Geometric gradient series

A geometric gradient series 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

Geometric gradient cash flow: $A_1=100$, growth $g=5$%/yr. Each year’s payment is $A_1(1+g{)}^{n-1}$.

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

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 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 $P=N{A}_1/(1+i)$.

Most of the time you evaluate these in a spreadsheet rather than by factor lookup. Type out the cash flow column $A_1,A_1(1+g),A_1(1+g{)}^2,\dots$, discount each one with $1/(1+i{)}^t$, then sum. Mechanical, and you don’t have to worry about which factor applies.

Geometric gradients fit cash flows that escalate with Inflation (prices and wages usually grow by a percentage each year, not a constant dollar amount), revenues in a growing market, or costs in a learning-curve industry, where each doubling cuts cost by a constant fraction (a negative geometric gradient on cost).

Special case: if each period’s cash flow scales with general inflation, then the real present worth, using the Real interest rate $i^{\prime }$ instead of nominal $i$, strips out the growth factor and the geometric gradient becomes a pure annuity in real terms. This is why 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.