Evaluate Loans, Annuities, And Bonds Using Present And Futur

Evaluate loans, annuities, and bonds using present and future value calculations

This assignment will allow you to demonstrate the following objectives: · Calculate the annual payment on a loan using the present value of an annuity. · Use discounting to determine the present value of an annuity. · Calculate the future value of an annuity and periodic annuity payments. · Determine the present value of a bond.

Answer the questions directly on this document. When finished, save the document as specified and upload it accordingly. Show all work for each problem.

Paper For Above instruction

Financial analysis of loans, annuities, and bonds is essential for making informed decisions in corporate finance and personal investments. This paper evaluates a variety of financial calculations including loan payments, present and future values of annuities, and bond valuation, illustrating practical applications of financial mathematics concepts.

1. Calculating the Annual Payment on a Business Loan

A supervisor has tasked us with evaluating a business loan of $400,000 at an 8% interest rate, amortized over five years. To compute the annual payment, we utilize the Present Value Interest Factor of Annuity (PVIFA), which helps relate the present value of an annuity to its periodic payments.

The PVIFA from Table 9.4 at 8% interest over 5 years is approximately 3.993. The formula to calculate the annual payment (PMT) is:

PMT = Present Value / PVIFA

Substituting the known values:

PMT = $400,000 / 3.993 ≈ $100,124.17

The annual payment on this loan is approximately $100,124.17.

This payment seems reasonable because it reflects a typical amortization payment, ensuring the entire principal and interest are paid off over five years. The calculation demonstrates fair loan structuring considering the interest rate and term, balancing manageable payments and total interest paid over the period.

2. Borrowing for a Condo: Loan Amortization Calculations

Dan plans to borrow $500,000 at 8% interest to purchase a condo. Calculations for monthly payments are as follows:

a) 30-Year Fixed-Rate Mortgage

The monthly interest rate (i) is 8% / 12 = 0.0066667. The number of payments (n) is 30 years * 12 months = 360 months. Using the mortgage formula or financial calculator, the monthly payment (PMT) is:

PMT = P × i / (1 - (1 + i)^-n)

Plugging in the values:

PMT = $500,000 × 0.0066667 / (1 - (1 + 0.0066667)^-360) ≈ $3,668.77

This is the monthly payment needed to amortize the loan over 30 years.

b) 15-Year Fixed-Rate Mortgage

For a 15-year term: n = 15 × 12 = 180 months, with the same monthly interest rate.

PMT = $500,000 × 0.0066667 / (1 - (1 + 0.0066667)^-180) ≈ $4,785.09

Shorter loan periods result in higher monthly payments but less total interest paid over the life of the loan.

c) Effect of a Smaller Loan Period

Reducing the loan term from 30 to 15 years increases the monthly payment, making it more costly monthly but substantially decreasing the total interest paid over the life of the loan. Shorter loans also help borrowers build equity faster and reduce overall debt burden, but require more immediate financial commitment.

3. Present Value Calculations for Retirement Savings

Using financial software or calculator, we evaluate Melanie’s retirement savings goal of $600,000 in 20 years with an 11% discount rate:

a) Present Value for 20 Years

The present value (PV) is calculated as:

PV = FV / (1 + r)^n

PV = $600,000 / (1 + 0.11)^20 ≈ $600,000 / 8.062 = $74,342.86

Melanie needs approximately $74,343 today to reach her goal of $600,000 in 20 years at 11% discount rate.

b) Present Value for 15 Years

PV = $600,000 / (1 + 0.11)^15 ≈ $600,000 / 5.113 ≈ $117,317.25

With a shorter 15-year horizon, the present value increases to approximately $117,317, reflecting the lesser time for discounting and thus higher current value necessary to meet the future goal.

The impact demonstrates that the longer the period, the less current capital is required, but a higher future value needs to be accumulated.

4. Annuities and Investment Growth

Anne plans to invest $400 annually for four years earning 6% interest annually, starting immediately. The future value (FV) of an annuity due considers investments made at the beginning of each period.

a) Future Value of Annuity Due

The formula for FV of an annuity due is:

FV = P × [(1 + r)^n - 1] / r × (1 + r)

Where P = 400, r = 0.06, n = 4

FV = 400 × [(1 + 0.06)^4 - 1] / 0.06 × (1 + 0.06) ≈ 400 × [1.2625 - 1] / 0.06 × 1.06 ≈ 400 × 4.383 ≈ $1,753.20

The future value of Anne’s annuity due is approximately $1,753.20.

b) Difference Between Annuity Due and Ordinary Annuity

An annuity due involves payments made at the beginning of each period, leading to higher future value due to the earlier compounding. An ordinary (or regular) annuity involves payments at the end of each period, resulting in slightly less future value because interest compounds over a shorter period for each payment.

5. Bond Valuation with Semiannual Payments

Jimmy's bond has a face value of $1,000, a 9.5% coupon rate paid semiannually, and a five-year maturity. Investors require a 14% yield.

a) Present Value of Bond

The semiannual coupon payment is:

Coupon = 1,000 × 9.5% / 2 = $47.50

Number of periods: 5 × 2 = 10

Market rate per period: 14% / 2 = 7% = 0.07

The present value (PV) is the sum of the present value of coupon payments and face value:

PV = C × [1 - (1 + r)^-n] / r + Face Value / (1 + r)^n

PV = 47.50 × [1 - (1 + 0.07)^-10] / 0.07 + 1,000 / (1 + 0.07)^10

Calculations:

PV of coupons: 47.50 × [1 - (1.07)^-10] / 0.07 ≈ 47.50 × 7.0236 ≈ $333.75

PV of face value: 1,000 / (1.07)^10 ≈ 1,000 / 1.967 = $508.41

Total PV ≈ $333.75 + $508.41 ≈ $842.16

The bond's present value is approximately $842.16.

b) Impact of Semiannual Payments

Paying interest semiannually increases the frequency of compounding, which generally leads to a slightly higher present value for the same nominal yield compared to annual payments, assuming the same nominal rate. This is due to more frequent compounding periods, which increases the effective yield and present value calculations.

Conclusion

Understanding how to calculate loan payments, present and future values of annuities, and bond pricing enables investors and financial managers to make sound financial decisions. These calculations highlight important trade-offs between loan terms, payment timing, and investment returns, which are crucial in planning and managing finances effectively.

References

  • Brigham, E. F., & Ehrhardt, M. C. (2016). Financial Management: Theory & Practice (15th ed.). Cengage Learning.
  • EuroFinance. (2020). Principles of Bond Valuation. Retrieved from https://www.eurofinance.com
  • Gitman, L. J., & Zutter, C. J. (2015). Principles of Managerial Finance (14th ed.). Pearson.
  • Rowling, D., & Smith, J. (2018). Time Value of Money Applications in Corporate Finance. Journal of Financial Education, 44, 134-142.
  • Investopedia. (2023). Present Value (PV) Definition. Retrieved from https://www.investopedia.com/terms/p/presentvalue.asp
  • Mun, K. (2018). Principles of Finance. Routledge.
  • Ross, S. A., Westerfield, R. W., & Jaffe, J. (2019). Corporate Finance (12th ed.). McGraw-Hill Education.
  • Stoltzfus, J. (2017). Bond Pricing and Yield. Journal of Financial and Quantitative Analysis, 52(2), 533–565.
  • SupplyChainDigital. (2021). Annuity Calculations Explained. Retrieved from https://www.supplychaindigital.com
  • Yan, J., & Liu, S. (2020). Financial Mathematics for Investors. Wiley.

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