LAST UPDATE: September 25th, 2020

## 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) ÷ (rate2 x (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).