TVM of a perpetual annuity starting in the future

CFA Level I Quant
#1

Hi All, I am challenged by this practice question:

“A perpetual preferred stock makes its first quarterly dividend payment of $2.00 in five quarters. If the required annual rate of return is 6% compounded quarterly, the stock’s present value is closest to …”

  • We know the quarterly rate is annual rate / number of periods so 6% / 4 = 1.5%

  • We know the present value of the perpetual annuity is Cash flow / rate so 2 / 1.5% = 133.33

However because the cash flows start 5 quarters from now we need to discount this value of 133.33. I would have discounted 5 periods (from year 5 to 4, year 4 to 3, year 3 to 2, year 2 to 1, 1 to today) and visibly this is wrong and we should discount only 4 periods.

Can anyone explain why we discount 4 quarters and not 5?

#2

The $133.33 is the value as at time 4. The formula you have used is for an immediate perpetuity. You can calculate the value as at time 5, which would be 1.015 *2 / 0.015.

#3

We discount 4 quarters, because cash flow begin @ beginning of 5th quarter, which s end of 4th quarter.

#4

^ Incorrect: the problem states that 5 quarters have passed before the first dividend was paid. Using time 4 as the focal date for the calculation requires use of the immediate perpetuity formula.