# H2 - ) 1 ( 1 1 r r r C PV (6) ∞-= 1 1 r C PV Or, finally,...

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Annuities and Perpetuities: Present Value William L. Silber I. The present value of an annuity, PV , can be written as the sum of the present values of each component annual payment, C , as follows: (1) t r C r C r C PV ) 1 ( ) 1 ( 1 2 + + + + + + = L where r is the single average interest rate per annum and t is the number of years the annuity is paid. This can be simplified as follows: (2) + + + + + = t r r r C PV ) 1 ( 1 ) 1 ( 1 1 1 2 L . Using a formula for the sum of a geometric progression (as long as 0 r ), we have: (3) + - = - r r C PV t ) 1 ( 1 , which is the same as: (4) + - = t r r r C PV ) 1 ( 1 1 II. Thus if you have a three-year annuity ) 3 ( = t that pays \$100 per annum ) 100 \$ ( = C and the average annual interest rate, r, is 6 percent, then from equation (4), we have:

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30 . 267 \$ ) 06 . 1 ( 06 . 1 06 . 1 00 . 100 \$ 3 = - = PV You can check that this is correct by calculating: 30 . 267 \$ ) 06 . 1 ( 100 \$ ) 06 . 1 ( 100 \$ 06 . 1 100 \$ 3 2 = + + = PV III. More interesting is what happens to the present value formula when the annual payments, C , continue forever. The annuity becomes a perpetuity as t and the formula in (4) becomes: (5) + - =
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Unformatted text preview: ) 1 ( 1 1 r r r C PV (6) ∞-= 1 1 r C PV Or, finally, (7) r C PV = IV. Equation (7) is very simple. It says that the present value of an annuity of C dollars per annum is C divided by r, where r is the average interest rate per annum. This makes considerable sense once you provide a numerical example. Suppose C =\$10 per annum and the interest rate is .05, or 5 percent. How many dollars, designated by the letter P, would you have to put away today so that it produces \$10 in each year forever? The answer is given by solving the following formula for P: . 200 \$ 05 . 10 \$ 10 \$ 05 . = = = × P P Investing \$200 at 5 percent generates \$10 in interest per year and continues to do so forever. Thus, if an annuity promises to pay \$10 forever and the annual interest rate is 5 percent, the value of that infinite stream of payments is \$200. If the annuity were priced in a competitive market its price should be \$200...
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## This note was uploaded on 02/04/2010 for the course ECON 106v taught by Professor Miyakawa during the Spring '08 term at UCLA.

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H2 - ) 1 ( 1 1 r r r C PV (6) ∞-= 1 1 r C PV Or, finally,...

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