# ch3 - CHAPTER 3 How to Calculate Present Values Answers to...

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11 CHAPTER 3 How to Calculate Present Values Answers to Practice Questions 1. a. PV = \$100 × 0.905 = \$90.50 b. PV = \$100 × 0.295 = \$29.50 c. PV = \$100 × 0.035 = \$ 3.50 d. PV = \$100 × 0.893 = \$89.30 PV = \$100 × 0.797 = \$79.70 PV = \$100 × 0.712 = \$71.20 PV = \$89.30 + \$79.70 + \$71.20 = \$240.20 2. a. PV = \$100 × 4.279 = \$427.90 b. PV = \$100 × 4.580 = \$458.00 c. We can think of cash flows in this problem as being the difference between two separate streams of cash flows. The first stream is \$100 per year received in years 1 through 12; the second is \$100 per year paid in years 1 through 2. The PV of \$100 received in years 1 to 12 is: PV = \$100 × [Annuity factor, 12 time periods, 9%] PV = \$100 × [7.161] = \$716.10 The PV of \$100 paid in years 1 to 2 is: PV = \$100 × [Annuity factor, 2 time periods, 9%] PV = \$100 × [1.759] = \$175.90 Therefore, the present value of \$100 per year received in each of years 3 through 12 is: (\$716.10 - \$175.90) = \$540.20. (Alternatively, we can think of this as a 10-year annuity starting in year 3.)

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12 3. a. = + = 0.88 r 1 1 DF 1 1 so that r 1 = 0.136 = 13.6% b. 0.82 (1.105) 1 ) r (1 1 DF 2 2 2 2 = = + = c. AF 2 = DF 1 + DF 2 = 0.88 + 0.82 = 1.70 d. PV of an annuity = C × [Annuity factor at r% for t years] Here: \$24.49 = \$10 × [AF 3 ] AF 3 = 2.45 e. AF 3 = DF 1 + DF 2 + DF 3 = AF 2 + DF 3 2.45 = 1.70 + DF 3 DF 3 = 0.75 4. The present value of the 10-year stream of cash inflows is (using Appendix Table 3): (\$170,000 × 5.216) = \$886,720 Thus: NPV = -\$800,000 + \$886,720 = +\$86,720 At the end of five years, the factory’s value will be the present value of the five remaining \$170,000 cash flows. Again using Appendix Table 3: PV = 170,000 × 3.433 = \$583,610 5. a. Let S t = salary in year t = = = 30 1 t t 1 t 30 1 t t t (1.08) (1.05) 20,000 (1.08) S PV = = = 30 1 t t 30 1 t t (1.029) 19,048 1.05) / (1.08 05) (20,000/1. \$378,222 (1.029) (0.029) 1 0.029 1 19,048 30 = × × = b. PV(salary) x 0.05 = \$18,911. Future value = \$18,911 x (1.08) 30 = \$190,295 c. Annual payment = initial value ÷ annuity factor 20-year annuity factor at 8 percent = 9.818 Annual payment = \$190,295/9.818 = \$19,382
13 6. Period Discount Factor Cash Flow Present Value 0 1.000 -400,000 -400,000 1 0.893 +100,000 + 89,300 2 0.797 +200,000 +159,400 3 0.712 +300,000 +213,600 Total = NPV = \$62,300 7. 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. PV = \$1 million × [Annuity factor at 8%, t = 15] = \$1 million × 8.559 PV = \$8.559 million Major refits cost \$2 million each, and will occur at times t = 5 and t = 10. PV = -\$2 million × [Discount factor at 8%, t = 5] PV = -\$2 million × [Discount factor at 8%, t = 10] PV = -\$2 million × [0.681 + 0.463] = -\$2.288 million Sale for scrap brings in revenue of \$1.5 million at t = 15. PV = \$1.5 million × [Discount factor at 8%, t = 15] PV = \$1.5 million × [0.315] = \$0.473 Adding these present values gives the present value of the entire project: PV = -\$8 million + \$8.559 million - \$2.288 million + \$0.473 million PV = -\$1.256 million 8. a. PV = \$100,000 b. PV = \$180,000/1.12 5 = \$102,137 c. PV = \$11,400/0.12 = \$95,000 d. PV = \$19,000 × [Annuity factor, 12%, t = 10] PV = \$19,000 × 5.650 = \$107,350 e. PV = \$6,500/(0.12 - 0.05) = \$92,857 Prize (d) is the most valuable because it has the highest present value.

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14 9. a. Present value per play is: PV = 1,250/(1.07) 2 = \$1,091.80 This is a gain of 9.18 percent per trial. If x is the number of trials needed to become a millionaire, then: (1,000)(1.0918) x = 1,000,000 Simplifying and then using logarithms, we find: (1.0918) x = 1,000 x (ln 1.0918) = ln 1000 x = 78.65 Thus the number of trials required is 79.
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• Spring '05
• Atkeson

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