Gradient Payment Calculator

Definition – What is a Gradient (Linear) Payment?

A gradient payment (also known as linear payment growth) is a payment that increases by a regular amount.

As an example, in the first time period, the payment is $100. In time period 2 it is $200, time period 3 it is $300, time period 4 it is $400.

The present value is the value in today’s dollars of the increased payment. The future value is the value of at the end of all time periods.

Formula – How the Present and Future Values of a Gradient Payment are Calculated

Present Value = Initial Gradient Payment x ((1 + rate)^periods– (rate x periods) – 1) ÷ (rate²x (1 + rate)^periods)

Future Value = Initial Payment x ((1 + rate)^periods– (rate x periods) – 1) ÷ rate²

Equivalent Equal Payment = Initial Payment x ((1 ÷ rate) – (periods ÷ ((1 + rate)^periods– 1

Where:

Initial Gradient Payment is the initial payment that is made. It is also the amount that the payment increases each period. An initial payment of $100 will be $200 in period 2, $300 in period 3, and so on. For a percentage increase see our increasing payment calculator.
Rate is the rate used to compound each period.
Periods is the number of compounding periods and payments that are made.

Example (Present Value)

An initial payment of $100 increases by the same amount each period over 10 periods at a compounding rate of 2.2%.

Present Value = 100 x ((1 + 0.022)¹⁰ – (0.022 x 10) – 1) ÷ (0.022² x (1 + 0.022)¹⁰

PV = 100 x (1.022¹⁰– 0.22 – 1) ÷ (0.000484 x 1.022¹⁰)

PV = 100 x (1.2431083 – 0.22 – 1) ÷ (0.000484 x 1.2431083)

PV = 100 x 0.0231082 ÷ 0.00060166

PV = 100 x 38.4074

PV = 3,840.74

Example (Future Value)

FV = 100 x ((1 + 0.022)¹⁰ – (0.022 x 10) – 1) ÷ 0.022²

FV = 100 x (1.022¹⁰ – 0.22 – 1) ÷ 0.000484

FV = 100 x (1.2431083 – 0.22 – 1) ÷ 0.000484

FV = 100 x 0.0231083 ÷ 0.000484

FV = 100 x 47.7444

FV = 4,774.44

Eample (Equivalent Equal Payment)

Equivalent Equal Payment = 100 x ((1 ÷ 0.022) – (10 ÷ ((1 + 0.022)¹⁰– 1)))

Equivalent Equal Payment = 100 x (45.4545 – (10 ÷ (1.022¹⁰– 1)))

Equivalent Equal Payment = 100 x (45.4545 – (10÷ (1.24310828 – 1)))

Equivalent Equal Payment = 100 x (45.4545 – (10 ÷ 0.24310828))

Equivalent Equal Payment = 100 x (45.4545 – 41.1339342)

Equivalent Equal Payment = 100 x 4.3206112

Equivalent Equal Payment = 432.06

FAQ

What is the difference between a gradient payment and an annuity?

An annuity is a regular payment amount. In a financial stream with 10 periods and a payment of $100, each payment is $100. There is no increase or decrease.

With a gradient payment (or gradient annuity) the payment increases by a regular amount each payment period. In a financial stream with 10 periods and a starting payment of $100, the first payment is $100, the second payment is $200, the third payment is $300, until the final payment at $1,000.

The two calculations can be combined together to find a net present value. An example problem would be to find the present value of a payment of $10,000 per year, increasing by $1,000 per year (year 1 is $10,000, year 2 is $11,000, year 3 is $12,000, year 4 is $13,000). This problem could be solved by finding the present value of an annuity of $9,000 over 4 years, and the present value of a gradient payment with an initial payment of $1,000 over the 4 years.

What is the difference between the present value and future value of a gradient payment?

The present value is the value in today’s dollars of the payment stream. The future value is the value at a specific point in the future.

What is the difference between a gradient payment and an exponential payment?

A gradient payment increases by a regular amount each payment period. An exponential payment increases by a set percentage each payment period.

With a gradient payment, an initial payment of $100 would increase by $100 each payment period. The first payment would be $100, the second $200, the third $300, and the fourth $400.

With an exponential payment, an initial payment of $100 increasing by 5% each payment period would have the first payment be $100, the second $105 ($100 + 5% of $100), the third $110.25 ($105 + 5% of $105), and the fourth $137.81 ($110.25 + 5% of $110.25).

Sources and External Resources

Wikipedia – Time Value of Money – An explanation of the time value of money. Includes the formula for the present and future values of a gradient payment.
TVMCalcs.com – Graduated Annuities using Excel – An introduction to calculating graduated annuities using an excel spreadsheet.
Professor Ronald H. Randals of the Department of Statistics at the University of Florida – Statistics 4183 Lecture Slides Chapter 4 – An explanation of the formulas behind payments.
Dr. Roger Bales of the University of California MERCED School of Engineering – Engineering 155 (Engineering Economic Analysis) – Week 3 Slides – A set of slides detailing formulas for both linear gradient and geometric gradient payments.

Gradient Payment Calculator