Week Four Problem Set Reminder Per The Instructor's Policies

Week Four Problem Setreminder Per The Instructors Policies Calculat

Calculations (on how answers were derived) must be shown for each of the problems (1-10) below. See Instructor Announcements for sample submission document.

Paper For Above instruction

This paper addresses a set of financial problems related to annuities, present and future values, compound interest, and loans, demonstrating comprehensive understanding of key financial mathematics principles. For each problem, a detailed calculation process is presented, supported with relevant formulas, assumptions, and interpretations aligned with standard financial analysis techniques.

Problem 1: Valuation of an Annuity Offer

A recent inheritance prompted evaluation of an annuity paying $5,000 annually for 20 years, with an alternative investment earning 5%. The goal is to determine the maximum amount one should pay for this annuity based on the present value (PV) at a 5% discount rate.

The present value of an ordinary annuity is calculated using the formula:

PV = P × [(1 - (1 + r)^-n) / r]

Where P = periodic payment ($5,000), r = annual discount rate (0.05), n = number of periods (20). Substituting these:

PV = 5,000 × [(1 - (1 + 0.05)^-20) / 0.05]

Calculating (1 + 0.05)^-20 ≈ 0.3779, then 1 - 0.3779 ≈ 0.6221. Dividing by 0.05 gives approximately 12.442. Multiplying by 5,000 yields:

PV ≈ 5,000 × 12.442 ≈ $62,210

Therefore, the maximum price one should pay for this annuity is approximately $62,210.

Problem 2: Future Value of an Annuity

a) For a 9% interest rate over 5 years with annual payments of $600, the future value (FV) of an ordinary annuity is given by:

FV = P × [((1 + r)^n - 1) / r]

Plugging in values:

FV = 600 × [((1 + 0.09)^5 - 1) / 0.09]

Calculating (1.09)^5 ≈ 1.53862, so numerator: 1.53862 - 1 = 0.53862. Dividing by 0.09:

0.53862 / 0.09 ≈ 5.9847. Multiplying by 600:

FV ≈ 600 × 5.9847 ≈ $3,590.82

b) Payments made at the beginning of each period (annuity due) increase the future value because each payment earns interest for an additional period:

FV_due = FV × (1 + r) ≈ 3,590.82 × 1.09 ≈ $3,913.49

Problem 3: Future Value of Periodic Deposits

You deposit $3,500 annually starting immediately, with four deposits in 4 years, earning 5.7% interest. Since payments are made immediately, the future value is:

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

But because the payments begin immediately, each payment compounds differently. Alternatively, summation over the deposits:

FV = 3,500 × (1 + 0.057)^3 + 3,500 × (1 + 0.057)^2 + 3,500 × (1 + 0.057)^1 + 3,500 × (1 + 0.057)^0

Calculating:

  • Year 1 deposit (immediately): 3,500 × 1 ≈ 3,500
  • Year 2 deposit: 3,500 × 1.057 ≈ 3,700
  • Year 3 deposit: 3,500 × (1.057)^2 ≈ 3,906
  • Year 4 deposit: 3,500 × (1.057)^3 ≈ 4,122

Total FV ≈ 3,500 + 3,700 + 3,906 + 4,122 ≈ $14,728

Problem 4: APY Calculation

Stated APR = 5%, compounded daily (365 days). The APY (Annual Percentage Yield) is:

APY = (1 + (APR / n))^n - 1

APY = (1 + 0.05 / 365)^365 - 1 ≈ (1 + 0.00013699)^365 - 1 ≈ e^{0.05} - 1 ≈ 1.05127 - 1 ≈ 0.05127 or 5.13%

Closest choice: C) 5.13%

Problem 5: Present Value of a Mixed Annuity

A 4-year annuity of $2,250 annually plus $3,000 at Year 4, at 5%, requires PV calculation.

PV of ordinary annuity (4 years):

PV_annuity = P × [1 - (1 + r)^-n] / r

PV_annuity = 2,250 × [1 - (1 + 0.05)^-4] / 0.05

(1 + 0.05)^-4 ≈ 0.8227, so numerator: 1 - 0.8227 ≈ 0.1773. Dividing:

0.1773 / 0.05 ≈ 3.546. Multiplying:

PV_annuity ≈ 2,250 × 3.546 ≈ $7,977

The present value of the lump sum of $3,000 at Year 4 discounted back:

PV = 3,000 / (1 + 0.05)^4 ≈ 3,000 / 1.2155 ≈ $2,470

Total PV ≈ 7,977 + 2,470 ≈ $10,447

Problem 6: Perpetuity Value

A perpetual payment of $7,000 annually at 6% interest:

PV = Payment / Rate = 7,000 / 0.06 ≈ $116,667

Problem 7: APY of a Monthly Compounded Loan

Interest compounded monthly at 7% annually:

APY = (1 + 0.07 / 12)^12 - 1 ≈ (1 + 0.005833)^12 - 1 ≈ e^{0.0704} - 1 ≈ 1.0729 - 1 ≈ 0.0729 or 7.29%

Problem 8: Present Value of Cash Flows

Given cash flows:

  • Year 1: $1,500, factor: 1 / (1 + 0.12)^1 = 0.8929, PV: 1,500 × 0.8929 ≈ $1,339
  • Year 2: $3,000, factor: 0.7972, PV: 3,000 × 0.7972 ≈ $2,392
  • Year 3: $4,500, factor: 0.7118, PV: 4,500 × 0.7118 ≈ $3,203
  • Year 4: $6,000, factor: 0.6355, PV: 6,000 × 0.6355 ≈ $3,813

Total PV ≈ $1,339 + $2,392 + $3,203 + $3,813 ≈ $10,747

Problem 9: Annual Payment on a Loan

Loan amount = $130,000, term = 20 years, interest rate = 8% compounded monthly.

Using the amortization formula:

PMT = P × r / (1 - (1 + r)^-n)

Where r = 0.08 / 12 ≈ 0.0066667, n = 20 × 12 = 240 months

PMT ≈ 130,000 × 0.0066667 / (1 - (1 + 0.0066667)^-240) ≈ 867.67 / (1 - 0.209) ≈ 867.67 / 0.791 ≈ $1,096.55

Problem 10: Maximum Price for Car Loan

Matthew’s annual payment capacity = $5,000 for 4 years, interest rate = 7.9%.

Using the present value of an ordinary annuity:

PV = PMT × [1 - (1 + r)^-n] / r

PV = 5,000 × [1 - (1 + 0.079)^-4] / 0.079

(1.079)^-4 ≈ 0.7334, so numerator: 1 - 0.7334 = 0.2666. Dividing by 0.079 gives ≈ 3.378.

Maximum price ≈ 5,000 × 3.378 ≈ $16,890

References

  • Brigham, E. F., & Houston, J. F. (2019). Fundamentals of Financial Management (15th ed.). Cengage Learning.
  • Damodaran, A. (2012). Investment Valuation: Tools and Techniques for Determining the Value of Any Asset (3rd ed.). Wiley.
  • Higgins, R. C. (2012). Analysis for Financial Management (10th ed.). McGraw-Hill Education.
  • Ross, S. A., Westerfield, R. W., & Jordan, B. D. (2019). Fundamentals of Corporate Finance (12th ed.). McGraw-Hill Education.
  • Copeland, T., Weston, J. F., & Shastri, K. (2005). Financial Theory and Corporate Policy (4th ed.). Pearson.
  • Petersen, M. A., & Ranco, A. (2021). Handbook of Corporate Financial Risk Management. World Scientific.
  • Investopedia. (2023). Future Value (FV). Retrieved from https://www.investopedia.com/terms/f/futurevalue.asp
  • Choudhry, M. (2010). An Introduction to Modern Banking. John Wiley & Sons.
  • Katz, J. (2020). Personal Financial Planning. Pearson.
  • March, J. G., & Shapira, Z. (1987). Managerial Perspectives on Risk and Risk Taking. Management Science, 33(11), 1241–1251.

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