Geometric Increasing Perpetuity

A geometric increasing perpetuity is a perpetuity whose payments grow at a constant rate g per period: 1, (1+g), (1+g)^2, \ldots forever. Paid at the end of each period, with the first payment one period from now, its present value at an effective rate i greater than g is

PV = \frac{1}{i-g}

• It is the geometric increasing annuity \left(1 - \left(\tfrac{1+g}{1+i}\right)^n\right)/(i-g) with n \to \infty: when g < i the ratio \left(\tfrac{1+g}{1+i}\right)^n goes to zero. When g \geq i the payments grow at least as fast as discounting shrinks them, and the perpetuity has no finite value.
• For a first payment P, multiply by P. Paid at the start of each period (a perpetuity-due, first payment now), each payment comes one period earlier, so the value is (1+i)/(i-g).
• The same formula values a stock whose dividends are expected to grow at a constant rate, with i the rate the investor requires.
• For payments rising by a fixed amount rather than a fixed percentage, see Increasing Annuity.

An Endowment Payout Growing with Inflation An endowment is to pay $5,000 at the end of the first year, with each later payment 2% more than the one before, forever. At i = 6\%, how much must be endowed today? How much if the first payment is made today instead?

Answer

\begin{align*}
PV &= \frac{5{,}000}{0.06 - 0.02} \\
&= 125{,}000 \\
PV_{\text{due}} &= 1.06 \times 125{,}000 \\
&= 132{,}500
\end{align*}

Without the growth the endowment would need only 5{,}000/0.06 = 83{,}333.33.

Level Payments, Then Growth A perpetuity-immediate pays $100 at the end of each of years 1 to 3. From year 4 on, each payment is 3% larger than the one before, so the year-4 payment is $103. Find its present value at i = 8\%.

Answer Value the growing part at time 3, one period before its first payment, then discount it to time 0:

\begin{align*}
PV_3 &= \frac{103}{0.08 - 0.03} \\
&= 2{,}060 \\
PV &= 100\,a_{\overline{3}|0.08} + 2{,}060\,(1.08)^{-3} \\
&= 100(2.577097) + 2{,}060(0.793832) \\
&= 257.71 + 1{,}635.29 \\
&= 1{,}893.00
\end{align*}