You deposit $1000 in an account that pays 8% interest compounded semiannually. After 2 years, the interest rate is increased to 8.40% compounded quarterly. What will be the value of the account after 4 years?
Pt = Po(1+r)^n.
r = Rate per compounding.
n = The # of compounding periods.
2 yrs @ 8% + 2 yrs @ 8.4%.
Pt = !000(1.04)^4 + Po2(1.021)^8,
Pt = 1169.8586 + 1169.8586(1.021)^8,
Pt = 1169.8586 + 1381.4631 = $2551.32.
To find the value of the account after 4 years, we need to calculate the compound interest for the first 2 years at an interest rate of 8% compounded semiannually, and then calculate the compound interest for the remaining 2 years at an interest rate of 8.40% compounded quarterly.
Step 1: Calculate the interest for the first 2 years compounded semiannually.
The formula for compound interest is given by:
A = P(1 + r/n)^(nt)
Where:
A = final amount
P = principal amount (initial deposit)
r = annual interest rate (in decimal form)
n = number of compounding periods per year
t = number of years
For the first 2 years compounded semiannually:
P = $1000
r = 8% = 0.08
n = 2 (compounded semiannually)
t = 2
Using the formula:
A1 = P(1 + r/n)^(nt)
A1 = $1000(1 + 0.08/2)^(2*2)
A1 = $1000(1 + 0.04)^4
A1 = $1000(1.04)^4
A1 ≈ $1162.43
Step 2: Calculate the interest for the remaining 2 years compounded quarterly.
For the remaining 2 years compounded quarterly:
P = A1 (the value after 2 years)
r = 8.40% = 0.084
n = 4 (compounded quarterly)
t = 2
Using the formula:
A2 = P(1 + r/n)^(nt)
A2 = $1162.43(1 + 0.084/4)^(4*2)
A2 = $1162.43(1 + 0.021)^8
A2 = $1162.43(1.021)^8
A2 ≈ $1339.86
Step 3: Calculate the total value of the account after 4 years.
The final value of the account after 4 years is the sum of the two previous amounts:
Total value = A1 + A2
Total value ≈ $1162.43 + $1339.86
Total value ≈ $2502.29
Therefore, the value of the account after 4 years is approximately $2502.29.
To find the value of the account after 4 years, we need to calculate the future value of the initial deposit of $1000 using the interest rates and compounding periods for the first 2 years and the subsequent 2 years.
Let's break down the problem into two parts:
1. First, we need to calculate the future value of the initial $1000 deposit after 2 years using the interest rate of 8% compounded semiannually. The formula to calculate the future value of an investment with compound interest is:
FV = P(1 + r/n)^(nt)
Where:
FV = future value
P = principal (initial deposit)
r = interest rate per compounding period
n = number of compounding periods per year
t = number of years
Plugging in the given values, we have:
FV1 = $1000(1 + 0.08/2)^(2*2)
= $1000(1 + 0.04)^4
= $1000(1.04)^4
≈ $1162.44
Therefore, after 2 years, the account will have a value of approximately $1162.44.
2. Next, we need to calculate the future value of $1162.44 after an additional 2 years using the interest rate of 8.4% compounded quarterly. Using the same formula, we have:
FV2 = $1162.44(1 + 0.084/4)^(4*2)
= $1162.44(1 + 0.021)^8
= $1162.44(1.021)^8
≈ $1383.67
Therefore, after 4 years, the account will have a value of approximately $1383.67.