Question

Problem 6 (10 points) Susan starts a saving account today and deposit $1000 at the beginning of every month. The saving account earns compounded interest continuously at an annual effective interest rate 6.37%. At the end of every year, Susan withdraws all interest earned from the saving account (while keeping the deposits in the saving account) into another investment fund which earns interest at an annual nominal interest rate 7.2% convertible monthly. Calculate the total amount of interest earned from these two accounts during the first 6 years.

Answer 1

To calculate the total interest earned over six years from a savings account with 6.37% continuous compounding and an investment fund with 7.2% nominal interest rate compounded monthly, one must calculate the interest from the savings account annually and add it to the fund, then compute the compounded interest for each annual contribution.

Step-by-step explanation:

To solve the problem of calculating the total amount of interest earned from two different investment accounts over the first six years, we need to consider the continuous compounding of interest for the savings account and the monthly compounding for the investment fund.

For the savings account with a continuous compounding interest rate of 6.37%, we use the formula for continuous compounding which is A = Pert, where A is the amount of money accumulated after time t, P is the principal amount, r is the annual interest rate, and e is Euler's number (approximately 2.71828). Given the nature of the deposits and withdrawals, we would need to calculate the interest for each deposit at the end of every year individually and sum them up.

For the second investment fund, which uses a nominal rate of 7.2% compounded monthly, the formula for compound interest is A = P(1 + r/n)nt, where A is the future value of the investment, P is the principal investment amount, r is the annual nominal interest rate, n is the number of times that interest is compounded per year, and t is the number of years the money is invested for.

To determine how much interest has been earned from both accounts combined over the six-year period, we would need to track the yearly interest withdrawals from the savings account and add them to the investment fund, then calculate the compound interest for each year's addition to this fund. The final total will represent the sum of interest earned in both accounts.

Answer 2

To calculate the total amount of interest earned from both accounts during the first 6 years, we need to consider the interest earned in the saving account and the interest earned in the investment fund. In the saving account, Susan makes deposits of $1000 at the beginning of every month. The interest is compounded continuously at an annual effective interest rate of 6.37%. In the investment fund, Susan withdraws all interest earned from the saving account at the end of each year, and it earns interest at an annual nominal interest rate of 7.2% compounded monthly.

Step-by-step explanation:

To calculate the total amount of interest earned from both accounts during the first 6 years, we need to break down the problem into two parts: interest earned in the saving account and interest earned in the investment fund.

Interest earned in the saving account:

1. Since Susan deposits $1000 at the beginning of every month, she will have made a total of 6 years x 12 months = 72 deposits.
2. To find the future value of these deposits, we can use the formula for compound interest: A = P * e^(rt), where A is the future value, P is the principal (the initial deposit), e is the mathematical constant approximately equal to 2.71828, r is the interest rate, and t is the time in years. In this case, P = $1000, r = 6.37% = 0.0637, and t = 6 years.
3. Using the formula, A = $1000 * e^(0.0637 * 6), we can calculate the future value of the deposits in the saving account.

Interest earned in the investment fund:

1. After each year, Susan withdraws all the interest earned from the saving account into another investment fund. This means that the initial deposit in the investment fund is equal to the interest earned in the previous year.
2. We can calculate the interest earned in each year using the formula for compound interest: A = P(1 + r/n)^(nt), where A is the future value, P is the principal (the initial deposit), r is the interest rate, n is the number of times interest is compounded per year, and t is the time in years. In this case, P = the interest earned in the previous year, r = 7.2% = 0.072, n = 12 (compounded monthly), and t = 1 year.
3. Using the formula, we can calculate the interest earned in each year and sum them up to find the total interest earned in the investment fund over 6 years.

Finally, we can add the total interest earned in the saving account and the investment fund to find the total amount of interest earned from these two accounts during the first 6 years.