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?
P = Po(1+r)^n.
b. r = (7%/2) / 100% = 0.035 = Semi-annual
% rate expressed as a decimal.
n = 2Comp./yr * 2.5yrs = 5 Compounding
periods.
P = 60,000(1.035)^5 = $71,261.18 = Bal.
after 2.5 yrs.
P = 71,261.18(1.028)^3 = $77,367.40 =
Bal. after 4 yrs.
To solve this problem, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A is the final amount
P is the principal amount (initial investment)
r is the annual interest rate (in decimal form)
n is the number of times that interest is compounded per year
t is the number of years
Since the interest is paid semiannually, n will be 2.
To find the interest accrued in the second year, we need to calculate the balance at the end of the second year and subtract the initial investment. Here's how to calculate each part:
(a) How much interest was accrued in the second year of the investment?
Step 1: Calculate the balance at the end of the second year.
- Calculate the interest earned during the first year using the formula:
Interest_1 = P(1 + r/n)^(nt) - P
- Calculate the investment balance at the end of the first year:
Balance_1 = P + Interest_1
- Now, calculate the interest earned during the second year:
Interest_2 = Balance_1(1 + r/n)^(nt) - Balance_1
(b) What will be the balance of Kate’s account at the end of the fourth year?
Step 2: Calculate the balance at the end of the fourth year.
- Calculate the interest earned during the third year using the formula:
Interest_3 = Balance_2(1 + r/n)^(nt) - Balance_2
- Calculate the investment balance at the end of the third year:
Balance_3 = Balance_2 + Interest_3
- Calculate the interest earned during the fourth year using the formula:
Interest_4 = Balance_3(1 + r/n)^(nt) - Balance_3
- Calculate the balance at the end of the fourth year:
Balance_4 = Balance_3 + Interest_4
Use the given values:
P = $60,000
r = 7% p.a. for the first 2½ years and 5.6% p.a. for the remaining time
n = 2 (interest is paid semiannually)
Plug these values into the formulas and calculate the answers using Excel or any suitable method.