Gradient Cash Flows: P/G

A uniform gradient of cash flows occurs when a series of cash flows increases or decreases by the same amount every period. For example, if costs increase by $200 every year for the next 8 years, we have a uniform gradient.

Gradients are either positive or negative, as shown below.

Gradients can be confusing, although in the generic cases shown above we'd say that the gradient occurs in years 1 through N there is no cash flow in year 1.

If we had the cash flow given below, we can use the P/G factor to find the present equivalent (i=8%).

P0 = 50(P/G, 8%, 5)

For more complex cases, it is usually easiest to decompose a cash flow into two parts: an annuity and a gradient so that

P = A(P/A, i, N) + G(P/G, i, N)


A = the magnitude and sign of the annual cash flow (equal to the magnitude of the first observed cash flow,

G = the magnitude and sign of the gradient cash flow

i = the interest rate

N = the number of periods over which the annuity and gradient occur.