PS1 - ECO 362 October 5 2007 PROBLEM SET 1 FINANCIAL...

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ECO 362 October 5, 2007 PROBLEM SET 1 FINANCIAL INVESTMENTS (ECO 362) Professor: Burt Malkiel Exercise 1 (a) (USING CALCULATOR) For the fi rst project we have 7 consecutive payments from year 1 to 7 of 330 and a last payment of 1 , 000 at year 8. Let i be the interest rate, so that NPV = F 0 + 8 X j =1 C i (1 + i ) j = F 0 + C 1 1 + i + C 1 (1 + i ) 2 + · · · + C 1 (1 + i ) 8 = 2 , 000 + 330 × 7 X j =1 1 (1 + i ) j + 1 (1 + i ) 8 × 1 , 000 = 2 , 000 + (1 + i ) 7 1 i (1 + i ) 7 × 330 + 1 (1 + i ) 8 × 1 , 000 since 7 X j =1 1 (1 + i ) j = 1 (1+ i ) 1 (1+ i ) 8 1 1 (1+ i ) = (1 + i ) 7 1 i (1 + i ) 7 . Hence, substituting by i = 12% , we get NPV = 2 , 000 + 330 × (1 . 12) 7 1 0 . 12(1 . 12) 7 + 1 (1 . 12) 8 × 1 , 000 , = 90 . 077 . (USING TABLES) The best way (in terms of time) is by decomposing the cash fl ows in two components —as we did above—: (a) annuity of 330 , and (b) payo ff of 1 , 000 at the end (of the project). Looking at Table B, for seven periods and an interest rate of 12% , we get 4 . 5638 . Alternatively, if we look at the Table for annuities (Table C) we obtain that the factor for 7 periods at a 12% is 10 . 089 . But this is the value of the annuity at t = 7; so, we have to discount that payment, 10 . 089 (1 . 12) 7 = 4 . 5638 . Next, we have to fi nd the discount rate for the last payment: (Table A) 0 . 4039 . Therefore, NPV = 2 , 000 + 330 × 4 . 5638 + 1 , 000 × 0 . 4039 = 90 . 046 where the di ff erence between using calculator and tables is due to the table approximation. In summary, 1
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ECO 362 October 5, 2007 Project 1 Project 2 Project 3 Project 4 NPV 90 . 08 173 . 04 30 . 41 92 . 96 Hence, Project 3 has the highest rank. (b) Recall that the IRR is the discount rate that makes the sum of discounted cash fl ows equal to zero, that is, the rate r such that 0 = I 0 + C 1 1 + r + · · · + C n (1 + r ) n = I 0 + n X j =1 C j (1 + r ) j . We have to use either Excel (or alternatively, a fi nancial calculator e.g. HP 17BII, HP 19BII, Texas Instruments BA II, etc.), or to interpolate i.e. compute the NPV for di ff erent values of the discount rate until we get something close to zero. Using Excel, we get 10 . 9% , 11 . 3% , 12 . 3% and 11% for projects 1, 2, 3 and 4, respectively. So the ranking becomes 3, 2, 4 and 1. Note that in some special cases, one can obtain r directly. For instance, for project 2 we have that 0 = 2 , 000 + 10 , 000 (1 + r ) 15 = (1 + r ) 15 = 10 , 000 2 , 000 = r = 5 1 / 15 1 = 11 . 33% .
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