Find the balance at the end of 6 years of $5000 investment in an account with a nominal annual rate of interest of 2.5% compounded quarterly over the first two years and then grows according to nominal annual rate of discount of 6% convertible monthly over the remaining 4 years.
Please explain this question step by step.
To find the balance at the end of 6 years of a $5000 investment, we need to follow these steps:
Step 1: Calculate the balance after the first two years with quarterly compounding.
- Convert the nominal annual interest rate of 2.5% to a quarterly interest rate. Divide it by 4 (since there are four quarters in a year): 2.5% / 4 = 0.625%.
- Convert the number of years into quarters for the first two years. Since each year has four quarters, 2 years will be 2 * 4 = 8 quarters.
- Use the compound interest formula to calculate the balance after 2 years:
Balance after 2 years = Principal * (1 + Interest rate per period)^Number of periods
Substituting the values: Balance after 2 years = $5000 * (1 + 0.00625)^8.
Step 2: Calculate the balance after the remaining 4 years with monthly discounting.
- Convert the nominal annual discount rate of 6% to a monthly discount rate. Divide it by 12 (since there are twelve months in a year): 6% / 12 = 0.5%.
- Convert the number of years into months for the remaining 4 years. Since each year has twelve months, 4 years will be 4 * 12 = 48 months.
- Use the compound discounting formula to calculate the balance after 4 years:
Balance after 4 years = Principal / (1 - Discount rate per period)^Number of periods
Substituting the values: Balance after 4 years = $5000 / (1 - 0.005)^48.
Step 3: Sum up the balances from Step 1 and Step 2.
- Add the balance after 2 years (from Step 1) to the balance after 4 years (from Step 2).
Total balance after 6 years = Balance after 2 years + Balance after 4 years.
Follow these steps using a calculator or spreadsheet to find the final balance after 6 years of the given investment.