Problem 6: Determine The Present Value If $5,000 Is Received

Problem 6: Determine the Present Values if $5,000 is Rec

Determine the present values if $5,000 is received in the future (i.e., at the end of each indicated time period) in each of the following situations: 5 percent for ten years, 7 percent for seven years, 9 percent for four years.

Paper For Above instruction

Understanding the concept of present value (PV) is fundamental in finance, as it allows investors and financial managers to evaluate the worth of future cash flows in today's terms. The present value of a future sum of money is calculated by discounting it at an appropriate interest rate, which reflects the time value of money and risk factors. The formula for calculating the present value of a lump sum is PV = FV / (1 + r)^n, where FV is the future value, r is the discount rate per period, and n is the number of periods.

Given the problem involves receiving $5,000 at the end of each period for multiple years, the scenario pertains to the present value of an annuity. The appropriate formula in this case is the present value of an ordinary annuity, expressed as:

PV = P × [(1 - (1 + r)^-n) / r]

where P is the periodic payment ($5,000), r is the discount rate per period, and n is the number of periods.

Calculations for Each Scenario

1. Discount rate = 5% for 10 years

Using the present value of an annuity formula:

PV = 5,000 × [(1 - (1 + 0.05)^-10) / 0.05]

Calculating:

(1 + 0.05)^-10 = (1.05)^-10 ≈ 0.61391

1 - 0.61391 ≈ 0.38609

0.38609 / 0.05 ≈ 7.7218

Therefore, PV ≈ 5,000 × 7.7218 ≈ $38,609

2. Discount rate = 7% for 7 years

PV = 5,000 × [(1 - (1 + 0.07)^-7) / 0.07]

Calculating:

(1 + 0.07)^-7 ≈ (1.07)^-7 ≈ 0.62997

1 - 0.62997 ≈ 0.37003

0.37003 / 0.07 ≈ 5.286

PV ≈ 5,000 × 5.286 ≈ $26,430

3. Discount rate = 9% for 4 years

PV = 5,000 × [(1 - (1 + 0.09)^-4) / 0.09]

Calculating:

(1 + 0.09)^-4 ≈ (1.09)^-4 ≈ 0.70843

1 - 0.70843 ≈ 0.29157

0.29157 / 0.09 ≈ 3.2408

PV ≈ 5,000 × 3.2408 ≈ $16,204

Conclusion

These calculations illustrate how the present value of future cash flows diminishes as either the discount rate increases or the number of periods extends. The present value calculations provide essential insights for financial decision-making, enabling investors to assess whether future cash flows justify current investments. Understanding the appropriate application of annuity formulas is vital in corporate finance, personal investing, and valuation scenarios.

References

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