To find the value of the account after 4 more years, we first need to calculate the value of the account after the initial 4 years at an interest rate of 8% compounded semiannually.
The formula to calculate the future value (FV) of an account with compound interest is:
FV = P * (1 + r/n)^(n*t)
Where:
FV = future value
P = principal amount (initial deposit)
r = annual interest rate (as a decimal)
n = number of times the interest is compounded per year
t = number of years
For the first 4 years at an 8% interest rate compounded semiannually:
P = $2,000
r = 8% = 0.08
n = 2 (semiannually compounded)
t = 4
Plugging these values into the formula, we get:
FV = $2,000 * (1 + 0.08/2)^(2*4)
= $2,000 * (1 + 0.04)^8
= $2,000 * (1.04)^8
β $2,000 * 1.36048928
β $2,720.98
So, after the initial 4 years, the value of the account will be approximately $2,720.98.
Next, we need to calculate the value of the account for the next 4 years, but with the interest rate increased to 8.28% compounded quarterly. We will use the same formula, but with the new values.
For the next 4 years at an 8.28% interest rate compounded quarterly:
P = $2,720.98 (value after the initial 4 years)
r = 8.28% = 0.0828
n = 4 (quarterly compounded)
t = 4
Plugging these values into the formula, we get:
FV = $2,720.98 * (1 + 0.0828/4)^(4*4)
= $2,720.98 * (1 + 0.0207)^16
β $2,720.98 * (1.0207)^16
β $2,720.98 * 1.3732372
β $3,740.44
Therefore, the value of the account after 4 more years will be approximately $3,740.44.