Lecture_Note_on_growing__annuities_and_H2 - HAAS SCHOOL OF...

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AAS S CHOOL OF B USINESS U NIVERSITY OF C ALIFORNIA AT B ERKELEY UGBA 103 S UMMER 2008 © A VINASH V ERMA T EACHING N OTE : P ERPETUITIES , A NNUITIES , AND G ROWING A NNUITIES Please make sure you read Homework 2 at the end of this note. A perpetuity is a stream of constant cash flows every period that goes on for ever. Suppose you have $50,000 and you deposit it in a bank account that pays interest at 6% a year for all time to come. Then you, and, after you, your heirs, will get $3000 [$50,000 times 6%] every year in perpetuity. You have therefore exchanged $50,000 now at t=0 for $3000 every year in perpetuity staring at t= 1. Since both you and the bank entered into the transaction voluntarily, it must be the case that the value today of that perpetual stream of payments is $50,000. Suppose we now want to find out what amount of deposit would generate annual cash flows of $12,000 in perpetuity. We shall arrive at the answer by following in reverse the steps by which we arrived at the amount of annual interest. That is, we shall divide the targeted cash flow of $12,000 by the annual interest rate of 6%, and get the answer as $200,000. To recapitulate, when we want to find out the annual (per period) interest payment, denoted C , we multiply the amount deposited at t= 0 by the interest rate per year (period). Therefore, when we are given the annual cash flow and the annual interest rate and want to work out the amount of deposit that would generate those cash flows, or, in other words, the present value of those cash flows, we shall of course have to reverse the process. It thus follows that: ( 29 r C C PV = perpetuity in period every $ Notice that we have used no algebra for arriving at this result. 1 The important thing to remember about this very simple result is that the point of valuation is one period before the point in time when the first cash flow occurs. The economic intuition is obvious: our 1 The result does have both an origin and a foundation in algebra: ( 29 ( 29 ( 29 ( 29 + + + + + + = ... 1 1 1 perpetuity in period every $ 3 2 r C r C r C C PV is an infinite geometric progression that converges for 0 r . The formula for sum of an infinite geometric progression is Σ k a * f k = a /(1 – f ) where a is the first term of the series, f (<1) is the factor by which terms in the series increase, and k is an index that runs from 0 to ∞. In our case, the first term is a = C/(1+r), and the factor by which the terms increase is f = 1/(1+r). Substituting these in the formula, we get the same result. Page 1 of 6
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This note was uploaded on 02/12/2012 for the course UGBA 101A taught by Professor Mccullough during the Spring '08 term at University of California, Berkeley.

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Lecture_Note_on_growing__annuities_and_H2 - HAAS SCHOOL OF...

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