Growing Annuity - How do I find the growth rate of a growing annuity?

Question

I need help understanding the solution given for this problem. Some alternate ways to do the problem would be helpful as well. I need to find the growth rate of a growing annuity.

Problem: Assume that you will be saving each year for 3 years, starting next year. If your first year savings is $2,500, at what constant rate must your savings grow each year to hit your target of $12,000 at the end of four years if your savings earn 5% per annum?

Solution given to me: 2,500(1+.05)^3 + 2,500(1+g)(1+.05)^2*(1+.05)^1 = 12,000

(1+g)^2 + 1.05(1+g)-3.46989 = 0

g = 41.01%

It's great that they gave me the solution yet I have no idea how they got it. I can't seem to solve that equation for g. If anyone could help, I'd greatly appreciate it.

Is there an easier way to do this? There has to be a way to calculate for g using a financial calculator. Thanks in advance!

Answer 1

You can understand the solution like this: The first year's savings, compounded at 5% for three years are:

2500 (1 + 0.05)^3 = 2894.0625

The second year's savings, increased by g%, compounded for two years are:

2500 (1 + g) (1 + 0.05)^2 = 2756.25 (1 + g)

The third year's savings, increased again by g%, with one year's growth at 5% are:

2500 (1 + g)^2 (1 + 0.05) = 2625. (1 + g)^2

You can solve the total for g by using the formula for a quadratic equation:

total savings = 2894.0625 + 2756.25 (1 + g) + 2625. (1 + g)^2 = 12000

∴ g^2 + 3.05 g - 1.41893 = 0      <- Quadratic form

∴ g = (-3.05 + (3.05^2 + 4 * 1.41893)^(1/2)) / 2 = 0.410085 = 41.01%

To solve this type of calculation in the general case

The summation of the above calculation can be written like so:

This is the general form. It can be used for a calculation over any number of years:

The general form can be converted to a formula for s by induction:

The equation cannot be expressed as a formula for g but it can be used to solve for g by plotting or using a solver.

i = 0.05
p = 2500
n = 3

s = ((1 + i) * ((1 + g)^n - (1 + i)^n) * p) / (g - i)

Using Excel's solver add-in.