eco204_HW_17_solution

eco204_HW_17_solution - University of Toronto Department of...

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Unformatted text preview: University of Toronto, Department of Economics, ECO 204 2008‐2009 S. Ajaz Hussain ECO 204 2008‐2009 Ajaz Hussain HW 17 Solutions Question 1 This question is based on the “Resource allocation problem across 3 divisions” in Lecture 19. In that example, we recognized the importance of making decisions on the basis of marginal analysis (see Lecture 18 and Practice Problems 19) Ajax Private Equity (PAE) has unlimited funds to invest owing to Ajax’s uncle Sad‐damn Hussain having left Ajax vast sums of money before he was unceremoniously hanged. Ajax’s financial advisor, “Trust me, my name is Madoff”, has identified the following seven projects as investment opportunities: Project A B C D E F G Initial Investment $1m $0.4m $0.3m $0.1m $0.2m $0.2m $0.1m (a) Given that you have unlimited funds which project should you invest in? 1 NPV $2m $1.4m $1.2m $0.6m $0.5m $0.3m $0.05m University of Toronto, Department of Economics, ECO 204 2008‐2009 S. Ajaz Hussain Answer: Since Ajax is lucky to have unlimited funds, not only can he afford the good things in life, he can and should invest in all seven projects since each of has a NPV > 0. (b) Due to a typo in Uncle Sad‐damn’s will, Ajax finds out that he only has $1m. Unable to afford Madoff Advisors he engages an MBA from McGrill University in Montreal who tells Ajax that since he only has $1m he should invest in the feasible project (i.e. less than or equal to $1m outlay) with the highest NPV. Is he correct? Hint: look at the NPV per $ invested. Answer: No. This is because as we say in the Division A, B, C example, what matters is the incremental gain, not the total gain. To see this, note that if you invest in project A, with a $1m investment you will get a NPV of $2m. But you can do better than this by looking at the projects in terms of NPV per $ invested (the analogue of Marginal ∏ in the Division A, B, C example in Lecture 19): Project A B C D E F G Initial Investment $1m $0.4m $0.3m $0.1m $0.2m $0.2m $0.1m Observe how project D has the highest NPV per dollar invested. Of the $1m, Ajax should allocate funds first to project D. Thereafter, with $0.9m left, he should invest in the project with the next highest NPV per dollar invested: project C. With investments in projects D and C he has $0.4m leftover. Using the logic of NPV per dollar Ajax should look next project B followed by project E. Together, projects D, C, B and E will require an investment of $1m and will yield a total NPV of $3.7m. Observe how thinking on the margin (“where should Ajax spend the next dollar?”) is a superior analytical tool. Too bad they don’t teach that at McGrill University. NPV $2.0m $1.4m $1.2m $0.6m $0.5m $0.3m $0.05m NPV per $ Invested 2.0 3.5 4.0 6.0 2.5 1.5 0.5 2 University of Toronto, Department of Economics, ECO 204 2008‐2009 S. Ajaz Hussain Question 2 (Summer 2008 Test 3 Question) After graduating from UT’s Commerce program, you fulfill your life‐long dream to produce a movie. You’ve just finished work on “The Dark Monopolist”, based on the life of a nefarious criminal Sad van Dam Hussain. The movie cost $50m to produce. Assume the MC of “printing” movies on films is negligible. If the movie is released into theaters the revenues as a function of the number of weeks in theaters is: R(w) = 10w ‐ 0.25w2, where R is millions of dollars and w is the number of weeks the movie is shown in theaters. (a) What is the optimal number of weeks the movie should be shown in theaters? Show all steps clearly. Answer: Since MC is $0, you want to maximize revenues. Set MR = 0, where MR means the additional revenue from showing the movie for another week in theaters: R(w) = 10w ‐ 0.25w2 → dR(w)/dw = MR = 10 ‐ 0.5w Note what this expression says: with each passing week, revenues fall by $0.5m. To maximize revenues, set MR = 0 → 10 ‐ 0.5w = 0 → w = 20 To maximize revenues the movie should be shown for 20 weeks in theaters. (b) Movies are often released into the DVD rental market after being first released in theaters. Suppose the movie yields $4m in profits per week in the DVD rental market. What is the optimal number of weeks that movie should be shown in theaters? Assume the movie cannot be shown in theaters and released on DVDs simultaneously. Show all steps clearly. Answer: Each week that the movie is shown in theaters means giving up $4m in profits by not releasing it into the DVD market. Thus, there is an opportunity of $4m. Solve for the number of weeks the movie should be shown in theaters: MR = MC + M(opportunity cost) 3 University of Toronto, Department of Economics, ECO 204 2008‐2009 S. Ajaz Hussain → 10 ‐ 0.5 w = 0 + 4 → 0.5 w = 6 → w = 12 Thus, with the DVD rental market, the movie should be shown for only 12 weeks in theaters after which it should be released into the DVD market. Question 3 In Lecture 19 we discussed the Net Present Value (NPV) and Internal Rate of Return (IRR) rules for investment decisions where the decision is to either invest or not invest. We discussed that in micro and finance the value of not investing is $0 and not the interest you’d get by putting the investment money into (say) the bank because the opportunity cost of investment ‐‐ putting money into the bank ‐‐ is built into the PV calculations. Let’s see if this is true. You have the opportunity to buy land for $10,000. You are sure that five years from now it will be worth $20,000. Suppose you can earn 8% a year by investing your money in the bank. (a) Should you buy the land? Use the NPV rule and verify your answer by comparing the Future Values (FV) of the land with the FV of putting your money in the bank at 8% interest. Answer: If you buy land, your revenue and cost streams are: Time Revenues Costs 0 $10,000 1 2 3 4 5 $20,000 Now, the NPV is = ‐$10,000 + $20,000/(1 + 0.08)5 = $3,612 Since the NPV > 0 you should use the $10,000 to buy the land instead of putting it into a bank giving you 8% interest. Let’s see if this is indeed true. Had you put $10,000 into the bank, the 5 years from now, at an interest rate of 8% you’d have a future value of: FV5 = $10,000(1 + 0.08)5 = $14,693 The FV of the land 5 years from now is $20,000 while that of putting the money in the bank is $14,693. Clearly, the land is a better investment. Of course, this depends on you being right 4 University of Toronto, Department of Economics, ECO 204 2008‐2009 S. Ajaz Hussain about the $20,000 future value. (b) Should you buy the land? Use the IRR rule. Answer: Recall that the IRR is that “interest rate” at which the NPV of investment is zero. Thus: NPV = ‐$10,000 + $20,000/(1 + r)5 = 0 → $20,000/(1 + r)5 = $10,000 → 2/(1 + r)5 = 1 → 2= (1 + r)5 → 21/5= 1 + r → r = 21/5 ‐ 1 → r ≈ 0.1486 or 14.86% Now, the opportunity cost of your capital ‐‐ if the next best alternative is to put the money in the bank ‐‐ is 8%. The IRR of 14.86% from buying the land is greater than 8% return from the bank and therefore you should buy the land. Intuitively, depositing $10,000 in land yields a greater return than the bank. Note: As you’ll see in corporate finance, when the decision is {Invest, Not Invest} the NPV and IRR rules will be in agreement. But when there are multiple investment choices, say, {Invest in A, Invest in B, Not Invest}, the NPV and IRR rules can disagree. Here is a simple example: the chart below shows the NPV of two projects versus the interest rate (see Lecture 19 for how we derived this chart): 5 University of Toronto, Department of Economics, ECO 204 2008‐2009 S. Ajaz Hussain Using the IRR criterion, both projects A and B look attractive: their IRRs are each greater than the opportunity cost of capital, say, the bank interest rate. On the basis of the IRR rule, you’d choose project B as its IRR higher than that of project B. But at the opportunity cost of capital note project A has a higher NPV than project B. Thus, the NPV and IRR rules disagree. You should also look at the additional NPV problems in practice problems 19. 6 ...
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This note was uploaded on 05/02/2011 for the course ECO 204 taught by Professor Hussein during the Fall '08 term at University of Toronto.

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