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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 β ( r ) = 1 ( 1 + r ) (7) Note that if r > , then < β < 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 = β t P t (8) • Defnition: A Geometric Series is the sum T ∑ t = β t = 1 + β + β 2 + ··· + β T (9) • A geometric series ∑ T t = β t represents the present value of a stream of payments of $1 per year from now until year T at interest rate r = 1 / β 1 . 5 Useful Facts about Geometric Series • Fact 1 T ∑ t = β t = 1 β T + 1 ( 1 β ) (10) • Proof of Fact 1 ( 1 β ) T ∑ t = β t = T ∑ t = o β t β T ∑ t = β t = h 1 + β + ··· + β T i β h 1 + β + ··· + β T i = h 1 + β + ··· + β T i h β + β 2 ··· + β T + 1 i = 1...
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This note was uploaded on 10/25/2011 for the course ECON 326 taught by Professor Hulten during the Spring '08 term at Maryland.
 Spring '08
 Hulten

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