You Wish To Accumulate 25,000 In 5 Years Time To Do So You P

you Wish To Accumulate 25 000 In 5 Years Time To Do So You Plan T

1. You wish to accumulate $25,000 in 5 years. You plan to make semi-annual deposits, with the first deposit made today. If you can earn 6% interest compounded semi-annually, determine the amount needed per deposit.

2. Gary deposits $300 at the end of each month into an account earning 6% interest compounded monthly. He makes these deposits for 10 years, except that in the fifth year, he was unable to make deposits. Find the value of the account two years after the last deposit.

3. What sum of money should be set aside to provide an income of $800 per month for 3 years, if the money earns interest at 6% compounded monthly, and the first payment is received 2 years from now?

4. Find the monthly deposit needed over 5 years to fund a perpetuity of $400 per month, starting 2 years after the last deposit. The interest rate changes from 8% compounded monthly to 9% compounded monthly on that date.

5. A company borrows $100,000 at 9% interest compounded monthly. The loan is to be repaid with monthly payments of $1550 until fully paid. Answer the following:

• What is the final payment amount that fully repays the loan?
• What is the outstanding balance after 48 payments?
• What is the interest paid in the 49th payment of $1550?
• What is the principal portion of the 49th payment?
• What is the total cost of debt over the first four years?
• Construct a partial amortization schedule showing details for the last three payments and totals.

Paper For Above instruction

Financial mathematics plays a vital role in personal and corporate financial planning, helping individuals and organizations make informed decisions about saving, investing, borrowing, and debt management. The problems outlined above showcase various fundamental concepts such as future value of annuities, present value calculations, loan amortization, and perpetuity funding, all of which rely heavily on the principles of compound interest and time value of money. This essay explores each of these financial scenarios, illustrating the application of key formulas and concepts in real-world contexts, supported by academic sources and standard financial practices.

Accumulating a Target Sum through Semi-Annual Deposits

The first scenario involves determining the semi-annual deposit needed to accumulate $25,000 in five years, given a 6% interest rate compounded semi-annually. Since the first deposit is made today, this can be viewed as a combination of an immediate present deposit and subsequent regular deposits. Using the future value of an annuity due, which accounts for payments made at the beginning of each period, and the present value of a lump sum, the calculation requires understanding the formulas:

FV = P * [(1 + r)^nt - 1] / r + P (because of the immediate deposit), where P is the deposit amount, r is the interest rate per period, n is the number of periods per year, and t is the total years. Solving for P involves algebraic manipulations, ensuring the deposits grow sufficiently to meet the future goal of $25,000.

Monthly Deposits and Future Value with Irregular Contributions

Gary’s situation introduces complexities with irregular deposit schedules—monthly contributions for ten years with a pause in the fifth year. The problem employs the future value of an ordinary annuity for the first four years and the last five years (excluding the paused year), plus the accumulation of deposits made before the pause. Calculating the value two years after the last deposit involves summing the accumulated value of deposits and the interest accrued during the post-deposit period using compound interest formulas.

The formulas involve the calculation of each deposit's future value at the end of the period, considering their timing and the interest rate. This problem underscores the importance of understanding how irregular deposits affect the total accumulated amount and demonstrates the use of the future value of multiple annuities with varying periods.

Funding Future Income via Present Value Calculations

To determine the lump sum needed now to generate an $800 monthly income for three years, starting two years from now, involves the present value of an ordinary annuity. The calculation considers the delay in start time (two years) and the monthly interest rate derived from 6% annual interest compounded monthly. Discounting the future payments to the present gives the required initial sum, which can be computed using the present value of annuity formula:

PV = Payment * [1 - (1 + i)^(-n)] / i, where i is the monthly interest rate and n is total number of payments.

Perpetuity Funding with Changing Interest Rates

The problem regarding funding a perpetuity of $400 per month with changing interest rates involves understanding the concept of perpetuity and the effect of changing discount rates. Since the interest rate changes from 8% to 9% after two years, the funding strategy must account for the different discount factors over the periods. Computing the monthly deposit over five years requires solving the present value of a perpetuity adjusted for the interest rate change, essentially blending two different discounting periods and interest rates, using the formulas for perpetuities and annuities with variable rates.

Loan Amortization and Final Payments

Amortizing a loan of $100,000 at 9% interest compounded monthly with monthly payments of $1550 entails calculating the amortization schedule, including the outstanding balance after a specified number of payments, the interest component in a specific payment, and the principal paid. The outstanding balance calculation involves subtracting the principal portion from the remaining balance after each payment and considering interest accrued each month. The final payment is typically smaller, covering any remaining balance after the scheduled payments. The total interest paid over the initial period, as well as the detailed schedule, provides insights into the cost of borrowing and repayment structure.

Constructing a partial amortization schedule requires meticulous calculation for each payment's interest and principal portions, summarized for the last three payments, offering a clear view of how debt diminishes over time and the effects of amortization on overall loan costs.

Conclusion

The breadth of these financial problems underscores the importance of mastering the principles of compound interest, annuities, present and future value calculations, and amortization schedules. These tools enable effective financial planning, allowing both individuals and corporations to optimize savings, manage debt, and plan sustainable income streams. The integration of changing interest rates, irregular contributions, and complex payment structures demonstrates the adaptability and depth of financial mathematics, essential for sound financial decision-making in real-world scenarios.

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