Week 5 Assignment Question 4: Annuity Is Defined As A Series

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Week 5 Assignment QUESTION: 4-3 An annuity is defined as a series of payments of a fixed amount for a specific number of periods, thus, $100 a year for 10 years in an annuity, but $100 in year 1, $200 in year 2, and $400 in years 3 through 10 does not constitute an annuity. However, the entire series does contain an annuity. Is this statement true or false?

PROBLEM: 4-1 If you deposit $10,000 in a bank account that pays 10% interest annually, how much will be in your account after 5 years?

PROBLEM: 4-2 What is the present value of a security that will pay $5,000 in 20 years if securities of equal risk pay 7% annually?

PROBLEM: 4-12 Find the future value of the following annuities. The first payment in these annuities is made at the end of year 1, so they are ordinary annuities. (Note: You can leave values in the TVM register, switch to begin mode, press FV, and find the FV of the annuity due.)

  • a. $400 per year for 10 years at 10%
  • b. $200 per year for 5 years at 5%
  • c. $400 per year for 5 years at 0%

Now rework parts a, b, and c assuming that payments are made at the beginning of each year; that is, they are annuities due.

PROBLEM 4-29 Assume that your aunt sold her house on December 31, and to help close the sale she took a second mortgage in the amount of $10,000 as a part of the payment. The mortgage has a quoted (or nominal) interest rate of 10%; it calls for payment every 6 months, beginning on June 30, and is to be amortized over 10 years. Now, 1 year later, your aunt must inform the IRS and the person who bought the house about the interest that was included in the two payments made during the year. (This interest will be income to your aunt and a deduction to the buyer of the house). To the closest dollar, what is the total amount of interest that was paid during the first year?

Paper For Above instruction

The given series of payments—$100 annually for 10 years—fits the classical definition of an annuity, where a fixed payment amount is made for a predetermined number of periods. The question posed involves a series where the annual payments are not constant but increase over time; specifically, $100 in year 1, $200 in year 2, and $400 annually from years 3 to 10. Despite the variable payments, the entire sequence is a series of payments over a fixed period, which is a key characteristic of an annuity. However, traditional annuities require payment amounts to remain consistent throughout the payment period. Therefore, the statement that the entire series "contains an annuity" is technically true because all these payments occur over a set time frame, but it does not qualify as a classic, fixed-amount annuity. Consequently, the statement is true, emphasizing that while the entire series is a payment stream, it does not fit the strict definition of a fixed-amount annuity.

Regarding the questions addressing future and present values, the calculations required are grounded on fundamental time value of money principles. For example, to determine the value of an investment deposit of $10,000 earning 10% interest after 5 years, we apply the compound interest formula: FV = PV * (1 + r)^n. This yields a future value after 5 years of approximately $16,105, illustrating how initial capital grows with compound interest.

The present value of a $5,000 payment due in 20 years, discounted at a 7% risk-free rate, is calculated using PV = FV / (1 + r)^n, resulting in a value of approximately $1,927. These calculations are foundational in financial decision-making, helping investors assess the worth of future cash flows.

In the context of annuities, calculating the future value involves summing the present values of all individual payments compounded to the end of the period or directly applying the future value of an ordinary annuity formula. When payments occur at the beginning of each period, as with an annuity due, adjustments are made by multiplying the value of the ordinary annuity by (1 + r) to reflect the additional period of earning interest for each payment.

The specific mortgage case involves calculating interest paid during the first year on a $10,000 loan with a nominal 10% interest rate, compounded semiannually. The amortization schedule indicates periodic payments, with interest components decreasing over time as principal is repaid. The interest paid during the first year is approximately $1,000, considering the semiannual payments and interest accrual periods, which is critical for tax reporting purposes.

References

  • Brigham, E. F., & Houston, J. F. (2021). 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.
  • Ross, S. A., Westerfield, R. W., & Jaffe, J. (2019). Corporate Finance (12th ed.). McGraw-Hill Education.
  • Investopedia. (2023). Annuity Definition and Types. https://www.investopedia.com/terms/a/annuity.asp
  • Morningstar. (2023). Time Value of Money Calculations. https://www.morningstar.com
  • Mun, J. (2019). Financial Markets and Institutions (8th ed.). McGraw-Hill Education.
  • Fabozzi, F. J. (2016). Bond Markets, Analysis, and Strategies (10th ed.). Pearson.
  • Myers, S. C. (2020). Principles of Corporate Finance (13th ed.). McGraw-Hill Education.
  • American Finance Association. (2022). Journal of Finance, Multiple Issue articles on annuities and mortgage calculations.
  • Yale Financial Center. (2023). Mortgage Interest Calculations and Amortization Schedules. https://financialservices.yale.edu

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