4. Present Value of Growing Annuity

In the last step of this section, we consider the valuation of annuities whose cash flows exhibit a constant rate of growth g, so-called growing annuities. Let's start with an example.

 

Example 8

In one year, a project makes a payment of 200'000. Thereafter, you expect this payment to grow each year at a rate of growth of 5%. The last payment is expected 3 years from now and the cost of capital is 10%. Based on this information, what's the present value of the cash flow stream?

 

Now we are dealing with a growing ordinary annuity. As in the case of level annuities, we can build a replicating portfolio of perpetuities to "quickly" compute the present value of  a growing annuity:

 

today year 1 year 2 year 3 year 4 year 5
Growing PerpetuityStart today \( C \) \( C(1+g) \) \(C(1+g)^2 \)  \(C(1+g)^3 \) ...
Growing PerpetuityStart year 3 \( -C(1+g)^3 \) ...
Growing Ordinary Annuity (3 years) \( \bf{C} \) \( \bf{C(1+g)} \) \(\bf{C(1+g)^2} \)  0 0

   

As the table shows, the  growing ordinary annuity can be replicated with a portfolio of two growing ordinary perpetuities:

  • Growing ordinary perpetuity that starts today, with an initial payment of C in 1 year and a growth rate of g
     
  • Growing ordinary perpetuity that starts in T = 3 years, with an initial payment of −C × (1+g)T in 4 years and a rate of growth of g

  

Consequently, the present value of these annuities is:

  

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

  

If we rearrange this expression, we get:

 

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

 

with C = first cash flow; R = discount rate; g = annual rate of growth; T = duration of the annuity.

  

In our example above, we have:

  • C = 200'000
  • R = 10%
  • g = 5%
  • T = 3 years.

 

Consequently, the present value of this growing annuity is:

  

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

  

\( PV_{\text{Growing Ordinary Annuity}} = \frac{200'000}{0.1-0.05} \times \left( 1- \left( \frac{1.05}{1.1} \right)^3 \right) = 521'037 \) 

 

Note that this approach with the replicating portfolio only works for annuities with a growth rate smaller than the cost of capital (g < R). If the annuity in question has a higher growth rate, the replicating portfolio does not work and we need to explicitly map the cash flows and compute their present values.

   

Example 9

For the coming year, a client of yours has an expected salary of $200'000. Going forward, her salary is expected to grow at a rate of 3% per year. For the following 20 years, her goal is to save 10% of her salary, starting in exactly 1 year. The expected rate of return on the investment portfolio is 7% (cost of capital). Based on this information, what's the present value of your client's future savings?

 

This is a growing perpetuity with the following characteristics:

  • C = Initial cash flow of 20'000 (10% of the salary)
  • g = annual growth rate of 3%
  • R = cost of capital of 7%
  • T = investment horizon of 20 annual payments.

 

We can use the equation above to compute the present value of this investment proposal:

  

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

   

\( PV_{\text{Growing Ordinary Annuity}} = \frac{20'000}{0.07-0.03} \times \left( 1- \left( \frac{1.03}{1.07} \right)^{20} \right) = 266'633 \) 

 

Based on this information, the present value of the expected savings is $266'633. 

 

Example 10

What if the client from the previous example started her saving payments in 3 months as opposed to 1 year?

 

As we have already seen in the case of perpetuities and level annuities, the approach above over-discounts the future cash flows by (1-n) years if the first payment is made in n years as opposed to 1 year. To adjust the present value, we compound it over (1-n) years.

 

The generalized version to compute the present value of growing annuities therefore is:

 

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

 

with C = first cash flow; R = discount rate; g = annual rate of growth; T = duration of the annuity; n = timing of the first cash flow (number of years periods from today).

 

In our example, if the cash flow arrives after 3 months (n = 0.25 years):

 

\( PV_{\text{Growing Annuity}} = \frac{20'000}{0.07-0.03} \times \left( 1- \left( \frac{1.03}{1.07} \right)^{20} \right) \times 1.07^{(1-0.25)} \) \( = 266'633 \times 1.07^{0.75} = 280'512 \)

 

If the payments already start in 3 months, the resulting present value is roughly $280'000.