On this page I introduced this compound interest formula:
We start with a principal or present value today and figure out how much we will accumulate sometime in the future assuming a fixed interest rate and a fixed number of compounding periods.
Now in many situations we want to answer the opposite question. We know that we will need a certain lump sum in the future, and we want to know how much we need today in order to accumulate this targeted future value. Well, applying a bit of algebra, we can convert earlier formula to get the formula we need to answer this question. Here is the converted formula.
When, practically speaking, would we need this inverted formula?
Suppose you are the parents of a newborn child and you want to ensure that you have saved up enough money by the time the kid turns 18 to attend college. Let’s keep it simple for now and suppose that we can safely assume that a four-year college will run $30,000 per year for a total of $120,000. Again, for simplicity we will laughably ignore any expected inflation in tuition expenses. Now suppose we can find a fixed income investment like a CD that yields an annual interest rate of 3% compounded annually. What does the inverted formula tell us? It tells, you to set aside a little over $70,000 today and with these given investment assumptions you will assure yourself of having the needed $120,000 eighteen years from now.
What did you say? You don’t have a spare $70,000 lying around today. Well maybe your kid really doesn’t need to go to college. What’s that? You think I did not really get the idea? Ah, the idea was for you to set aside a certain fixed amount each month into an investment account to get to the targeted $120,000 18 years from now. Oh, now I get it.
Well, that inverted formula I just showed you does not provide a solution to that problem. So, for our new problem what we really need is something different, we need something termed an annuity formula. An annuity is simply a series of fixed payments over a period of time. There are two kinds called an ordinary annuity and an annuity due. In the annuity due formula, it is assumed that a deposit will be made at the beginning of the period and the interest received at the end of the period.
We begin with this annuity due formula to compute the future value of a series of regular deposits.
In this formulation it is assumed that we already know the value of the periodic deposits, the “d” term and the amount of the initial deposit the “P” term. Notice that the first term on the right-hand side, the P , is the simple compound interest formula we saw in a previous page showing how an initial deposit will grow with fixed interest and compounding periods.
Now in our tuition example we already know what the desired future value is: $120,000. Here we are looking for the amount of the periodic payments needed to arrive at this future value., the “d” in the above annuity due formula. And for simplicity we will assume that there will be no initial lump sum deposit. By deleting the initial P term and performing a little algebra we can derive this formula for figuring out the needed periodic deposit to reach the desired future value.
Returning to our example, assuming that deposits will be made and interest credited monthly we derive a required deposit of about $421 per month to assure that $120,000 will be available when our child reaches 18.
Now in the interest of full disclosure I have made a few simplifying assumptions in solving this parental dilemma. First off, I ignored the potential impact of inflation on tuition. I used what it costs today in tuition expenses for a four-year degree. The issue of how and when to deal with inflation in time value applications is a big one and I am going to cover it in on this page.
In the example I ignored income taxes. If the investment vehicle chosen yields taxable interest, then the tax has to be accounted for somehow. One way is to assume that any income tax due is paid for separately by the parents. In other words, the parents set up a segregated tuition savings account and pay the taxes out of a separate account.
Another way to account for the impact of taxes is to assume that the taxes will be paid for out of the tuition account and simply adjust the expected after-tax return by the tax rate.
So, in our example we assumed that the annual rate of return was 3% before taxes. If the interest income is taxed at a rate of 20%, we can adjust our computation by reducing this rate by 20% and using a 2.4% annual rate (3% x 80%). Here is our revised required deposit.
Of much greater impact is the assumption about interest income rates. At the time I am preparing this book in December, 2020 a projected interest rate return of 3% on a safe long-term bond or bond fund is not unrealistic. But what if our new parents were not that risk averse and were willing to make investments in equities which have historically outperformed fixed income securities by a significant amount? What rate of return should our parents apply into the formula in this case?
Well suppose they choose a highly diversified equity index fund as their investment vehicle. As an example, let’s consider the Fidelity 500 Index Fund. Here is the annual performance of the fund from 2010 to 2020. You can see that the year-to-year returns vary significantly from a low of -4.4% to a high of 32.37%. This table also shows the average rate of return over this eleven-year period to be 14.13%.
Now as parents looking at funding a college fund an average 14% or so return looks a whole lot better than a paltry 3%. Let’s assume that these equity returns are taxable again at a rate of 20%. This would leave us with a tax adjusted return of 11.3%. So, plugging this into our formula the monthly amount needed to be set aside would be about $174 per month. This compared to the $446 per month needed with the 3% investment would make the investment decision seem like a no brainer.
But as you can see there is quite a bit of variability in the equity investment and the formula we are using requires that we apply the same fixed rate of return each compounding period. So, using an average rate of return seems like our best option. But note the last two columns of this table. The second to last columns shows how $100 invested in the beginning of 2010 actually grew. The last column shows how an initial investment of $100 in the beginning of 2010 would have grown had the fund achieved a smooth annual average instead of the up and down ride it actually experienced. Note that the actual performance is only about 95% of the projected average performance. Now that does not seem that bad, but a similar variance would leave the parents about $6,000 short at the end of 18 years.
The takeaway caveat here is that almost all-time value formulas assume equal rates of return over equal compounding periods. If you substitute an average rate of return for a rate of return that is highly volatile you may be far off, and painfully far off in achieving the targeted future value.