## Increasing Payment Calculator

## Definition – What is an Increasing Payment?

An increasing payment is a payment that increases by the same percentage each payment period.

As an example, in the first time period, the payment is $100 and increases by 5%. In time period 1 the payment is $100. In time period 2 it is $105 ($100 + 5% of $100) , in time period 3 it is $110.25 ($105 + 5% o $105), in time period 4 it is $115.76 ($110.25 + 5% of 110.25).

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 Value of an Increasing Payment is Calculated

If the rate of increase is NOT equal to the compounding rate:

Part 1 = (1 + Rate of Increase) ÷ (1 + Rate)

Present Value = Initial Payment x (Part 1^{Number of Periods}-1) ÷ (Rate of Increase – Rate)

If the rate of increase IS equal to the compounding rate:

Present Value =Initial Payment x (Periods ÷ (1 + Rate of Increase))

Where:

- Initial Payment is the initial payment that is made.
- Payment increase is the percentage (entered as a decimal) amount that the payment increases each payment period.
- Rate is the rate used to compound each period.
- Periods is the number of compounding periods and payments that are made.

## FV of an Increasing Payment Formula – How the Future Value of an Increasing Payment is Calculated

If the rate of increase is NOT equal to the compounding rate:

Future Value = Initial Payment x ((1 + Payment Increase)^{Periods}-(1 + Rate)^{Periods}) ÷ (Payment Increase – Rate)

If the rate of increase IS equal to the compounding rate:

Future Value = Initial Payment x (Periods ÷ (1 + Payment Increase))

Where:

- Initial Gradient Payment is the initial payment that is made. It is also the amount that the payment increases each period.
- 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 0.5% each period over 10 periods at a compounding rate of 2.2%. Find the present value.

Part 1 = (1 + 0.005) ÷ (1 + 0.022) = 1.005 ÷ 1.022 = 0.9833659

Present Value = 100 x (0.0933659^{10 }– 1) ÷ (0.005 – 0.022)

Present Value = 100 x (0.84557365 – 1) ÷ (0.005 – 0.022)

Present Value = 100 x (-0.154426 ÷ -0.017)

Present Value = 100 x 9.09388

Present Value = 908.39

### Example (Future Value)

An initial payment of $100 increases by 0.5% each period over 10 periods at a compounding rate of 2.2%. Find the future value.

Future Value = 100 x ((1 + 0.005)^{10} – (1 + 0.022)^{10}) ÷ (0.005 – 0.022)

Future Value = 100 x (1.005^{10}– 1.022^{10}) ÷ -0.017

Future Value = 100 x (1.051140 – 1.243108) ÷ -0.017

Future Value = 100 x -0.191968 ÷ -0.017

Future Value = 100 x 11.292235

Future Value = 1129.22

## FAQ

### What is the difference between an increasing 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 an increasing payment the payment increases by a specific percentage each payment period. In a financial stream with 4 periods and a starting payment of $100 increasing by 5% each period, the first payment is $100, the second payment is $105 ($100 + 5%), the third payment is $110.25 ($105 + 5%), and the fourth and final payment of $115.75 ($110.25 + 5%).

### What is the difference between the present value and future value of an increasing 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 an increasing payment and a gradient 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.