Determine The Present Value Of $5,000 If Received

To Kim Woodsdetermine The Present Values Pvs If 5000 Is Received In

To Kim Woods determine The Present Values (PVs) if 5,000 is received in the future (i.e., at the end of each indicated time period) in each of the following situations: (a) 5% for ten years; (b) 7% for seven years; (c) 9% for four years. Additionally, determine the present value (PV) if 15,000 is to be received at the end of eight years with a 9% discount rate. Assess how the PV changes if the receipt is delayed to six years instead of eight. Use a financial calculator or computer software to solve for the future values of investments at specified rates and periods, and address how different compounding periods (semiannual versus quarterly) influence the calculations.

Paper For Above instruction

The calculation of present value (PV) is fundamental in financial mathematics, serving as the basis for valuation of future cash flows. This financial principle relies on the concept that money available today is worth more than the same amount received in the future due to its potential earning capacity. The PV calculations are especially relevant in investment analysis, loan amortization, and capital budgeting, guiding investors and financial managers in informed decision-making.

Part 1: Present Value of Future Cash Flows at Different Rates and Durations

In the initial segment, we evaluate the PV of a sum of 5,000 received at the end of different periods and under varying interest rates. Using the PV formula:

\[ PV = \frac{FV}{(1 + r)^n} \]

where \(FV\) is the future value, \(r\) is the interest or discount rate per period, and \(n\) is the number of periods, we find:

- For 5% over ten years:

\[ PV = \frac{5000}{(1 + 0.05)^{10}} = \frac{5000}{1.6289} \approx 3071.80 \]

- For 7% over seven years:

\[ PV = \frac{5000}{(1 + 0.07)^{7}} = \frac{5000}{1.6058} \approx 3114.22 \]

- For 9% over four years:

\[ PV = \frac{5000}{(1 + 0.09)^{4}} = \frac{5000}{1.4116} \approx 3544.28 \]

These calculations demonstrate how higher interest rates or longer periods diminish the present value, aligning with the concept of discounting.

Part 2: Present Value of a Future Sum

Next, considering a future receipt of 15,000 at the end of eight years with a 9% discount rate:

\[ PV = \frac{15,000}{(1 + 0.09)^{8}} = \frac{15,000}{1.9990} \approx 7,502.75 \]

If the receipt is delayed to six years:

\[ PV = \frac{15,000}{(1 + 0.09)^{6}} = \frac{15,000}{1.6771} \approx 8,937.74 \]

This illustrates how the PV increases as the period shortens, reflecting less discounting.

Part 3: Future Value Calculations

Using a financial calculator or software (such as Excel), the future value (FV) of an investment can be calculated with the FV function:

- For an initial investment of 15,555 earning 14.5% over seven years:

\[ FV = 15,555 \times (1 + 0.145)^{7} \]

Calculating:

\[ FV \approx 15,555 \times 2.6236 \approx 40,812.15 \]

- For an initial amount of 19,378 invested for eight years at 18%:

\[ FV \approx 19,378 \times (1 + 0.18)^{8} = 19,378 \times 4.029 \approx 78,097.18 \]

Regarding the impact of compounding frequency:

- Semiannual compounding involves dividing the annual rate by 2 and multiplying the number of years by 2:

\[ FV = PV \times \left( 1 + \frac{r}{2} \right)^{2n} \]

- Quarterly compounding divides the annual rate by 4 and multiplies the periods by 4:

\[ FV = PV \times \left( 1 + \frac{r}{4} \right)^{4n} \]

This adjustment results in slightly higher FV due to more frequent interest calculations, emphasizing the importance of compounding frequency in future value estimations.

Part 4: Present Value of a Lump Sum at Different Durations

Calculating PV of 359,000 received in 23 years at an 11% discount rate:

\[ PV = \frac{359,000}{(1 + 0.11)^{23}} = \frac{359,000}{10.704} \approx 33,520.71 \]

For 20 years:

\[ PV = \frac{359,000}{(1 + 0.11)^{20}} = \frac{359,000}{8.062} \approx 44,498.65 \]

This demonstrates that the earlier the payout is received, the higher its present value due to less discounting.

Part 5: Impact of Compounding Frequency

Finally, for 19,378 invested over eight years at 18%, the FV with semiannual compounding:

\[ FV = PV \times \left( 1 + \frac{0.18}{2} \right)^{2 \times 8} \]

\[ FV \approx 19,378 \times (1 + 0.09)^{16} \approx 19,378 \times 3.808 \approx 73,777.28 \]

If quarterly compounding is used:

\[ FV = PV \times \left( 1 + \frac{0.18}{4} \right)^{4 \times 8} \]

\[ FV \approx 19,378 \times (1 + 0.045)^{32} \approx 19,378 \times 4.135 \approx 80,155.87 \]

Quarterly compounding yields a higher FV compared to semiannual because interest is compounded more frequently, leading to a greater accumulation over the same period.

Conclusion

Calculating present and future values under different interest rates, durations, and compounding frequencies is vital in financial decision-making. Understanding the mathematical principles behind these calculations enables investors and financial managers to appropriately value investment opportunities, assess the impact of changes in interest rates, and plan for future financial needs efficiently.

References

  • Brealey, R. A., Myers, S. C., & Allen, F. (2020). Principles of Corporate Finance (13th ed.). McGraw-Hill Education.
  • Hatch, P. (2018). Financial Mathematics: A Very Short Introduction. Oxford University Press.
  • Ross, S. A., Westerfield, R., & Jaffe, J. (2019). Corporate Finance (12th ed.). McGraw-Hill Education.
  • Damodaran, A. (2012). Investment Valuation: Tools and Techniques for Determining the Value of Any Asset. Wiley Finance.
  • Investopedia. (2023). Present Value (PV). Retrieved from https://www.investopedia.com/terms/p/presentvalue.asp
  • Investopedia. (2023). Future Value (FV). Retrieved from https://www.investopedia.com/terms/f/futurevalue.asp
  • Gallagher, D. R. (2019). Financial Accounting. Cambridge University Press.
  • Craig, J. (2021). Time Value of Money Calculations in Excel. Journal of Financial Planning, 34(2), 45-52.
  • Investopedia. (2023). Compounding Frequency. Retrieved from https://www.investopedia.com/terms/c/compoundingfrequency.asp
  • Brigham, E. F., & Ehrhardt, M. C. (2021). Financial Management: Theory & Practice (16th ed.). Cengage Learning.

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