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# Economics: Annuities and Perpetuities

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A perpetuity-immediate pays \$10 at the end of every 3-year interval, with the first imbursement at the end of year 6, and a present value (PV1) of \$32.  Find the present value (PV2) of a perpetuity-immediate paying \$1 at the end of every 4 months, with the same effective interest rate.

Jul 9, 2017

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A perpetuity-immediate pays \$10 at the end of every 3-year interval, with the first imbursement at the end of year 6, and a present value (PV1) of \$32.  Find the present value (PV2) of a perpetuity-immediate paying \$1 at the end of every 4 months, with the same effective interest rate.

This is a strange perpetuity!!. It only pays \$10 every 3 years? It is obviously here for the purpose of doing the calculations involved.

Since the first part has PV of \$32, that means that in order for the perpetuity to earn \$10 at the end of year 6, it has to earn: \$10/\$32=0.3125 or 31.25% over the first 6 years. Or 1.3125^1/6 =4.64% compounded annually. But for subsequent years, your perpetuity would have to earn: 1.3125^1/3 =9.49% in order to earn \$10 at the end of year 3, after the first payment, and continue that perpetually.

The rate of 9.49% is an annual rate. Will have to convert this to a 4-months rate for the 2nd part of your question. So, 1.0949^1/3 =3.07% compounded every 4 months, or 3 times a year.

So the PV of your 2nd  perpetuity would be: \$1/0.0307 =\$32.59

Jul 9, 2017
edited by Guest  Jul 9, 2017
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The solution should resolve to \$39.84.

Melody and the obsequious Morgan Tud present two methods to solve this question. Melody’s method is straightforward and easier to understand. Tud derives a quadratic equation from the two relations. It’s a brilliant method, but not intuitive for most persons trying to solve these types of questions.

https://web2.0calc.com/questions/perpetuity_1

(The dialogue between Chirurgeon Tud and Sir CPhill is quite funny, too.)

GA

Jul 10, 2017
edited by Guest  Jul 10, 2017