You deposit $4000 in an account that pays 8 % interest compounded semiannually. After 2 years, the interest rate is increased to 8.48 % compounded quarterly. What will be the value of the account after a total of 4 years?
4000 * (1 + .08/2)^(2*2) * (1 + .0848/4)^(4*2)
To solve this problem, we need to calculate the value of the account after each interest rate change and then add them up to get the total value after 4 years.
Step 1: Calculate the value of the account after 2 years with an interest rate of 8% compounded semiannually.
To calculate this, we can use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the account
P = the initial deposit
r = the interest rate (as a decimal)
n = the number of times interest is compounded per year
t = the number of years
In this case, P = $4000, r = 8% = 0.08, n = 2 (semiannually), and t = 2 years.
Plugging in these values, we get:
A = 4000(1 + 0.08/2)^(2*2)
A = 4000(1 + 0.04)^4
A = 4000(1.04)^4
A ≈ $4413.02
So, after 2 years, the value of the account is approximately $4413.02.
Step 2: Calculate the value of the account after another 2 years with an interest rate of 8.48% compounded quarterly.
Using the same formula, but with the new interest rate (8.48% = 0.0848) and the new compounding period (quarterly = 4 times per year), we get:
A = 4413.02(1 + 0.0848/4)^(4*2)
A = 4413.02(1 + 0.0212)^8
A = 4413.02(1.0212)^8
A ≈ $5216.60
So, after a total of 4 years, the value of the account will be approximately $5216.60.
To find the value of the account after 4 years, we need to calculate the future value of the initial deposit of $4000 with the two different interest rates: 8% compounded semiannually for the first 2 years, and 8.48% compounded quarterly for the remaining 2 years.
Let's break down the calculation into two steps:
Step 1: Calculate the value after 2 years with 8% interest compounded semiannually.
First, we need to determine the number of compounding periods (n) and the interest rate per compounding period (r):
n = number of years x compounding frequency
n = 2 years x 2 compounding periods per year
n = 4 compounding periods
r = annual interest rate / compounding frequency
r = 8% / 2
r = 4% per compounding period
Now, we can use the formula for the future value of a compound interest calculation:
Future Value = Initial Deposit x (1 + r)^n
Future Value = $4000 x (1 + 4%)^4
Future Value = $4000 x (1 + 0.04)^4
Future Value = $4000 x (1.04)^4
Future Value = $4000 x 1.169858
Future Value = $4679.43 (rounded to the nearest cent)
Step 2: Calculate the value after the remaining 2 years with 8.48% interest compounded quarterly.
Now, we need to determine the number of compounding periods and the interest rate per compounding period for the remaining 2 years:
n = number of years x compounding frequency
n = 2 years x 4 compounding periods per year
n = 8 compounding periods
r = annual interest rate / compounding frequency
r = 8.48% / 4
r = 2.12% per compounding period
Using the same formula as before, we can calculate the future value:
Future Value = $4679.43 x (1 + 2.12%)^8
Future Value = $4679.43 x (1 + 0.0212)^8
Future Value = $4679.43 x (1.0212)^8
Future Value = $4679.43 x 1.178625
Future Value = $5514.18 (rounded to the nearest cent)
Therefore, the value of the account after a total of 4 years will be approximately $5,514.18.