# ch2 - CHAPTER 2 Present Value and the Opportunity Cost of...

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1 CHAPTER 2 Present Value and the Opportunity Cost of Capital Answers to Practice Questions 1. Let INV = investment required at time t = 0 (i.e., INV = -C 0 ) and let x = rate of return. Then x is defined as: x = (C 1 – INV)/INV Therefore: C 1 = INV(1 + x) It follows that: NPV = C 0 + {C 1 /(1 + r)} NPV = -INV + {[INV(1 + x)]/(1 + r)} NPV = INV {[(1 + x)/(1 + r)] – 1} a. When x equals r, then: [(1 + x)/(1 +r)] – 1 = 0 and NPV is zero. b. When x exceeds r, then: [(1 + x)/(1 + r)] – 1 > 0 and NPV is positive. 2. The face value of the treasury security is \$1,000. If this security earns 5%, then in one year we will receive \$1,050. Thus: NPV = C 0 + [C 1 /(1 + r)] = -1000 + (1050/1.05) = 0 This is not a surprising result, because 5 percent is the opportunity cost of capital, i.e., 5 percent is the return available in the capital market. If any investment earns a rate of return equal to the opportunity cost of capital, the NPV of that investment is zero.

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2 3. NPV = -\$1,300,000 + (\$1,500,000/1.10) = +\$63,636 Since the NPV is positive, you would construct the motel. Alternatively, we can compute r as follows: r = (\$1,500,000/\$1,300,000) – 1 = 0.1538 = 15.38% Since the rate of return is greater than the cost of capital, you would construct the motel. 4. Investment NPV Return 1) \$5,000 1.20 18,000 10,000 = + 80.0% 0.80 10,000 10,000 18,000 = = 2) \$2,500 1.20 9,000 5,000 = + 80.0% 0.80 5,000 5,000 9,000 = = 3) \$250 1.20 5,700 5,000 = + 14.0% 0.14 5,000 5,000 5,700 = = 4) \$1,333.33 1.20 4,000 2,000 = + 100.0% 1.00 2,000 2,000 4,000 = = a. Investment 1, because it has the highest NPV. b. Investment 1, because it maximizes shareholders’ wealth. 5. a. NPV = (-50,000 + 30,000) + (30,000/1.07) = \$8,037.38 b. NPV = (-50,000 + 30,000) + (30,000/1.10) = \$7,272.73 Since, in each case, the NPV is higher than the NPV of the office building (\$7,143), accept E. Coli’s offer. You can also think of it another way. The true opportunity cost of the land is what you could sell it for, i.e., \$58,037 (or \$57,273). At that price, the office building has a negative NPV. 6. The opportunity cost of capital is the return earned by investing in the best alternative investment. This return will not be realized if the investment under consideration is undertaken. Thus, the two investments must earn at least the same return. This return rate is the discount rate used in the net present value calculation.
3 7. a. NPV = -\$2,000,000 + [\$2,000,000 × 1.05)]/(1.05) = \$0 b. NPV = -\$900,000 + [\$900,000 × 1.07]/(1.10) = -\$24,545.45 The correct discount rate is 10% because this is the appropriate rate for an investment with the level of risk inherent in Norman’s nephew’s restaurant. The NPV is negative because Norman will not earn enough to compensate for the risk. c. NPV = -\$2,000,000 + [\$2,000,000 × 1.12]/(1.12) = \$0 d. NPV = -\$1,000,000 + (\$1,100,000/1.12) = -\$17,857.14 Norman should invest in either the risk-free government securities or the risky stock market, depending on his tolerance for risk. Correctly priced securities always have an NPV = 0. 8. a. Expected rate of return on project = This is equal to the return on the government securities. b. Expected rate of return on project = This is less than the correct 10% rate of return for restaurants with similar risk. c. Expected rate of return on project = This is equal to the rate of return in the stock market.

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## This note was uploaded on 03/06/2008 for the course ECON 106F taught by Professor Atkeson during the Spring '05 term at UCLA.

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ch2 - CHAPTER 2 Present Value and the Opportunity Cost of...

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