Kate is thinking about investing $60 000 for 4 years. She deposits her money into an account which earns interest paid semiannually at a rate of 7% p.a. After 2½ years, the interest rate drops to 5.6% p.a. and stays constant for the remainder of the investment period.
Use Excel or another suitable method to solve the problems below.
(a) How much interest was accrued in the second year of the investment?
(b) What will be the balance of Kate’s account at the end of the fourth year?
amount after the first 2½ years
= 60000(1.035)^7
= 76336.756
I will assume that the rate for the remaining 1½ of 5.6% is also compounded semiannually
amount at end of 4 years
= 76336.756(1.028)^3
= 82930.26
Don't know how I got 7 as the exponent in the first part
of course it should have been 5 ( for 5 half years in 2 1/2 years)
Please make the necessary corrections.
A woman has a total of $9,000 to invest. She invests part of the money in an account that pays 8% per year and the rest in an account that pays 11% per year. If the interest earned in the first year is $840, how much did she invest in each account?
To solve these problems, we need to use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment
P = the principal amount (initial deposit)
r = annual interest rate (expressed as a decimal)
n = number of times the interest is compounded per year
t = number of years
In this case, the interest is compounded semiannually, so n = 2. The interest rates change after 2½ years, so we need to calculate the interest for different periods.
(a) To find the interest accrued in the second year of the investment, we need to find the difference between the future value of the investment after two years and the future value after one year.
1. Calculate the future value after two years:
P = $60,000
r = 7% p.a. -> 0.07
n = 2
t = 2
Using the compound interest formula:
A2 = 60000(1 + 0.07/2)^(2*2)
= 60000(1.035)^4
≈ $70,381.37
2. Calculate the future value after one year:
P = $60,000
r = 7% p.a. -> 0.07
n = 2
t = 1
Using the compound interest formula:
A1 = 60000(1 + 0.07/2)^(2*1)
= 60000(1.035)^2
≈ $64,218.75
3. Calculate the interest accrued in the second year:
Interest = A2 - A1
= $70,381.37 - $64,218.75
≈ $6,162.62
Therefore, the interest accrued in the second year of the investment is approximately $6,162.62.
(b) To find the balance of Kate's account at the end of the fourth year, we need to use a similar calculation.
1. Calculate the future value after four years:
P = $60,000
r = 7% p.a. -> 0.07 (for the first 2½ years)
r = 5.6% p.a. -> 0.056 (for the remaining 1½ years)
n = 2
t = 4
Using the compound interest formula:
A4 = 60000(1 + 0.07/2)^(2*2.5) * (1 + 0.056/2)^(2*1.5)
≈ $76,092.66
Therefore, the balance of Kate's account at the end of the fourth year will be approximately $76,092.66.