# HOW TO CALCULATE PRESENT VALUES

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1 Chapter 2 HOW TO CALCULATE PRESENT VALUES Brealey, Myers, and Allen Principles of Corporate Finance 11 th Global Edition McGraw-Hill Education Copyright 2014 by The McGraw-Hill Companies, Inc. All rights reserved. Calculating Future Values Future Value Amount to which investment will grow after earning interest Present Value Value today of future cash flow 2-2 1

2 Future Value of \$100 = 1 st basic financial principle: a dollar today is worth more than a dollar tomorrow. Example: FV FV \$100(1 t r) What is the future value of \$100 if interest is compounded annually at a rate of 7% for two years? FV \$100(1.07) (1.07) \$ FV \$100(1.07) 2 \$ FIGURE 2.1 FUTURE VALUES WITH COMPOUNDING 2-4 2

3 Present value = PV PV= discount factor C Discount factor = DF = PV of \$1 Discount factors can be used to compute present value of any cash flow 2-6 3

4 Given any variables in the equation, one can solve for the remaining variable Prior example can be reversed 2 DF2 1/ (1.07) PV DF 1 (1.07) C 2 2 PV FIGURE 2.2 PRESENT VALUES WITH COMPOUNDING 2-8 4

5 Valuing an Office Building Step 1: Forecast Cash Flows Cost of building = C 0 = \$700,000 Sale price in year 1 = C 1 = \$800,000 (certain) Step 2: Estimate Opportunity Cost of Capital If one-year safe assets (U.S. Treasury debt securities) offer a return of 7%, then cost of capital = r = 7% 2-9 Valuing an Office Building Step 3: Discount future cash flows C PV (1r ) (1.07) 1 \$800,000 \$747,664 Step 4: Go ahead if PV of payoff exceeds investment NPV \$747,664 \$700, 000 \$47,

6 Net Present Value NPV = PV required investment NPV = C 0 C1 1 r 2-11 Risk and Present Value Higher risk projects require a higher rate of return Higher required rates of return (opportunity costs) cause lower PVs PV of C \$800,000 at 12% 1 \$800,000 PV \$714,

7 Risk and Net Present Value NPV=PV required investment NPV=\$714,286 \$700,000 \$14,286 2 nd basic financial principle: a safe dollar is worth more than a risky dollar Net Present Value Rule Accept investments that have positive net present value Using the original example: Should one accept the project given a 10% expected return? NPV = \$700,000 + \$800, \$27,

8 Rate of Return Rule Accept investments that offer rates of return in excess of their opportunity cost of capital In the project listed below, the opportunity cost of capital is 12%. Is the project a wise investment? Return profit investment \$800,000 \$700, , \$700,000 or 14.3% 2-15 Multiple Cash Flows Discounted Cash Flow (DCF) formula: PV 2 C1 C (1 r) (1 r) C t t (1 r)

9 FIGURE 2.5 NET PRESENT VALUES PERPETUITIES AND ANNUITIES Perpetuity Financial concept in which cash flow is theoretically received forever Return r cash flow present value C PV

10 2-2 PERPETUITIES AND ANNUITIES Perpetuity PV of cash flow cash flow discount rate PV 0 C 1 r PERPETUITIES AND ANNUITIES Present Value of Perpetuities What is the present value of \$1 billion every year, for eternity, if the perpetual discount rate is 10%?

11 2-2 PERPETUITIES AND ANNUITIES Present Value of Perpetuities What if the investment does not start making money for 3 years? PERPETUITIES AND ANNUITIES Annuity Asset that pays fixed sum each year for specified number of years Asset Perpetuity (first payment in year 1) Year of Payment 1 2..t t + 1 Present Value C/r Perpetuity (first payment in year t + 1) C/[r(1+r) t ] Annuity from year 1 to year t C C 1 t r r (1 r)

12 2-2 PERPETUITIES AND ANNUITIES Example: Tiburon Autos offers payments of \$5,000 per year, at the end of each year for 5 years. If interest rates are 7%, per year, what is the cost of the car? PERPETUITIES AND ANNUITIES PV of annuity 1 C r 1 t r 1 r

13 2-2 PERPETUITIES AND ANNUITIES Annuity Example: The state lottery advertises a jackpot prize of \$365 million, paid in 30 yearly installments of \$ million, at the end of each year. Find the true value of the lottery prize if interest rates are 6%. 1 1 Lottery Value Value \$167,500, PERPETUITIES AND ANNUITIES If the lottery is paid at the beginning of each period, this will be an annuity due Lottery value = PV = (1+0.06) = million. 1 1 PV of annuity due C (1 r) t r r1 r

14 2-2 PERPETUITIES AND ANNUITIES Future Value of an Annuity FV of annuity C 1 r r t PERPETUITIES AND ANNUITIES Future Value of an Annuity What is the future value of \$20,000 paid at the end of each of the following 5 years, assuming investment returns of 8% per year? FV 20,000 \$117,

15 2-3 GROWING PERPETUITIES AND ANNUITIES Constant Growth Perpetuity PV 0 r C 1 g g = the annual growth rate of the cash flow This formula can be used to value a perpetuity at any point in time PV t C t 1 r g GROWING PERPETUITIES AND ANNUITIES Constant Growth Perpetuity What is the present value of \$1 billion paid at the end of every year in perpetuity, assuming a rate of return of 10% and constant growth rate of 4%? 1 PV \$ billion

16 2-3 GROWING PERPETUITIES AND ANNUITIES Growing Annuities Golf club membership is \$5,000 for 1 year, or a single payment today of \$12,750 for three years membership. Find the better deal given annual payment at the end of the year and 6% expected annual increase, discount rate 10%. PV= \$5000 x [1/( )] x ([1 - (1.06) 3 ]/(1.10) 3 ) = \$5,000 x = \$13, HOW INTEREST IS PAID AND QUOTED Effective Annual Interest Rate (EAR) Interest rate annualized using compound interest Annual Percentage Rate (APR) Interest rate annualized using simple interest

17 2-4 HOW INTEREST IS PAID AND OUTLAID Given a monthly rate of 1%, what is the (EAR)? What is the (APR)? m 12 EAR=[1+(r/m)] -1 (1+.01) 1= 12 EAR=(1+.01) 1=.1268, or 12.68% r APR=.01 12=.12, or 12.00% HOW INTEREST IS PAID AND OUTLAID Continuous compounding As m approaches infinity, [1+(r/m)] m approaches (2,718) r e= 2,718: the base for natural logarithms Therefore, \$1 invested at a continuously compounded rate of r will grow to e r = (2,718) r by the end of the first year. At the end of t years it will grow to e rt = (2,718) rt

18 2-4 HOW INTEREST IS PAID AND OUTLAID Example: Suppose you invest \$1 at a continuously compounded rate of 11% ( r.11) for one year ( t=1). The end-year value is e.11, or \$ In other words, investing at 11% a year continuously compounded is exactly the same as investing at 11.6% a year annually compounded PROBLEMS-ANSWERS 1. At an interest rate of 12%, the six-year discount factor is.507. How many dollars is \$.507 worth in six years if invested at 12%? Answer:.507 x = \$1 2. If the PV of \$139 is \$125, what is the discount factor? Answer: DF x 139 = 125. Therefore, DF =125/139 = If the cost of capital is 9%, what is the PV of \$374 paid in year 9? Answer: PV = 374/(1.09) 9 =

19 PROBLEMS-ANSWERS 4. An investment costs \$1,548 and pays \$138 in perpetuity. If the interest rate is 9%, what is the NPV? Answer: NPV = 1, /.09 = (cost today plus the present value of the perpetuity). 5. A common stock will pay a cash dividend of \$4 next year. After that, the dividends are expected to increase indefinitely at 4% per year. If the discount rate is 14%, what is the PV of the stream of dividend payments? Answer: PV = 4/(.14.04) = \$ PROBLEMS-ANSWERS 6. The interest rate is 10%. What is the PV of an asset that pays \$1 a year in perpetuity? Answer: PV = 1/.10 = \$ The cost of a new automobile is \$10,000. If the interest rate is 5%, how much would you have to set aside now to provide this sum in five years? Answer: PV = 10,000/( ) = \$7, (assuming the cost of the car does not appreciate over those five years)

20 PROBLEMS-ANSWERS 8. You have to pay \$12,000 a year in school fees at the end of each of the next six years. If the interest rate is 8%, how much do you need to set aside today to cover these bills? Answer: The six-year annuity factor [(1/0.08) 1/{(0.08 x (1+.08) 6 )}] = You need to set aside (12,000 six-year annuity factor) = 12, = \$55, cont. You have invested \$60,476 at 8%. After paying the above school fees, how much would remain at the end of the six years? Answer: At the end of six years you would have (60,476-55,475) = \$7, PROBLEMS-ANSWERS 9. You are quoted an interest rate of 6% on an investment of \$10 million. What is the value of your investment after four years if interest is compounded: a. Annually? FV = 10,000,000 x (1.06) 4 = 12,624,770. b. Monthly? FV = 10,000,000 x (1 +.06/12) (4 x 12) = 12,704,892. c. Continuously? FV = 10,000,000 x e (4 x.06) = 12,712,

21 PROBLEMS-ANSWERS 10. a. If the one-year discount factor is.905, what is the one-year interest rate? 1 DF 1 r r 1 = = 10.50% 1 b. If the two-year interest rate is 10.5%, what is the two-year discount factor? 1 1 DF 2 2 (1 r ) (1.105) c. Given these one- and two-year discount factors, calculate the two-year annuity factor. AF 2 = DF 1 + DF 2 = = PROBLEMS-ANSWERS d. If the PV of \$10 a year for three years is \$24.65, what is the three-year annuity factor? PV of an annuity = C [Annuity factor at r% for t years] Here: \$24.65 = \$10 [AF 3 ] AF 3 = e. From your answers to (c) and (d), calculate the three-year discount factor. AF 3 = DF 1 + DF 2 + DF 3 = AF 2 + DF = DF 3 DF 3 =

22 PROBLEMS 11. Halcyon Lines is considering the purchase of a new bulk carrier for \$8 million. The forecasted revenues are \$5 million a year and operating costs are \$4 million. A major refit costing \$2 million will be required after both the fifth and tenth years. After 15 years, the ship is expected to be sold for scrap at \$1.5 million. If the discount rate is 8%, what is the ship s NPV? 2-43 ANSWERS We can break this down into several different cash flows, such that the sum of these separate cash flows is the total cash flow. Then, the sum of the present values of the separate cash flows is the present value of the entire project. (All dollar figures are in millions.) Cost of the ship is \$8 million PV = \$8 million Revenue is \$5 million per year, operating expenses are \$4 million. Thus, operating cash flow is \$1 million per year for 15 years. 1 1 PV \$1million \$8.559million (1.08) Major refits cost \$2 million each, and will occur at times t = 5 and t = 10. PV = (\$2 million)/ (\$2 million)/ = \$2.288 million Sale for scrap brings in revenue of \$1.5 million at t = 15. PV = \$1.5 million/ = \$0.473 million Adding these present values gives the present value of the entire project: NPV = \$8 million + \$8.559 million \$2.288 million + \$0.473 million NPV = \$1.256 million

23 PROBLEMS 12. A mortgage requires you to pay \$70,000 at the end of each of the next eight years. The interest rate is 8%. a. What is the present value of these payments? b. Calculate for each year the loan balance that remains outstanding, the interest payment on the loan, and the reduction in the loan balance ANSWERS 1 1 a. PV \$70,000 \$402, (1.08) b. Yea r Beginning-of- Year Balance (1) Year-end Interest on Balance (2)= (1)*8% Total Yearend Payment (3) 1 402, , , , , , , , , , , , , , , , , , , , , , , , Amortizatio n of Loan (4)=(3)-(2) End-of-Year Balance (5)=(1)-(4) 37, , , , , , , , , , , , , , ,

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