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# answers-5 - in&nite amounts to get in&nite...

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Assignment for Week Five: Answer Key Anton Cheremukhin October 30, 2008 Di/erence Equations a) x t = 2 x t ° 1 + 1 steady-state: x ss = ° 1 , unstable b) x t = 0 : 5 x t ° 1 ° 2 steady-state: x ss = ° 4 , stable 1

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c) x t = ° 2 x t ° 1 + 1 steady-state: x ss = 1 = 3 , unstable d) x t = ° 0 : 5 x t ° 1 ° 2 steady-state: x ss = ° 4 = 3 , stable 2
e) x t = x t ° 1 + 1 no steady-state. f) x t = ° x t ° 1 ° 2 steady-state: x ss = ° 1 , cycle, neither stable or unstable. 3

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g) x t = x t ° 1 steady-state: any point, neither come back to it nor explode. Arrow Securities A1) P = Q A ± D A + Q B ± D B + Q C ± D C = 0 : 2 ± 1 + 0 : 6 ± 1 + 0 : 1 ± 1 = 0 : 9 A pure discount bond pays you 1 dollar in each state of the world, so its price is equal to the discount factor. A2) P = Q A ± D A + Q B ± D B + Q C ± D C = 0 : 2 ± 10 + 0 : 6 ± 10 + 0 : 1 ± 0 = 8 B) You can sell such a bond for 1\$, and then use 90 cents to buy the three arrow securities, which would make sure you get back a dollar tomorrow, which you then pay to the person you sold the bond to. You are left with 10 cents today for free. In real markets if there are other instruments, you should buy something else (which would discount future), and sell this discount bond. Both in
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Unformatted text preview: in&nite amounts, to get in&nite pro&ts today. Perpetuity P = c 1+ r + c (1+ r ) 2 + c (1+ r ) 3 + ::: = & 1 t =1 c (1+ r ) t = c 1+ r 1 1 & 1 1+ r = c r = : 1 : 1 = 1 A perpetuity pays you a coupon forever. A price of a perpetuity is equal to the price of a bond that pays a coupon for T periods and then returns the principal (see next question). Ten-year Bond Assume the coupon is 10% of the principal, which implies a principal of 1000\$. PV ( coupon ) = & T t =1 c (1+ r ) t = c 1+ r 1 & ( 1 1+ r ) T 1 & 1 1+ r PV ( principal ) = c=r (1+ r ) T PV ( bond ) = c 1+ r 1 & ( 1 1+ r ) T 1 & 1 1+ r + c=r (1+ r ) T = c r & 1 ± ± 1 1+ r ² T ³ + c=r (1+ r ) T = c r = 100 : 1 = 1000 Here, & 10 t =1 100 (1+0 : 1) t = 614 : 46 , 100 = : 1 (1+0 : 1) 10 = 385 : 54 ; Hence, PV ( bond ) = 614 : 46 + 385 : 54 = 1000 4...
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