Idea Transcript
Delayed Perpetuities and Annuities The equations for a perpetuity and annuity are derived from the assumption that the first cash flow will occur 1-‐period ahead in time. Perpetuity equation:
PV = Annuity equation: !" =
C1 r!g
! 1 1− ! (1 + !)!
We typically think of “PV” as the value of the cash flows today, or time t=0. Note however, “PV” can occur at different points in time. Most simply, if you are to receive $1,000 10 years from today, what is it’s value 9 years from today? You would discount that 1K back one period and have PV9 = C10/(1+r)1, or PV9 = 1000/(1+r)1. This is a “PV” that is not at time t=0. So, just keep in mind that “present values” doesn’t always refer to “today.” A “delayed perpetuity” is a perpetuity that does not start its cash flow stream one period from today. Suppose a $1000 perpetuity starts at 4 years from today, and r=5% and g=2%. If one were to simply follow the above formula and solve 1000/(.05-‐.03) = $50,000, you would NOT have the PV today. Instead, the $50,000 amount is the PV of the stream at t=3. So, the PV of the stream today (t=0) is 50,000/(1.05)3 = $43,191.88. It is a two-‐step process. In general, the PV today (at time=0) of a perpetuity beginning at time period t is:
" Ct % $ ' #r ! g& PV0 = where the numerator is the PV of the perpetuity at time t-‐1, so we have to discount (1+ r)t!1 that amount back by the same number of periods to get the PV today. We would handle a delayed annuity in the same manner. If we have an annuity starting at time period t, using the above formula for an annuity would give you the PV at time t-‐1, and so we would then have to discount it back by t-‐1 periods to get the PV at time t=0.
Example question: Module 1.4 (#6):
This is a perpetuity delayed just 1-‐period (it starts at time t=2). PV at t1 = 175,000/(.1-‐.035) = 2,692,307.69. So, PV 1 today (at t=0) is PV = 2692307.69/(1.1) = $2,447,552.45