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.
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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.
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.
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
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.
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.
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)
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