Find The Following Values For A Lump Sum Assuming Annual Com

Find The Following Values For A Lump Sum Assuming Annual Compoundi

Determine the present value, future value, or both for different cash flow scenarios involving lump sums and annuities, assuming annual compounding or discount rates as specified. Calculate the future and present values for specified investments, as well as the valuation of uneven cash flow streams and lottery winnings, using appropriate financial formulas.

Paper For Above instruction

The task involves calculating the present and future values of various cash flow arrangements under the assumption of annual compounding and discounting. These calculations are fundamental in financial management for assessing investments, project evaluations, and valuing cash flow streams. The key formulas involved are rooted in the concepts of future value (FV) and present value (PV), which are highly useful for making informed financial decisions (Ross, Westerfield, & Jordan, 2019).

Future and Present Values of Lump Sums

For lump sums, the calculations hinge on the compound interest formula:

FV = PV × (1 + r)^n

and its inverse:

PV = FV / (1 + r)^n

where PV is the present value, FV is the future value, r is the annual interest rate, and n is the number of years.

Applying these formulas:

  • Part a: Future value of $500 at 8% for 1 year:

    FV = 500 × (1 + 0.08)^1 = 500 × 1.08 = $540

  • Part b: Future value of $500 at 8% for 5 years:

    FV = 500 × (1 + 0.08)^5 = 500 × 1.4693 ≈ $734.65

  • Part c: Present value of $500 to be received in 1 year at 8%:

    PV = 500 / (1 + 0.08)^1 = 500 / 1.08 ≈ $462.96

  • Part d: Present value of $500 in 5 years at 8%:

    PV = 500 / (1 + 0.08)^5 = 500 / 1.4693 ≈ $340.39

Valuation of Annuities

For regular annuities, the formulas are:

PV = P × [(1 - (1 + r)^-n) ) / r]

FV = P × [((1 + r)^n - 1) / r]

where P is the payment amount per period, n is the number of periods, and r is the interest rate per period (Brigham & Ehrhardt, 2019).

Given the parameters:

  • Part a: Present value of $400 per year for 10 years at 10%:

    PV = 400 × [(1 - (1 + 0.10)^-10) / 0.10] ≈ $400 × 6.145 = $2,458

  • Part b: Future value of $400 per year for 10 years at 10%:

    FV = 400 × [((1 + 0.10)^10 - 1) / 0.10] ≈ $400 × 15.937 = $6,375

  • Part c: Present value of $200 per year for 5 years at 5%:

    PV = 200 × [(1 - (1 + 0.05)^-5) / 0.05] ≈ $200 × 4.329 = $866

  • Part d: Future value of $200 per year for 5 years at 6%:

    FV = 200 × [((1 + 0.06)^5 - 1) / 0.06] ≈ $200 × 5.637 = $1,127

Valuation of Uneven Cash Flows

Uneven cash flow streams are valued by summing the present values of each cash flow, discounted at the opportunity cost rate. The general formula is:

PV = Σ CF_t / (1 + r)^t

where CF_t is the cash flow at time t, r is the discount rate, and t is the period.

For the specific cash flows:

  • Part a (Year 0): If cash flow at year 0 is $a, and the discount rate is 10%, the present value is simply $a, since it occurs immediately.
  • Part b (Year 0): The present value is the sum of all discounted cash flows from each year:

    PV = CF_1 / (1+0.10)^1 + CF_2 / (1+0.10)^2 + ... + CF_n / (1+0.10)^n.

Valuation of Annuity Payments and Future Values of Cash Flows

Similarly, the future value at Year 5 of these streams is computed by compounding each cash flow forward to Year 5:

FV = Σ CF_t × (1 + r)^{n - t}

where each cash flow is grown to Year 5 terms.

Lottery Winnings Valuation

The lottery provides 20 annual payments of $1.75 million, starting immediately. To value this, treat it as an annuity with immediate payments (an annuity due). The present value is computed as:

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

where P = 1.75 million, n = 20 years, r = 6% (the comparable risk rate). The value of this immediate annuity must account for the fact that payments start immediately (annuity due), thus multiplying by (1 + r).

Specifically:

PV = 1.75 million × [ (1 - (1 + 0.06)^-20) / 0.06 ] × 1.06 ≈ 1.75 million × 12.547 × 1.06 ≈ $23.347 million

This calculation indicates the discounted value of the lottery payments, which provides a benchmark for whether to take the payments or a lump sum today.

Conclusion

Accurate valuation of lump sums, annuities, and uneven cash flows requires applying the correct formulas based on whether cash flows are received or paid at the beginning or end of periods and the appropriate interest or discount rates. These methods underpin sound financial planning, investment analysis, and valuation decisions, offering critical insights into the worth of future cash flows and investment opportunities. Mastery of these calculations enabling financial managers and investors to optimize decision-making processes (Damodaran, 2012).

References

  • Brigham, E. F., & Ehrhardt, M. C. (2019). Financial Management: Theory & Practice. Cengage Learning.
  • Damodaran, A. (2012). Investment Valuation: Tools and Techniques for Determining the Value of Any Asset. Wiley.
  • Ross, S. A., Westerfield, R. W., & Jordan, B. D. (2019). Fundamentals of Corporate Finance. McGraw-Hill Education.
  • Padgett, J. R. (2017). Financial Mathematics: Actuarial and Financial Applications. Springer.
  • Hull, J. C. (2018). Options, Futures, and Other Derivatives. Pearson.
  • Benninga, S. (2014). Financial Modeling, 4th Edition. The MIT Press.
  • Levy, H., & Sarnat, M. (2018). International Financial Management. Pearson.
  • Fabozzi, F. J. (2016). Bond Markets, Analysis, and Strategies. Pearson.
  • Shapiro, A. C. (2019). Multinational Financial Management. Wiley.
  • Cresta, J. (2017). Essentials of Financial Management. Cengage Learning.

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