# Present value of perpetual GROWING periodic payments

There are multiple sites that post a formula for the present value (PV) of a perpetual EQUAL periodic payment:

PV = a / ((1 + i)^t - 1)

where a (in \$) is the value of the periodic payment, and t (in years) is the period. In other words, the instrument generates a payment of \$a every t years. i is the discount interest rate (as a decimal fraction).

My current task differs slightly. The payment is generated periodically (every t years), but the payment amount grows at an annual growth rate (g). For the avoidance of doubt, the growth rate g is ANNUAL.

Unfortunately, I do not have the math skills to construct a formula for my case. My guess would be that the annual growth rate g can be subtracted from the discount rate i. Hence:

PV = a / ((1 + i - g)^t - 1)

Is this correct? Thank you very much.

• I am curious about the background of this question. Is a bank or broker offering you such a product or is this hypothetical? – JTP - Apologise to Monica Feb 8 '20 at 22:22
• @JTP I'm pretty sure the lotteries' 30 year annuity option pays like this. maybe OP is a jackpot winner – Daniel Feb 9 '20 at 13:52
• Maybe, but 30 yrs is not quite perpetual. – JTP - Apologise to Monica Feb 9 '20 at 14:02
• @ JTP. I am toying with selling a conservation easement on my tree farm. If I sell it, I will not be allowed to participate in the forest riparian easement program. Hence, I will be losing the income from that program. The program pays a certain percent of the price of the timber left in the riparian area. This income can be generated at every harvest (i.e., every t years, where t is a rotation cycle). My first payment is scheduled to come very soon (“now”). Long-term, the prices for timber grow (g). – Tree Fever Feb 9 '20 at 18:34
• Is the growth rate multiplicative or additive? That is, is a certain amount being added to the payments each year, or are the payments being multiplied by a certain amount each year? – Acccumulation Feb 9 '20 at 19:35

## 3 Answers

The only reason you can calculate the PV of perpetual payments is because of the discount rate; although you have an infinite number of payments, the present value of each payment is decreasing, leading to the values summing to a finite total. If you have growing payments, then if the growth exceeds the discount rate, then overall the present value of each payment is more than the previous, so the total will be infinite.

If the growth rate is less than the discount rate, then it's the ratio, not the difference, that should be used.

We are dealing with a geometric series. The formula for an infinite geometric series is:

a/(1-r)

Where r is the amount by which each term is multiplied. Suppose the payment at year 0 is a, and at year 1 it's a+ag. That's equivalent to a(1+g); the payment is being multiplied by (1+g). When we discount the payment, on the other hand, we divide; we should have a/(1+i). So the total factor for one year is (1+g)/(1+i). For t years, it's [(1+g)/(1+i)]^t. So the formula for the total r is:

r = [(1+g)/(1+i)]^t

and the formula for the sum is:

a/(1-[(1+g)/(1+i)]^t)

• Nanoman, and Chris Degnen: Thank you very much for your help! – Tree Fever Feb 10 '20 at 17:51

With the assumption that the payments are made at the end of each t-year period (i.e., the first payment is made, not now, but t years from now), and that the growth at rate g starts now (i.e., the first payment equals a(1 + g)^t), your formula is roughly correct. The effect of annual growth on valuation corresponds to a reduction in the discount rate.

However, if these rates are not very small, they should be combined multiplicatively rather than additively. That is, in place of (1 + i - g)^t you should have (1 + i)^t / (1 + g)^t.

Also, if the first payment (t years from now) equals a rather than a(1 + g)^t, then simply divide the entire PV formula by (1 + g)^t.

• Dear Nanoman: Thank you very much for your help. The payments are scheduled every t years, with the first one being due immediately, and the annual growth g of payments starts immediately. Now (time 0): Payment a t years: Payment a*(1+g)^t 2t years: Payment a*(1+g)^2t Etc. Did I understand you correctly that PV in this case should be calculated as follows: PV = a + a / ((1 + i )^t / (1 + g)^t - 1) = a (1 + 1/((1 + i )^t / (1 + g)^t - 1) ? Thank you again for your kind help. – Tree Fever Feb 9 '20 at 18:07
• Sorry, the formating in commets is rather restrictive. Let's me re-post mine: Dear Nanoman: Thank you very much for your help. The payments are scheduled every t years, with the first one being due immediately, and the annual growth g of payments starts immediately. Now (time 0): payment a; t years: payment a*(1+g)^t; 2t years: payment a*(1+g)^2t; etc. Did I understand you correctly that PV in this case should be calculated as follows: PV = a + a / ((1 + i )^t / (1 + g)^t - 1) = a (1 + 1/((1 + i )^t / (1 + g)^t - 1) ? Thank you again for your kind help. – Tree Fever Feb 9 '20 at 18:20

Where `a` is paid every `t` years, and `i` is the annual effective interest rate, the periodic rate `r` is

``````r = (1 + i)^t - 1
``````

Then the present value of a perpetuity with constant payments is given by these two equivalent formulae. The second matches the OP's first formula.

$\sum_{k=1}^{\infty }\frac{a}{(1+r)^k}=\frac{a}{r}$

$\sum_{k=1}^{\infty }\frac{a}{(1+i)^{k\cdot t}}=\frac{a}{(1+i)^t-1}$

The present value of a perpetuity with growing payments is given by these two equivalent formulae, where `g` is the annual growth rate and `h` is the periodic growth rate.

``````h = (1 + g)^t - 1
``````

$\sum_{k=1}^{\infty }\frac{a(1+h)^{k-1}}{(1+r)^k}=\frac{a}{r-h}$

$\sum_{k=1}^{\infty }\frac{a(1+h)^{k-1}}{(1+i)^{k\cdot t}}=\frac{a}{(1+i)^t-1-h}$

See also Perpetuity with growth formula which matches the third formula: `PV = a/(r - h)`.

Conclusion

The formula you require can be obtained by substituting `h` in the fourth summation or formula.

$\sum_{k=1}^{\infty }\frac{a(1+g)^{t(k-1)}}{(1+i)^{k\cdot t}}=\frac{a}{(1+i)^t-(1+g)^t}$

``````PV = a/((1 + i)^t - (1 + g)^t)
``````

or even more simply, from `PV = a/(r - h)`

where `r = (1 + i)^t - 1` and `h = (1 + g)^t - 1`.

The summations help clarify how the PV formulas are arrived at.

• Dear Chris Degden: Thank you very much for your help. Based on Nanoman’s message, the formula I am looking for seems to be: PV = a + a / ((1 + i )^t / (1 + g)^t - 1) = a (1 + 1/((1 + i )^t / (1 + g)^t - 1) Whereas based on your message, it is PV = a + a / ((1 + i )^t - (1 + g)^t - 1) = a (1 + 1/((1 + i )^t - (1 + g)^t - 1) In other words, the two of you seem to deal differently with how the two rates (i and g) interact. Which is the correct formula? Thank you very much! – Tree Fever Feb 9 '20 at 18:42
• If you add values, e.g. `a = 100, i = 0.1, g = 0.05, t = 3`, both my `PV = a/((1 + i)^t - (1 + g)^t)` and nanoman's `PV = (a/((1 + i)^t/(1 + g)^t - 1))/(1 + g)^t` produce the same result: 576.784. I think they are both correct. – Chris Degnen Feb 9 '20 at 19:07