Whether you are planning for a major purchase, financing an education or planning for retirement, an analysis structure that explicitly considers investment returns, cash flows and horizon is crucial to your eventual success. Expected inflation and variability of investment return present difficulties in any planning environment, and the longer the investment horizon, the more important these variables become.
This workshop outlines step-by-step procedures for evaluating pension investments and future investment goals. In each case, the example is explained using the tables provided, but for those of you who are mathematically inclined, the equations are also given.
Accumulating a Future Sum
Our first case concerns planning to accumulate a specific sum by a specific future date. For instance, let’s say that in 15 years, you want to have accumulated $100,000 in today’s dollars.
To start the planning process, you must make two educated “guesses”: you must estimate what your annual investment return before taxes will be over the period, and you must estimate the annual inﬂation rate over the period. Let’s anticipate that over the next 15 years, the annual investment return before taxes is 10% and the annual inﬂation rate is 3%.
Next, you must determine how much money in 15 years will be equivalent to $100,000 in today’s dollars. This is a function of the rate of inﬂation, which we are assuming to be 3%. To determine the future value, you simply check the compounding growth tables (see Table 1), which indicate how much $1 will grow over various time periods and various investment rates. (You can also do this mathematically, without referring to any tables. The equation is included in Table 1.) You then multiply that figure by today’s dollar figure:
$100,000 × 1.558 = $155,880
In order to have the purchasing power equivalent of $100,000 today available in 15 years, you must plan to accumulate $155,880.
How to Get There
The next question is: What kind of investment schedule will allow you to accumulate that future sum?
In order to determine this, you must decide the frequency of payments: In other words, will they be monthly, quarterly or annually?
Let’s assume you have decided to make annual payments. That means that every year, you will be contributing a set amount to your goal, and it will be earning a return of 10% (this was our previous investment assumption).
An annuity table will provide you with the proper figure for determining this amount (see Table 2, which also includes an equation if you want to do this calculation yourself). The tables indicate how much $1 invested each period will grow, given various investment rate assumptions. In this instance, Table 2 indicates that $1 invested at the end of each year for 15 years (i.e., over 15 periods) will grow to $31,772, if it is compounding at a rate of 10%. This $1, times the amount (unknown) that you must invest periodically, will produce your end value of $155,800:
31.772 × V0 = $155,800.
Next, rearrange the formula to solve for the variable:
$155,800 ÷ 31.772 = $4,903.69.
In order to amass $155,800 by the end of 15 years, $4,903.69 must be invested at 10% at the end of each year for 15 years.
If you wanted to make your payments semiannually, you will be looking at a different section of the table: You will be looking at 30 six-month periods, compounding at a rate of 5% (half of our 10% annual assumption, since you are looking at a semi-annual time period). Your semi-annual payments in this instance would be:
$155,800 ÷ 66.439 = $2,345.01.
Table 2 can also be used to determine the future value of an individual retirement account, assuming you contribute to it every year. Let’s look at an IRA’s future value at the end of 20 years, assuming you invest $2,000 annually and have an anticipated annual investment return of 10%.
First, you would find out the value of $1 invested at the end of each year for 20 years at a 10% compound investment rate: Table 2 shows this to be $57.275. Next, you multiply this by $2,000:
$2,000 × 57.275 = $114,550.
At the end of 20 years, the IRA would have reached $114,550 (this is, of course, before taxes, and represents the future value, and not the value in today’s dollars).
If you assumed an annual investment return that was higher—say 16%—the future value would be worth much more:
$2,000 × 115.37 = $230,740.
Over a 20-year period, the difference between the final value at a 10% investment return and a 16% investment return is substantial.
Withdrawing the Money
Once you’ve accumulated the sum, you’ll want to know how much you can spend: What is the stream of income that can be derived for a given period?
Let’s assume, for instance, that you have accumulated the $155,800 in 15 years, and you would like to make quarterly withdrawals from these savings over a five-year period. Again, you must assume an annual rate of return for this period—in this example, let’s assume 12%. Table 3 gives the present value of a $1 periodic investment withdrawal for various investment rates—it is, in other words, the present value of Table 2. (The equation to derive these figures is also presented in Table 3). Quarterly withdrawals over a five-year period result in a total of 20 periods; the quarterly interest rate is 3% (12% ÷ 4). Table 3 shows the present value figure, given these variables, to be 14.8775. To determine the amount you can withdraw, you solve for the variable:
$155,800 ÷ 14.8775 = $10,472.19.
Withdrawing $10,472.19 at the end of each quarter for five years would exhaust the original sum of $155,800. If you had annual withdrawals for five periods at 12%, the amount you can withdraw annually is $96,881.78:
$155,800 ÷ 3.6048 = $43,220.15.
These techniques and the tables are very useful planning tools. They allow you to view the sensitivity of your financial plans to varying investment return assumptions and planning periods. For longer periods, the numbers can differ substantially for different return assumptions and there are no close substitutes for those calculations.
This article was written by John Markese for the April 1985 issue of the AAII Journal. At the time, Markese was director of research at AAII. He is also a former president of AAII and currently serves as chairman.