Perpetuity Question

Q. 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:

  1. $31.
  2. $126.
  3. $133.

B is correct. The value of the perpetuity one year from now is calculated as:

PV = A/r, where PV is the present value, A is the annuity, and r is expressed as a quarterly required rate of return because the payments are quarterly.PV = $2.00/(0.06/4)PV = $133.33.

The value today is (where FV is future value) PV = FV N*(1 + _ r ) N; _ PV = $133.33(1 + 0.015)–4;

PV = $125.62 ≈ $126.

Question: I am confused about why we discount only 4 periods instead of 5 periods in the last step? Can someone help please?

the solution first calculates the value of the annuity at time t = 4 (where t = 0, 1, 2, … are in quarters) using the shortcut formula for perpetuity-immediate. it then discounts to the present value at t = 0.

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The formula PV = A/r is based on the first cash flow occuring one period from now (i.e. ordinary annuity). In other words V(0) = A(1)/r.

So if your first cash flow occurs five quarters from now, A(5), then:

A(5)/r = V(4)

And to go from V(4) to V(0), you have to discount it 4 quarters back.

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Got it! Thanks a lot!

When in doubt, draw a time line.

When not in doubt, draw a time line anyway, just in case.

yup! Thanks for the advice!