Geometric gradient series

A geometric gradient series is a cash-flow pattern that grows (or shrinks) by a constant percentage $g$ each period. If the first-period cash flow is $A_1$ at time 1, then the cash flow in period $t$ is

$A_t=A_1(1+g{)}^{t-1}.$

The total present worth at rate $i$ over $N$ periods has a closed form. Let $g^{\prime }=\frac{1+i}{1+g}-1$ (the adjusted interest rate that’s left after stripping growth). Then

$P=\frac{A_1}{1+g}\cdot (P/A,g^{\prime },N)\text{(when }g\ne i).$

When $g=i$ exactly (growth matches interest), the discount and growth exactly cancel and the present worth is $P=N{A}_1/(1+i)$.

In practice geometric gradients are usually evaluated in a spreadsheet rather than by factor lookup. Type out the cash flow column $A_1,A_1(1+g),A_1(1+g{)}^2,\dots$ and discount each one with $1/(1+i{)}^t$, then sum. This is mechanical, transparent, and avoids confusion about which factor applies.

Geometric gradients are the natural model for cash flows that escalate with Inflation (prices and wages typically grow by a percentage each year, not a constant dollar amount), for revenues in a growing market, or for costs in a learning-curve industry (where each doubling reduces cost by a constant fraction — a negative geometric gradient on cost).

Special case: if the geometric gradient describes a stream where each period’s cash flow scales with general inflation, then the real present worth of the stream — using the Real interest rate $i^{\prime }$ instead of nominal $i$ — has the growth factor stripped out, and the geometric gradient becomes a pure annuity in real terms. This is one reason inflation analyses often switch between real and actual dollars: a geometric gradient in actual dollars can be a uniform annuity in real dollars.

For the linear cousin (constant additive increase per period), see Arithmetic gradient series.