## 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) ÷ (rate^{2 }x (1 + rate)^{periods})

Future Value = Initial Payment x ((1 + rate)^{periods }– (rate x periods) – 1) ÷ rate^{2 }

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) ÷ (0.022^{2} x (1 + 0.022)^{10}

PV = 100 x (1.022^{10 }– 0.22 – 1) ÷ (0.000484 x 1.022^{10})

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) ÷ 0.022^{2}

FV = 100 x (1.022^{10} – 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 ÷ (1.022^{10}– 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.