# Future Value of a Growing Annuity

Updated: April 12, 2022

The future value of a growing annuity is the total value of a series of payments that are growing (or declining) at a constant rate during a certain time period.

To better understand this terminology, it is helpful to first understand the definition of an annuity and its types. An annuity is simply a series of cash instalments that are paid over a certain period of time, accompanied by an interest rate. A growing annuity is a series of equal payments over time that grow at a constant rate. It is sometimes referred to as a graduated annuity or an increasing annuity.

We are looking at the future value of these growing payments. The word “value” here means the financial limit that a series of payments can reach. The cash payments will grow at a constant rate that will be separate from the interest rate of the investment. The increased cash payments will then grow again within the investment because of the investment’s rate of increase.

Imagine if someone invests $1000 for payment 1, but then decides they can pay more than that every other time. We want to calculate how that would make a difference to the ending value of the investment. However, the cash flow must be increasing at a constant rate. Meanwhile, the interest rate should remain the same. If we know these rates, we can plug it easily into the formula. ## Future Value of a Growing Annuity Formula $$FV = C \times \bigg[ \dfrac{(1 + r)^{n} - (1 + g)^{n}}{r - g} \bigg]$$ • C = cash value of the first payment • r = interest rate • g = growth rate • n = number of periods In this equation, the first payment (C) would need to be made within the first period. Then, each subsequent payment would increase by the growth rate. So the growth rate is referring specifically to the rate that the periodic cash payments would increase annually. In an ordinary growing annuity, the payments are made at the end of each period. Also, the frequency of the interest rate should match the frequency of the payments. If you are talking about monthly payments, then the interest rate should be monthly. The interest rate is often referred to here as the discount rate, or rate of return. ### FV of Growing Annuity if r = g You might come across a situation where the interest rate (r) and the growth rate (g) are the same. In this case the above formula would not work any you’d get an error. The solution is to calculate the future value of the annuity without the growth rate using the below formula: $$FV = C \times n \times (1 + r)^{n - 1}$$ ## Future Value of a Growing Annuity Example Greg is considering opening an investment account. It has an annual interest rate of 7%. This year at work he received a$3000 bonus, and he’s trying to decide if he should put it towards the account in its first year. His company has committed to increasing his bonus by 2% each year until he retires. He plans to put that money into the retirement account every year. If he works for 20 additional years before he retires, what would be the future value of this growing annuity?

Let’s break it down to identify the meaning and value of the different variables in this problem.

• Cash value of the payment made in the first period (C): 3000
• Interest rate (r): 7% or 0.07
• Number of Payments (n): 20
• The growth rate of the payments (g): 2% or 0.02
• Future Value of a Growing Annuity (FV): Unknown

We can apply the values to our variables and calculate the future value of his growing investment account.

$$FV = 3000 \times \bigg[ \dfrac{(1 + 0.07)^{20} - (1 + 0.02)^{20}}{0.07 - 0.02} \bigg] = \143{,}024.22$$

### What is a growing annuity?

A growing annuity is a periodic payment made over time that changes in value due to the changing interest rate. In a growing annuity, you can choose to have larger payments at the end of each pay period or make smaller payments throughout each pay period and receive more money overall.

### What is an example of a growing annuity?

An example of a growing annuity is purchasing an item on credit with an initial payment followed by regular installments. This type of purchase would have the initial payment plus the growth rate in the formula.