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Present and Future Value Calculation in Cash Flows

This chapter explores calculating the present value and future value of cash flows at different discount rates. It also covers perpetuity values and how to calculate effective annual rate and APR for loans.

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Present and Future Value Calculation in Cash Flows

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  1. Chapter 6 Questions and Problems

  2. 1. Present Value and Multiple Cash Flows • Manila Office Products has identified an investment project with the following cash flows, denominated in millions of pesos. If the discount rate is 8 percent, what is the present value of these cash flows? What is the present value at 16 percent? At 26 percent?

  3. 3. Future Value and Multiple Cash Flows • Bogata Bean Farm has identified an investment project with the following cash flows, denominated in thousands of pesos. If the discount rate is 8 percent, what is the future value of these cash flows in Year 4? What is the future value at a discount rate of 11 percent? At 24 percent?

  4. 10. Calculating Perpetuity Values • The Perpetual Life Insurance Co. is trying to sell you an investment policy that will pay you and your heirs $15,000 per year forever. If the required return on this investment is 10 percent, how much will you pay for the policy?

  5. 10. • This cash flow is a perpetuity. To find the PV of a perpetuity, we use the equation: • PV = C / r • PV = $15,000 / .10 = $150,000.00

  6. 11. Calculating Perpetuity Values • In the previous problem, suppose the Perpetual Life Insurance Co. told you the policy costs $195,000. At what interest rate would this be a fair deal?

  7. 11. Here we need to find the interest rate that equates the perpetuity cash flows with the PV of the cash flows. Using the PV of a perpetuity equation: • PV = C / r • $195,000 = $15,000 / r • We can now solve for the interest rate as follows: • r = $15,000 / $195,000 = 7.69%

  8. 12. Calculating EAR • Find the EAR in each of the following cases:

  9. 12. For discrete compounding, to find the EAR, we use the equation: • EAR = [1 + (APR / m)]m – 1 • EAR = [1 + (.11 / 4)]4 – 1 = 11.46% • EAR = [1 + (.07 / 12)]12 – 1 = 7.23% • EAR = [1 + (.09 / 365)]365 – 1 = 9.42%

  10. 13. Calculating APR • Find the APR. or stated rate, in each of the following cases:

  11. 13. Here we are given the EAR and need to find the APR. Using the equation for discrete compounding: • EAR = [1 + (APR / m)]m – 1 • We can now solve for the APR. Doing so, we get: • APR = m[(1 + EAR)1/m – 1] • EAR = .081 = [1 + (APR / 2)]2 – 1 • APR = 2[(1.081)1/2 – 1] = 7.94%

  12. 19. EAR versus APR • Big Dom's Pawn Shop charges an interest rate of 25 percent per month on loans to its customers. Like all lenders. Big Dom must report an APR to consumers. What rate should the shop report? What is the effective annual rate?

  13. 19. The APR is simply the interest rate per period times the number of periods in a year. In this case, the interest rate is 30 percent per month, and there are 12 months in a year, so we get: • APR = 12(25%) = 300% • To find the EAR, we use the EAR formula: • EAR = [1 + (APR / m)]m – 1 • EAR = (1 + .25)12 – 1 = 1,355.19 %

  14. 20. Calculating Loan Payments • You want to buy a new sports coupe for €56,850, and the finance office at the dealership has quoted you an 5.6 percent APR loan for 60 months to buy the car. What will your monthly payments be? What is the effective annual rate on this loan?

  15. 20. We first need to find the annuity payment. We have the PVA, the length of the annuity, and the interest rate. Using the PVA equation: • PVA = C({1 – [1/(1 + r)]t } / r) • €56,850 = €C[1 – {1 / [1 + (.056/12)]60} / (.056/12)] • Solving for the payment, we get: • C = €56,850 / 52.226 = €1,088.53

  16. To find the EAR, we use the EAR equation: • EAR = [1 + (APR / m)]m – 1 • EAR = [1 + (.056 / 12)]12 – 1 = 5.75%

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