.03 (as a decimal)
Number of times interest is compounded per year (n) = 2 (compounded semiannually)
Number of years (t) = 1
Using the formula:
A = 12000(1 + 0.03/2)^(2*1)
A = 12000(1.015)^2
A = 12000(1.030225)
A = $12,363.00 (rounded to the nearest cent)
For Bank B:
Principal amount (P) = $12,000
Annual interest rate (r) = 2.75% or 0.0275 (as a decimal)
Number of times interest is compounded per year (n) = infinity (compounded continuously)
Number of years (t) = 1
Using the formula for continuous compound interest:
A = Pe^(rt)
Where e is a mathematical constant approximately equal to 2.71828.
A = 12000 * e^(0.0275 * 1)
A = 12000 * e^(0.0275)
A = 12000 * 1.027941524
A = $12,335.30 (rounded to the nearest cent)
Based on the calculations, the value of the account at the end of the year would be higher in Bank A, which is $12,363.00 compared to the value of Bank B, which is $12,335.30. Therefore, I would recommend choosing Bank A for higher returns.