Investment Theory and Risk Management, + Website

Pension Logic

Here is a practical savings problem that involves compounding. Suppose you are 25 years old and earn a salary set at the beginning of the year to $Y. Your plan for retirement is to save a fraction, d, of each year's salary. This savings, $dY will earn interest each year equal to r. Also, your salary is expected to grow at rate g annually. Derive an expression for the cumulative savings if you work N years.

It makes sense to break this problem into parts. The first year's savings is dY dollars, and this earns interest, compounded annually, for N years. Therefore at the end of N years, you should have . At the beginning of the second year, your salary has grown to and you save the same fraction of this salary, which earns interest for N – 1 years (and so on). Thus, this part grows to . Let's now extend this logic. The total S of all your savings over the N years until retirement will be equal to the sum:

We solve this like any other geometric series (see the appendix to this chapter). In this case, we multiply both sides of this relation by yielding:

Now, subtract this ...