midterms solution - Midterm 1 Econ 138 1 Question 1(NPV 8...

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Midterm 1, Econ 138 February 21, 2008 1 Question 1 (NPV - 8 points) Consider an in fi nite-horizon investment project that generates return R t in period t 0 . Denote the (constant) interest rate by r . 1.1 Part a (1 point) Write down a basic formula for the NPV of this project. Solution: P t =0 R t (1 + r ) t [1 point for correct formula, including speci fi ced t = 0 and . ] 1.2 Part b (2 points) Suppose r = . 05, and R t = 1 for all t 0. What is the most you would pay for such this project, i.e., how much would you be willing to invest today, at t = 0, to get the returns to this project? Solution: Derivation of in fi nite geometric series ( not required ): P t =0 R ( 1 1+ r ) t = R + R 1 1+ r + R ³ 1 1+ r ´ 2 + R ³ 1 1+ r ´ 3 + ... 1 1+ r P t =0 R ( 1 1+ r ) t = R 1 1+ r + R ³ 1 1+ r ´ 2 + R ³ 1 1+ r ´ 3 + ... = r 1+ r P t =0 R ( 1 1+ r ) t = R ⇐⇒ P t =0 R ( 1 1+ r ) t = 1+ r r R Here: $ 1 . 05 0 . 05 = $21. [1 point for correct formula, with or without numbers plugged in; 1 point for correct solution.] 1.3 Part c (2 points) Now suppose instead that R t = 5 t 0. Find an r such that this project is exactly as pro fi table as the one in the Part b, i.e., such that you would be willing to pay/invest exactly 1
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the same amount for this project. Solution: 1+ r r · $5 = $21 ⇐⇒ $5 = r ($21 $5), so r = 5 16 . [1 point for correct set-up; 1 point for correct solution.] 1.4 Part d (3 points) Let’s return to the initial assumption that r = 0 . 05, and suppose R t = 10 for all periods 0 t T and R t = 0 for all periods thereafter ( t > T ). Find the smallest value of T such that this asset is more valuable than the one in Part b . Hint 1: Derive the formula and then try out T = 0 , T = 1, ... etc. Hint 2: 1 1 . 05 = 0 . 952380952; ³ 1 1 . 05 ´ 2 = 0 . 907029478; ³ 1 1 . 05 ´ 3 = 0 . 863837599; ³ 1 1 . 05 ´ 4 = 0 . 822702475 Solution: Derivation of fi nite geometric series ( not required ): P T t =0 R ( 1 1+ r ) t = R + R 1 1+ r + R ³ 1 1+ r ´ 2 + R ³ 1 1+ r ´ 3 + ... + R ³ 1 1+ r ´ T 1 1+ r P T t =0 R ( 1 1+ r ) t = R 1 1+ r + R ³ 1 1+ r ´ 2 + R ³ 1 1+ r ´ 3 + ... + R ³ 1 1+ r ´ T +1 = r 1+ r P T t =0 R ( 1 1+ r ) t = R R ³ 1 1+ r ´ T +1 ⇐⇒ P T t =0 R ( 1 1+ r ) t = 1+ r r R 1 ³ 1 1+ r ´ T +1 ¸ Hence, the formula is 1 . 05 0 . 05 · 10 1 ³ 1 1 . 05 ´ T +1 ¸ 21 ⇐⇒ 1 ³ 1 1 . 05 ´ T +1 ¸ 1 10 ⇐⇒ 9 10 ³ 1 1 . 05 ´ T +1 ⇐⇒ 0 . 9 (0 . 952380952) T +1 T = 0 , 1 fail, but T = 2 works, i.e., the smallest T is T = 2. [1 point for correct formula, with or without numbers plugged in; 1 point for solving it in terms of numbers; 1 point for fi nding the right T .] 2 Question 2 (Basic Moral Hazard - 55 points) Suppose a fi rm has an investment project costing I and cash C < I . The fi rm is run by a risk-neutral manager who, for simplicity, also owns the entire fi rm. The manager can either work hard, which gives no private bene fi t, or shirk, which yields a private bene fi t equal to $ B . The project will either return R or 0 dollars, and the probability of a return of R is p H if the manager works hard, and p L if she shirks. As always, we assume that the market for external fi nancing is competitive, that the interest rate is zero, and that the manager’s liability is limited (i.e., she cannot be asked to pay money out of her private wealth.) 2
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