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How It Works
A uniform gradient is a cash flow pattern where payments increase by a fixed amount (G) each period. The first payment is 0, the second is G, the third is 2G, and so on through (n−1)G in the final period. This models situations like escalating maintenance costs or step-raise salary schedules.
The three gradient factors convert this linearly increasing series into its future value, present value, or an equivalent uniform annuity. These are essential tools in engineering economics for evaluating projects with growing costs or revenues.
Example Problem
Gradient Future Worth: Maintenance costs increase by $100 each year for 5 years at 5% interest. What is the future value of this gradient?
- F = $100 × [(1.05)5 − 0.05(5) − 1] / 0.05²
- (1.05)5 = 1.27628, numerator = 1.27628 − 0.25 − 1 = 0.02628
- F = $100 × 0.02628 / 0.0025 = $1,051.01
Gradient Present Worth: Same gradient discounted to today:
- P = $100 × 0.02628 / (0.0025 × 1.27628) = $823.26
Equivalent Annuity: The equivalent uniform annual payment:
- A = $100 × [1/0.05 − 5/(1.055 − 1)] = $100 × 1.9025 = $190.25
Frequently Asked Questions
What is a uniform gradient in engineering economics?
A uniform gradient is a cash flow series where each payment increases by a constant amount G over the previous period. The series starts at 0 in period 1, G in period 2, 2G in period 3, and so on. It models linearly growing costs like escalating maintenance or step-raise contracts.
How is a gradient different from a uniform series?
A uniform series has equal payments every period (A, A, A, ...). A gradient has linearly increasing payments (0, G, 2G, 3G, ...). In practice, many cash flows combine both: a base annuity plus a gradient component. You can analyze them separately and add the results.
Why does the gradient start at zero in period 1?
By convention, the gradient amount G represents the increase per period, not the first payment. Period 1 has zero gradient; period 2 has G; period n has (n−1)G. If you need a base payment plus a gradient, add a separate uniform series calculation for the base amount.
When would I use the gradient uniform series factor?
Use A = G[1/i − n/((1+i)n − 1)] to convert a gradient into an equivalent level annuity. This is useful for comparing projects with growing costs against projects with constant costs, or for converting escalating payments into a single equivalent annual cost.
Related Calculators
- Compounding and Discount Factors Calculator — single payment and uniform series compound amount factors.
- Present Worth Calculator — discount future values back to today.
- Interest Rate Calculator — simple and compound interest computations.
- Profitability Index Calculator — evaluate project viability using discounted cash flows.
- Depreciation Calculator — straight-line and declining balance depreciation methods.
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Reference: Lindeburg, Michael R. 1992. Engineer In Training Reference Manual. Professional Publication, Inc. 8th Edition.