You deposit $2000
in an account that pays 7
%
interest compounded semiannually. After 2
years, the interest rate is increased to 7.32
%
compounded quarterly. What will be the value of the account after a total of 4
years?
amount after first 2 years = 2000(1.035)^4
then for the next 2 years we have i = .0732/4 = .0183
and n = 8 , number of quarters in 2 years
final amount = 2000(1.035)^4 * (1.0183)^8
= .....
You deposit2000 $ in an account that pays6 % interest compounded semiannually. After years, the interest rate is increased to6.84 % compounded quarterly. What will be the value of the account after a total of 2years?
To calculate the value of the account after 4 years, we need to calculate the future value of the initial deposit at an interest rate of 7% compounded semiannually for the first 2 years, and then at an interest rate of 7.32% compounded quarterly for the next 2 years.
Step 1: Calculate the future value after the first 2 years.
The formula to calculate the future value is: FV = P(1 + r/n)^(nt), where:
FV = future value
P = principal (initial deposit)
r = interest rate
n = number of compounding periods per year
t = number of years
For the first 2 years, the interest rate is 7% compounded semiannually.
P = $2000
r = 7% or 0.07
n = 2 (semiannual compounding)
t = 2 years
FV = 2000(1 + 0.07/2)^(2*2)
= 2000(1 + 0.035)^4
= 2000(1.035)^4
≈ $2285.96
After 2 years, the value of the account will be approximately $2285.96.
Step 2: Calculate the future value after the next 2 years.
The interest rate is now 7.32% compounded quarterly.
P = $2285.96
r = 7.32% or 0.0732
n = 4 (quarterly compounding)
t = 2 years
FV = 2285.96(1 + 0.0732/4)^(4*2)
≈ 2285.96(1.0183)^8
≈ $2567.99
After 4 years, the value of the account will be approximately $2567.99.
To calculate the value of the account after 4 years, we need to calculate the value of the initial deposit at the end of each compounding period and then sum them up.
First, let's calculate the value of the account after 2 years with a 7% interest rate compounded semiannually. We can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = The future value of the investment
P = The principal amount (initial deposit)
r = The annual interest rate (in decimal form)
n = The number of times the interest is compounded in a year
t = The number of years
For the first 2 years, n = 2 (semiannually compounded), r = 7% (0.07), and t = 2:
A1 = 2000(1 + 0.07/2)^(2*2)
= 2000(1 + 0.035)^4
= 2000(1.035)^4
≈ $2245.76
The value of the account after 2 years with a 7% interest rate compounded semiannually is approximately $2245.76.
Next, let's calculate the value of the account after another 2 years with a 7.32% interest rate compounded quarterly. Again, using the compound interest formula:
For the next 2 years, n = 4 (quarterly compounded), r = 7.32% (0.0732), and t = 2:
A2 = 2245.76(1 + 0.0732/4)^(4*2)
= 2245.76(1 + 0.0183)^8
= 2245.76(1.0183)^8
≈ $2538.29
The value of the account after another 2 years with a 7.32% interest rate compounded quarterly is approximately $2538.29.
Finally, we add the values of the account at the end of each period:
Total value after 4 years = A1 + A2
≈ 2245.76 + 2538.29
≈ $4784.05
The value of the account after a total of 4 years is approximately $4784.05.