Science Byte: Equations of Time

I used to teach financial mathematics to undergraduates, where “equations of time” involving compound interest routinely pop up. These equations may not have the elegance or aura of Euler’s identity or Einstein’s famous E = mc^2 , but they are enormously useful in an everyday sense and as such I consider them beautiful in their own way. Here’s one equation that has that quality:

This equation involves the simplest idea in financial mathematics: Time Value of Money. This is the idea (which we all understand informally) that with positive interest rates, a sum of money grows bigger over time. The reverse is also true: given a future sum, the “present value” of that amount is smaller than that future value. That’s basically all to time value of money! While there are details to it, these are mainly bells and whistles wrapped around this fundamental idea. For example, the above equation does not simply deal with one sum of money but a sequence of sums or payments increasing at a constant rate over a period of time. This sequence is technically known as an ordinary growing annuity.

In financial mathematics, it is standard to work with a basic annuity comprising of a sequence of payments of $1 each period for T periods. In the case of an ordinary annuity (also called annuity-in-arrear), the first payment occurs one period from now (e.g., one month, or one year from now depending on the context of the problem).

In the above equation, PVIFGA stands for Present Value of an ordinary Growing Annuity. To see the usefulness of this equation, consider the problem of estimating how much money one needs to have in the bank when one retires. To make the problem more realistic, will assume the following scenario. Suppose you are now 30 and you plan to retire at age 65. Assume also:

  • That in your first year of retirement, you wish to spend $36,000 in today’s dollars
  • That this amount will be drawn from your bank account immediately at 65
  • That you will live to age 90 (to be on the conservative side)
  • That the annual inflation rate is 3% in the pre-retirement period and 4% per annum during retirement
  • That the rate of return you will earn during retirement is 3% (I use a relatively small number since old people tend to be risk-averse).

The problem is this: given the above assumptions, how much do you need to set aside for your retirement at age 65? This is of course a problem of great importance. While there is no “right” answer for all people (it depends on the assumptions), it is still useful to have ballpark estimates, and this is what the PVIFGA equation delivers.

Before I go on, we need to make a little adjustment to the PVIFGA equation. Recall that this equation is for an ordinary annuity where the first payment occurs one period later. However, our assumption states that you wish to draw the first sum of money immediately at age 65. Technically, this means we are dealing with an annuity due rather than an ordinary annuity. No problem. If you can figure out the number for an ordinary annuity, multiplying that number by 1+r will give the corresponding figure for an annuity due. It’s that simple.

Back to the problem. You need $36,000 for the first year of retirement. In future dollars (i.e., adjusting for the corrosive effects of inflation), this amount is $101,299 which is $36,000 compounded for 35 years at the assumed inflation rate of 3%. Next, multiply $101,299 by PVIFGA(1+r). The result is:

$101,299 x PVIFGA(25, 2%, 2.5%)*(1.025) = $101,299 x 28.1415 = $2,850,705.

Many people are shocked when they are told that it takes millions in a bank account to retire comfortably and hence, one must save sufficiently while there is time. Based on our set of assumptions, the PVIFGA equation confirms this cautionary advice. Indeed, it gives a concrete number: $2.85 million. While you can challenge this estimate by challenging the underlying assumptions, having a concrete goal makes financial planning a lot more productive and interesting.

Taking a step back, you can now see why our equation is so powerful. It is powerful because it saves you the tedium of discounting and adding the present value of each and every payment over multiple periods (in our example, 25 years). One formula is all it takes!

How to accumulate $2.85 million in 35 years is the next big question. Again, having an equation to answer this question is useful. For this problem, I add three other assumptions:

  • That you will start saving immediately for your retirement in 35 years’ time
  • That the interest rate on your savings is a constant 2% per annum
  • That your savings will increase by 5% each year.

The following equation can be used to calculate how much you need to save every year from now till you reach 65:

Clearly, this equation won’t even make it to the quarter finals of an equation beauty contest, but it is insightful, and beautiful in its own ugly sort of way. The meaning of each term in this equation is shown here:

S is the amount you need to save yearly. Y is amount you want to spend in the first year of retirement. Therefore, the numerator of this equation is simply the present value of all the money you need during retirement. We have already calculated this amount (the $2,805,705 computed earlier).

To solve for S, divide $2,850,705 by the Future Value of a Growing Annuity Due, denoted by FVIFGA(1+r) in the denominator. Think of FVIFGA as the mirror image of PVIFGA. Like PVIFGA, it depends only on three variables – r, g and T. Plugging in the relevant numbers gives an FVIFGA(1+r) of 119.548. Therefore, S = 2,850,705/119.548 = $23,846. Divide this by your current annual income of $60,000 gives your savings rate: 23,846/60,000 = 40%. If your savings in subsequent years consistently pace income, as this equation assumes, your savings rate will stay constant at 40%.

Whether Einstein meant what he said about compound interest, the essence of his statement cannot be disputed. Those “ugly” equations in time value of money holds the key to long term financial security.

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