LAST UPDATE: October 15th, 2019

### 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.

### Gradient Payment Formula – How the Present and Future Values of a Gradient Payment are Calculated $\text{Present Value} = \text{Initial Gradient Payment} \times \frac{(1 + \text{rate})^{\text{periods}} - (\text{rate} \times \text{periods}) - 1}{\text{rate}^2 \times (1 +\text{rate})^{\text{periods}}}$ $\text{Future Value} = \text{Initial Payment} \times \frac{(1 + \text{rate})^\text{periods} - (\text{rate} \times \text{periods}) - 1}{\text{rate}^2}$ $\text{Equivalent Equal Payment} = \text{Initial Payment} \times (\frac{1}{\text{rate}} - \frac{\text{periods}}{(1 + \text{rate})^\text{periods} - 1})$

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%. $\text{Present Value} = \text{Initial Gradient Payment} \times \frac{(1 + \text{rate})^{\text{periods}} - (\text{rate} \times \text{periods}) - 1}{\text{rate}^2 \times (1 +\text{rate})^{\text{periods}}}$ $\text{PV} = 100 \times \frac{(1 + 0.022)^{10} - (0.022 \times 10) - 1}{0.022^2 \times (1 +0.022)^{10}}$ $\text{PV} = 100 \times \frac{1.022^{10} - 0.22 - 1}{0.000484 \times 1.022^{10}}$ $\text{PV} = 100 \times \frac{1.2431083 - 0.22 - 1}{0.000484 \times 1.2431083}$ $\text{PV} = 100 \times \frac{0.0231082}{0.00060166}$ $\text{PV} = 100 \times 38.4074$ $\text{PV} = 3,840.74$

### Example (Future Value) $\text{Future Value} = \text{Initial Payment} \times \frac{(1 + \text{rate})^\text{periods} - (\text{rate} \times \text{periods}) - 1}{\text{rate}^2}$ $\text{FV} = 100 \times \frac{(1 + 0.022)^{10} - (0.022 \times 10) - 1}{0.022^2}$ $\text{FV} = 100 \times \frac{1.022^{10} - 0.22 - 1}{0.000484}$ $\text{FV} = 100 \times \frac{1.2431083 - 0.22 - 1}{0.000484}$ $\text{FV} = 100 \times \frac{0.0231083}{0.000484}$ $\text{FV} = 100 \times 47.7444$ $\text{FV} =4,774.44$

### Eample (Equivalent Equal Payment) $\text{Equivalent Equal Payment} = \text{Initial Payment} \times (\frac{1}{\text{rate}} - \frac{\text{periods}}{(1 + \text{rate})^\text{periods} - 1})$ $\text{Equivalent Equal Payment} = 100 \times (\frac{1}{0.022} - \frac{10}{(1 + 0.022)^{10} - 1})$ $\text{Equivalent Equal Payment} = 100 \times (45.4545454 - \frac{10}{1.022^{10} - 1})$ $\text{Equivalent Equal Payment} = 100 \times (45.4545454 - \frac{10}{1.24310828 - 1})$ $\text{Equivalent Equal Payment} = 100 \times (45.4545454 - \frac{10}{0.24310828})$ $\text{Equivalent Equal Payment} = 100 \times (45.4545454 - 41.1339342)$ $\text{Equivalent Equal Payment} = 100 \times 4.3206112$ $\text{Equivalent Equal Payment} = 432.06$

### 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.