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Unformatted text preview: Intertemporal Choice John Rust Econ 425 University of Maryland <http://gemini.econ.umd.edu/jrust/econ425/intertemporal.pdf> March, 2008 1 Interest Rates and Present Values Defnition A simple loan contract involves lending a certain amount L to a borrower at the beginning of the year in exchange, the borrower repays L ( 1 + r ) at the end of the year, where r > is the interest rate Defnition The present value of a payment of L at the end of the year is the amount a lender would pay to a borrower now in exchange for receiving L at the end of the year. This is given by: PV ( L ) = L ( 1 + r ) (1) Notice that if the lender lent this amount at interest rate r , then the total amount to be repaid at the end of the year would be L ( 1 + r ) PV ( L ) = ( 1 + r ) L ( 1 + r ) = L (2) Thus, the present value of a payment of L one year from now assures that a lender will get the rate of return r on the amount L / ( 1 + r ) loaned today. 2 Present Values of Streams of Payments Suppose someone were to offer me a stream of payments of P today, P 1 one year from now, P 2 two years from now, etc. What is the present value of this stream of payments? PV ( P ,..., P T ) = P + P 1 ( 1 + r ) + P 2 ( 1 + r ) 2 + + P T ( 1 + r ) T = T t = P t ( 1 + r ) t (3) What is the present value of 5 equal annual payments of $300 at a 10% interest rate? PV ( , 300 , 300 , 300 , 300 , 300 ) = 300 ( 1 . 1 ) + 300 ( 1 . 1 ) 2 + + 300 ( 1 . 1 ) 5 = 1137 . 236 (4) 3 Internal Rates of Return DeFnition: The internal rate of return on a stream of payments ( P ,..., P T ) whose present value is PV ( P ,..., P T ) is the interest rate r that solves PV ( P ,..., P T ) = T t = P t ( 1 + r ) (5) Example Suppose a bank is willing to lend me $1,000 today in exchange for 5 annual payments of $300. What is the internal rate of return that the bank is earning on this loan? Solution: We seek the interest rate r that solves 1000 = 300 ( 1 + r ) + 300 ( 1 + r ) 2 + + 300 ( 1 + r ) 5 (6) You can check (using a calculator) that the r that solves this is r = . 15238237 . 4 Discount Factors and Geometric Series Defnition: The discount factor corresponding to interest rate r is ( r ) = 1 ( 1 + r ) (7) Note that if r > , then < < 1 . Discount actors are between and 1 We can write the present value of a stream of payments using discount factors: PV ( P ,..., P T ) = T t = t P t (8) Defnition: A Geometric Series is the sum T t = t = 1 + + 2 + + T (9) A geometric series T t = t represents the present value of a stream of payments of $1 per year from now until year T at interest rate r = 1 /  1 . 5 Useful Facts about Geometric Series Fact 1 T t = t = 1 T + 1 ( 1 ) (10) Proof of Fact 1 ( 1 ) T t = t = T t = o t T t = t = h 1 + + + T i h 1 + + + T i = h 1 + + + T i h + 2 + T + 1 i = 1...
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 Spring '08
 Hulten

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