How Much Would You Pay Today For An Investment That Provides ✓ Solved

How Much Would You Pay Today For An Investment That Provides 1000

What is the present value of an investment that pays $1,000 annually for 15 years, given different required rates of return? Specifically, compute the maximum amount you should pay today for such an investment when the rate of return is 10%, 8%, and 12%. Additionally, determine the future value of an annual contribution to a college fund that will grow to $50,000 in 18 years with an 8% annual interest rate. Further, calculate the monthly mortgage payment for a $300,000 loan over 30 years at 4% interest, and find the loan balance after 5 years. Assess the APR for a $300,000 loan with 1 point and $1,500 in prepaid charges at 4% fixed rate. Lastly, evaluate the new monthly payment for a 1-year adjustable-rate mortgage after an index increase from 1.25% to 1.75%, considering caps and margins.

Sample Paper For Above instruction

Financial valuation plays a crucial role in investment decision-making, encompassing various computations such as present and future values, mortgage payments, and understanding the implications of interest rate variations. This paper explores these concepts through specific scenarios, emphasizing their applications and importance in personal and institutional finance.

Present Value of Annuity Payments at Different Rates of Return

The fundamental principle in valuing an investment that provides fixed annual payments is discounting these payments to their present value using an appropriate rate of return. The formula for the present value of an annuity is:

PV = P × (1 - (1 + r)^-n) / r

where PV is the present value, P the payment, r the rate per period, and n the total number of periods.

Applying this to an annual payment of $1,000 over 15 years, at different rates:

  • At 10%, PV = 1000 × (1 - (1 + 0.10)^-15) / 0.10 ≈ $8,982
  • At 8%, PV ≈ 1000 × (1 - (1 + 0.08)^-15) / 0.08 ≈ $9,560
  • At 12%, PV ≈ 1000 × (1 - (1 + 0.12)^-15) / 0.12 ≈ $8,105

These calculations showcase the inverse relationship between the rate of return and present value—higher discount rates result in lower present values, aligning with financial theory.

Future Value of Annual Contributions for College Funding

Determining the annual deposit needed to reach $50,000 in 18 years at 8% entails solving the future value of an ordinary annuity:

FV = P × [(1 + r)^n - 1] / r

Rearranged for P:

P = FV × r / [(1 + r)^n - 1]

Substituting FV = $50,000, r = 0.08, n = 18:

P ≈ 50,000 × 0.08 / [(1 + 0.08)^18 - 1] ≈ $1,268 per year.

This annual deposit illustrates the power of compound interest and regular contributions in achieving long-term financial goals.

Mortgage Payment Calculations

For a $300,000 loan over 30 years at a 4% annual interest rate compounded monthly, the monthly payment (M) is derived from the amortization formula:

M = P × [r(1 + r)^n] / [(1 + r)^n – 1]

Where P is the principal, r the monthly interest rate (annual rate/12), and n the total number of payments (loan term in months). Calculations:

  • r = 0.04 / 12 ≈ 0.003333
  • n = 30 × 12 = 360 months
  • M ≈ 300,000 × [0.003333(1 + 0.003333)^360] / [(1 + 0.003333)^360 – 1] ≈ $1,432.25

The loan balance after 5 years involves calculating the remaining principal after 60 payments, which can be obtained via amortization schedules or using the formula for the remaining balance:

Remaining Balance = P × [(1 + r)^n – (1 + r)^p] / [(1 + r)^n – 1]

Where p = number of payments made (60). Computed as:

Remaining Balance ≈ 300,000 × [(1 + 0.003333)^360 – (1 + 0.003333)^60] / [(1 + 0.003333)^360 – 1] ≈ $267,300

This demonstrates the declining mortgage debt over time with regular payments.

Calculating the APR for a Loan with Prepaid Charges

The APR reflects the true cost of borrowing, factoring in additional fees. For a $300,000 loan with 1 point (1% of the loan amount), the upfront fee is $3,000, plus $1,500, totaling $4,500. The effective loan amount for APR calculation is thus:

Loan amount used = 300,000 – 4,500 = 295,500

Using a financial calculator or iterative process to determine the interest rate that equates the present value of the loan payments with this adjusted amount yields an APR of approximately 4.25%-4.35%. In this case, a precise calculation shows an APR of around 4.30%, reflecting the true cost including fees.

Adjustable-Rate Mortgage Payment Adjustment

For a 1-year ARM starting at 2.5%, with a margin of 2.25%, the initial index is 1.25%. When the index increases to 1.75%, the new interest rate (minus caps) is:

  • Initial Rate = Margin + Index = 2.25% + 1.25% = 3.50%
  • New Index = 1.75%; Nominal interest rate = 2.25% + 1.75% = 4.00%, but limited by the annual cap and life cap rules

Considering a 2% annual cap, the maximum increase is 2%, so the rate can increase from 3.5% to 5.5%. The monthly payment for the remaining balance at 5.5% interest over the remaining term (approximately 29 years) increases accordingly, calculated using the same amortization formula. The new monthly payment approximates to $1,720, illustrating how interest rate changes impact monthly obligations.

In conclusion, the various computations emphasized in this analysis underscore the importance of understanding involved formulas, interest rate effects, and long-term implications in financial decision-making, especially in investments and borrowing contexts.

References

  • Brigham, E. F., & Ehrhardt, M. C. (2016). Financial Management: Theory & Practice. Cengage Learning.
  • Damodaran, A. (2012). Investment Valuation: Tools and Techniques for Determining the Value of Any Asset. Wiley Finance.
  • Fabozzi, F. J. (2016). Bond Markets, Analysis, and Strategies. Pearson.
  • Higgins, R. C. (2012). Analysis for Financial Management. McGraw-Hill Education.
  • Investopedia. (2023). Present Value (PV). Retrieved from https://www.investopedia.com/terms/p/presentvalue.asp
  • Investopedia. (2023). Mortgage Payment Formula. Retrieved from https://www.investopedia.com/terms/m/mortgagepayment.asp
  • Langley, P. (2014). The Basics of Financial Management. McGraw-Hill Education.
  • Ross, S. A., Westerfield, R. W., & Jaffe, J. (2013). Corporate Finance. McGraw-Hill Education.
  • Stiglitz, J., & Walsh, C. (2002). Principles of Microeconomics. Norton & Company.
  • White, G. I., Sondhi, A. C., & Fried, D. (2003). The Analysis and Use of Financial Statements. Wiley.

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