6. Summary

This section has shown useful shortcuts for the valuation of annuities. These investment proposals deliver a constant (or a constantly growing) cash flow over a specific investment horizon. They are extremely popular in the context of saving and retirement planning. Moreover, many fixed-income securities such as corporate or government bonds take the form of annuities. 

 

In this section, we have considered two types of annuities:

  • Level Annuities, which deliver a constant cash flow C over T investment periods.
  • Growing Annuities, which deliver an initial cash flow C, which growth at a constant rate of g over T investment periods.

  

Moreover, we have seen how to handle different starting points of the annuity payments, in particular annuities that start in one investment period (so-called ordinary annuities) as well as annuities that start today (so-called annuities due). 

  

For all types of annuities, we have computed both the present values and the future values at the end of the investment horizon with a series of practical examples. The accompanying Excel file "Excel Solutions to Examples" contains the solutions to all the examples that we have solved in this section.

  

Moreover, the Excel File "Excel Templates for Annuities" contains useful templates to compute PVIFA and FVIFA as well as the present and future values of the different types of annuities.

   

Throughout the section, we have used the following notation:

  • C = Cash flow
  • R = Cost of capital
  • T = Investment horizon (number of years or periods)
  • g = Constant growth rate of cash flow
  • n = Timing of initial cash flow (years from today)

   

With this notation in mind, we can now summarize the main valuation formulas:

  

Annuities: Present Value

The present value of an annuity that starts with an initial payment of C in n years, makes a total of T payments, and is subject to a cost of capital of R is:
 

\( PV_0 = C \times PVIFA_{R,T} \times (1+R)^{(1-n)} \) 

  

\(\text{with:} \ PVIFA_{R,T}=\frac{1-(1+R)^{-T}}{R}\)

  

Annuities: Future Value

The future value of such an annuity is:

 

\( FV_{T} = C \times FVIFA_{R,T} \times (1+R)^{(1-n)} \)

 

\(\text{with:} \ FVIVA_{R,T} = \frac{(1+R)^T-1}{R} \)

 

Growing Annuities: Present Value

The present value of an annuity that starts with a cash flow of C in n years, which subsequently grows at a rate of g, makes a total of T payments, and is subject to a discount rate of R is:

  

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

  

Growing Annuities: Future Value

Finally, the future value of such an annuity is:

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

   

As pointed out above, the Excel File "Excel Templates for Annuities" provides a template for the valuation of these annuities. It also shows the specific implementation of Excel's built-in functions "PV" and "FV", which can be very useful in the valuation of annuities.