(L08)DynamProg_ns

(L08)DynamProg_ns - Topic 8: Dynamic Programming Reading:...

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Primbs/Investment Science 1 Topic 8: Dynamic Programming Reading: Luenberger Chapter 4, Sections 6 – 7 Chapter 5, Sections 3 – 4
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Primbs/Investment Science 2 Dynamic Programming Dynamic Programming Running Present Value The Basic Recursion A Fishing Example Including Probability Rolling a Die A Card Game
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Primbs/Investment Science 3 Running Present Value x 0 x 1 x 2 x 4 x 3 x 5 A recursive method for calculating present value.
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Primbs/Investment Science 4 Running Present Value x 0 x 1 x 2 x 4 x 3 x 5 Let PV(i) = present value of all cash flows occurring at time i and later. 5 ) 5 ( x PV = ) 1 ( ) 5 ( ) 4 ( 5 , 4 4 f PV x PV + + = ) 1 ( ) 4 ( ) 3 ( 4 , 3 3 f PV x PV + + = ) 1 ( ) 3 ( ) 2 ( 3 , 2 2 f PV x PV + + = ) 1 ( ) 2 ( ) 1 ( 2 , 1 1 f PV x PV + + = ) 1 ( ) 1 ( ) 0 ( 1 0 s PV x PV + + =
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Primbs/Investment Science 5 Present Value Updating The running present values satisfy the recursion: ) 1 ( ) 1 ( ) ( 1 , + + + + = k k k f k PV x k PV Ex: Time 0 1 2 3 Cash Flow 20 25 30 35 Discount 1/(1+f k,k+1 ) 0.943 0.935 0.93 PV(k) 98.73 83.48 62.55 35 55 . 62 ) 93 (. 35 30 ) 2 ( = + = PV
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Primbs/Investment Science 6 Present Value Updating The running present values satisfy the recursion: ) 1 ( ) 1 ( ) ( 1 , + + + + = k k k f k PV x k PV Ex: Time 0 1 2 3 Cash Flow 20 25 30 35 Discount 1/(1+f k,k+1 ) 0.943 0.935 0.93 PV(k) 98.73 83.48 62.55 35 48 . 83 ) 935 (. 55 . 62 25 ) 1 ( = + = PV
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Primbs/Investment Science 7 Present Value Updating The running present values satisfy the recursion: ) 1 ( ) 1 ( ) ( 1 , + + + + = k k k f k PV x k PV Ex: Time 0 1 2 3 Cash Flow 20 25 30 35 Discount 1/(1+f k,k+1 ) 0.943 0.935 0.93 PV(k) 98.73 83.48 62.55 35 48 . 83 ) 943 (. 48 . 83 20 ) 0 ( = + = PV
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Primbs/Investment Science 8 Application to Floating Rate Bonds A bond where the coupons are tied to the current short rate of interest. After every coupon payment, the coupon rate is reset to the current short rate. Exercise: Using running present value, show that the value of a floating rate bond is equal to par at any reset date.
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Primbs/Investment Science 9 Application to Floating Rate Bonds n n-1 n-2 n-3 Assume coupon payments are yearly: F n s F n s F = - + - + ) 1 ( 1 ) 1 ( 1 1 discount F F n s F n s F = - + - + ) 2 ( 1 ) 2 ( 1 1 discount F F n s F n s F = - + - + ) 3 ( 1 ) 3 ( 1 1 discount F ... where ) ( 1 t s is the short rate when you are at time t. F n s C n ) 1 ( 1 - = ) 1 ( 1 - n s ) 2 ( 1 - n s ) 3 ( 1 - n s ) 4 ( 1 - n s F F n s C n ) 2 ( 1 - = F n s C n ) 3 ( 1 - = F n s C n ) 4 ( 1 - = F n s C n ) 5 ( 1 - =
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Primbs/Investment Science 10 Dynamic Programming Dynamic Programming Running Present Value The Basic Recursion A Fishing Example Including Probability Rolling a Die A Card Game
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Primbs/Investment Science 11 Example: Fishing Every year we decide whether to fish in a lake or not. Fish Population stays the same Don’t fish Population doubles We can represent the possible decisions and outcomes on a graph. 10 20 10 20 40 20 40 80 10 10 Next, we associate cash flows with each branch. 7 0 0 7 14 0 14 0 28 0 7 0 Your profit is 70% of the initial number of fish in the lake if you fish. Don’t Fish Fish
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Primbs/Investment Science 12 Example: Fishing 10 20 10 20 40 20 40 80 10 10 7 0 0 7 14 0 14 0 28 0 7 0 Assume cash flows occur at the beginning of each year.
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(L08)DynamProg_ns - Topic 8: Dynamic Programming Reading:...

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