5. Future Value of Growing Annuity

Finally, the tools that we have developed in this section also allow us to compute the future value of a growing annuity. Using the same logic as in the case of the level annuity, we can write for a growing annuity: 

  

\( FV_{\text{Growing Annuity}} = PV_{\text{Growing Annuity}} \times (1+R)^T \)

 

From the previous section, we know that the present value of a growing annuity is:

   

\( PV_{\text{Growing Annuity}} = \frac{C}{R-g} \times \left( 1- \left( \frac{1+g}{1+ R} \right)^T \right) \times (1+R)^{(1-n)} \) 

 

So that the future value is:

 

\( FV_{\text{Growing Annuity}}= \) \( \frac{C}{R-g} \times \left( 1- \left( \frac{1+g}{1+ R} \right)^T \right) \times (1+R)^{(1-n)} \times (1+R)^T \)

  

This simplifies to:

  

\( \bf{FV_{\text{Growing Annuity}}= \frac{C}{R-g} \times \left( (1+ R)^T - (1+g)^T \right) \times (1+R)^{(1-n)}} \)

 
with: C = initial cash flow; R = discount rate; g = growth rate of cash flows; T = duration of investment project; n = timing of initial cash flow (years from today).

 

 This is the general expression to compute the future value of a growing annuity at the end of the annuity's last investment period (T periods from the start).

 

Example 11

Let's consider the investor from Example 9 before: 

  • C = Initial cash flow of 20'000
  • g = annual growth rate of 3%
  • R = discount rate of 7%
  • T = Investment horizon = 20 years
  • n = timing of initial cash flow = 1 year from now

 

Based on this information, what's the future value of the growing annuity?

 

We can plug the numbers in the expression above to obtain a future value of approximately 1.03 million: 

 

\( FV_{\text{Growing Annuity}}= \frac{20'000}{0.07-0.03} \times \left( 1.07^{20} - 1.03^{20} \right) \times 1.07^0 = 1'031'787\)

 

Example 12

What about the future value of the situation in Example 10 before, where the first payment is in n = 0.25 instead of n=1 years and all else remains the same?

 

The computation of the future value is almost identical to Example 11, with the exception that n takes a value of 0.25 instead of 1. Consequently, the future value of the growing annuity is roughly 1.09 million:

 

\( FV_{\text{Growing Annuity}}= \frac{20'000}{0.07-0.03} \times \left( 1.07^{20} - 1.03^{20} \right) \times 1.07^{0.75} = 1'085'495\)