midterms solution - Midterm 1, Econ 138 February 21, 2008 1...

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Midterm 1, Econ 138 February 21, 2008 1 Quest ion1(NPV-8po ints) Consider an in f 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 f ced t =0 and . ] 1.2 Part b (2 points) Suppose r = . 05, and R t =1fo ra l l 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 f 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
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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 =10fo ra l lp e r iod s 0 t T and R t =0fora l lper iodstherea fter( 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 f 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 f nding the right T .] 2 Question 2 (Basic Moral Hazard - 55 points) Suppose a f
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midterms solution - Midterm 1, Econ 138 February 21, 2008 1...

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