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Financial Mathematics Old FM Forum

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Old 05-31-2018, 05:36 PM
actsci123451 actsci123451 is offline
Join Date: Feb 2018
Posts: 11
Default PV perpetuity question

At an annual effective interest rate of i, i > 0%, the present value of a perpetuity paying 10 at the end of each 3-year period, with the first payment at the end of year 6, is 32.

At the same annual effective rate of i, the present value of a perpetuity- immediate paying 1 at the end of each 4-month period is X.

Calculate X.

Can someone please show me how to solve this using the formula for the PV of a perpetuity immediate, X/i ? The answer is 39.8

(I am aware the solutions solve it using a/1-r, just trying to understand the X/i formula better). Thanks in advanced
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Old 05-31-2018, 06:27 PM
Academic Actuary Academic Actuary is offline
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The X/i formula is based upon being one period away from the first payment where i is the effective rate per payment interval. General perpetuities are best solved by summing the infinite series to get a formula for the PV. In this case 10 v^6/(1-v^3) = 32. It looks like a quadratic where v^3 = X.

You can use the perpetuity formula to get X = 1/i' where i' is the effective rate for 4 months or i upper 3 over 3 which can be found from the v from the original equation.
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Old 06-01-2018, 12:00 PM
actsci123451 actsci123451 is offline
Join Date: Feb 2018
Posts: 11

Thanks for this, Academic Actuary. I have something else I was hoping I could get clarified as well.

I am having trouble understanding the idea that the PV of a perpetuity immediate following any payment is the same. Take the above problem for example: Since the payments start at time 6, if I were to use the X/i formula to get the PV at time 0, I would discount by v. If I were to get the PV of the payments from time 9 onwards, I would need to discount by v^2. How are these PVs the same?

Also, does summing the infinite series always give the PV of the perpetuity at time 0?

Thanks for any help.
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Old 06-01-2018, 12:36 PM
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Breadmaker Breadmaker is offline
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For a perpetuity immediate of 1 at an interest rate of 10%, the PV is 1/0.1 = 10. Now take that 10 and invest at 10% for 1 year. At the end of year 1, we have an accumulated balance of 11 and pay out 1 for an end balance of 10. Reinvest the 10 over and over and over and over...
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