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#1




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 3year 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 4month 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/1r, just trying to understand the X/i formula better). Thanks in advanced 
#2




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/(1v^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. 
#3




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. 
#4




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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