Case Application: Time Value Of Money

Case Applicationtime Value Of Moneyrichard e-mailed me that he and Mon

Richard e-mailed me that he and Monica differed about the impact of his extra spending over the past 15 years. He calculated it at about $3,000 a year. He said the total cost of $45,000 was well within his capability to make up. Monica said the cost was much higher and asked that they compute it. They had been offered an investment of $20,000 that would pay $70,000 in 20 years. They want to know if they should take it. Finally, Richard could sign up for an annuity at work. It would cost $100,000 at age 65 and provide payments of $8,000 per year over his expected 17-year life span. He wants to know if it is attractive. The appropriate market rate of return on investments is 7 percent after tax.

Paper For Above instruction

The scenario presented by Richard and Monica highlights key concepts in the time value of money (TVM), which are essential in evaluating financial decisions involving investments, savings, and retirement planning. Understanding how money’s value changes over time allows individuals to make informed choices that optimize their financial well-being. This paper explores the calculations and explanations necessary to assess their options, emphasizing the importance of interest compounding, investment returns, and present versus future value analyses.

1. Calculating the Future Value of the $3,000 Annual Deficit

Richard's claim that his extra spending has a $3,000 annual deficit over the past 15 years can be quantified using the future value (FV) formula for an ordinary annuity. Assuming he would have invested this amount annually at a 7% after-tax rate, the accumulated amount would reflect the potential growth of missed savings. The formula for FV of an ordinary annuity is:

FV = P × [(1 + r)^n - 1] / r

Where:

  • P = $3,000 (annual deficit)
  • r = 0.07 (annual rate of return)
  • n = 15 years

Applying these values:

FV = 3,000 × [(1 + 0.07)^15 - 1] / 0.07

Calculating (1 + 0.07)^15 ≈ 2.75903, then FV ≈ 3,000 × (2.75903 - 1) / 0.07 ≈ 3,000 × 1.75903 / 0.07 ≈ 3,000 × 25.1376 ≈ $75,412.80

Therefore, if Richard had invested his $3,000 yearly deficit at 7%, it would have grown to approximately $75,413 after 15 years. This demonstrates the significant impact of compounding over time in converting small regular savings into a substantial future sum.

2. Explanation of Compounding and Its Effect on the Cumulative Amount

Compounding refers to the process where earned interest is reinvested to earn additional interest in subsequent periods. Essentially, it is "interest on interest," causing the growth of an investment to accelerate over time. Unlike simple interest, which is computed solely on the original principal, compound interest considers the accumulated interest from previous periods, resulting in exponential growth.

In the context of question 1, compounding magnifies the value of regularly invested amounts. At a 7% rate, each year's investment gains interest, and this interest is then included in the subsequent year's principal, leading to a geometric increase in the total accumulated amount. The longer the investment period, the more pronounced the effect of compounding, which underscores its importance in personal financial planning. The calculation of $75,413 illustrates how relatively modest annual savings, when compounded over 15 years at a reasonable rate, can grow into a substantial nest egg, highlighting the importance of early and consistent saving.

3. Evaluation of the Proposed $20,000 Investment

The investment opportunity offers $20,000 today, which is projected to grow to $70,000 in 20 years. To evaluate whether this is attractive, one must calculate its internal rate of return (IRR) or investment return rate and compare it to the market rate of 7%. The future value (FV) formula can be rearranged to find the rate of return:

FV = PV × (1 + r)^n

Where PV = $20,000, FV = $70,000, n = 20 years. Solving for r:

r = (FV / PV)^(1/n) - 1

r = (70,000 / 20,000)^(1/20) - 1 = (3.5)^(0.05) - 1

Calculating (3.5)^(0.05) ≈ e^{0.05 × ln(3.5)} ≈ e^{0.05 × 1.2528} ≈ e^{0.06264} ≈ 1.0646

Therefore, r ≈ 1.0646 - 1 = 0.0646 or 6.46%.

This approximate return of 6.46% is slightly below the market rate of 7%. Given that market rates reflect the opportunity cost of capital, accepting this investment depends on Richard and Monica’s risk tolerance and alternative options. If they prioritize higher returns or safer investments with comparable yields, they might reject this offer. Conversely, if they are comfortable with a slightly lower return and trust the investment's predictability, they might accept.

Furthermore, other factors include the risk profile of the investment, inflation expectations, and how the return compares with inflation rates and other opportunities in the market. This calculation underscores the importance of evaluating investments not just based on nominal figures but also considering the annualized return relative to market benchmarks.

4. Expected Return on the Annuity and Its Investment Attractiveness

The proposed annuity costs $100,000 at age 65, with annual payments of $8,000 over 17 years. The main question is whether its internal rate of return (IRR) exceeds the market rate of 7%, which would make it a financially attractive option.

Using the present value (PV) of an annuity formula:

PV = P × [1 - (1 + r)^-n] / r

Where:

  • P = $8,000
  • n = 17 years
  • PV = $100,000 (cost of the annuity)

We solve for r, the IRR:

100,000 = 8,000 × [1 - (1 + r)^-17] / r

This requires iterative or financial calculator methods, but for an approximate estimate, we can use the trial method or specialized software. Alternatively, we can estimate IRR by trial:

If r = 5%:

PV ≈ 8,000 × (1 - (1 + 0.05)^-17) / 0.05 ≈ 8,000 × (1 - 0.416) / 0.05 ≈ 8,000 × 0.584 / 0.05 ≈ 8,000 × 11.68 ≈ $93,440

Less than $100,000, so IRR > 5%. Now try 6%:

PV ≈ 8,000 × (1 - (1 + 0.06)^-17) / 0.06 ≈ 8,000 × (1 - 0.393) / 0.06 ≈ 8,000 × 0.607 / 0.06 ≈ 8,000 × 10.12 ≈ $80,960

Lower than $100,000, so IRR is between 5% and 6%. Fine-tuning suggests IRR ≈ 5.5%.

Since this IRR (~5.5%) is below the market rate of 7%, the annuity may be less attractive. However, it provides a guaranteed income stream, which has value in retirement planning. If Richard’s primary objective is income stability rather than maximizing return, the annuity might still be attractive. Conversely, if maximizing growth is the goal, he might reject it in favor of investments yielding higher returns.

5. Explaining the Time Value of Money Using the Analysis

The calculations above vividly demonstrate the core principle of the time value of money: a dollar today is worth more than a dollar in the future because of its potential earning capacity. For Richard and Monica, understanding how investing their money at a market rate of 7% can grow savings significantly over time is crucial for making informed financial decisions. The FV calculations show that small, consistent investments can compound over the years, creating substantial wealth — emphasizing the importance of early and continuous investing.

The evaluation of the $20,000 investment and the annuity illustrates the necessity of comparing their respective returns to market benchmarks. If the expected return on an investment exceeds the market rate, it offers a compelling opportunity; if not, the value of the money in terms of purchasing power and future income must be carefully considered. Additionally, understanding that delaying savings reduces the potential accumulation due to less opportunity for compounding reinforces the importance of timely financial planning.

By integrating concepts such as present value, future value, internal rate of return, and compounding, individuals can assess the opportunity cost of spending versus saving, enabling them to prioritize financial goals such as retirement security or wealth accumulation. Communication of these principles to Richard and Monica underscores the significance of applying the time value of money in everyday financial decisions, helping them to achieve a more secure financial future.

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.
  • Higgins, R. C. (2018). Analysis for Financial Management. McGraw-Hill Education.
  • Ross, S. A., Westerfield, R. W., & Jaffe, J. (2019). Corporate Finance. McGraw-Hill Education.
  • Mishkin, F. S., & Eakins, S. G. (2018). Financial Markets and Institutions. Pearson.
  • Gitman, L. J., & Zutter, C. J. (2018). Principles of Managerial Finance. Pearson.
  • Investopedia. (2023). Time Value of Money (TVM). Retrieved from https://www.investopedia.com/terms/t/timevalueofmoney.asp
  • Corporate Finance Institute. (2023). Present Value (PV). Retrieved from https://corporatefinanceinstitute.com/resources/knowledge/finance/present-value-pv/
  • Federal Reserve Bank. (2023). Interest Rates and Their Impact. Retrieved from https://www.federalreserve.gov/
  • Harvard Business Review. (2019). The Power of Compound Interest. Retrieved from https://hbr.org/

Related Assignments