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.
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
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
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
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
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.
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.
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).