A commonly asked question in investments is, “how long will it take to double (triple, etc.) my money?” While this question is fun to think about, and the math needed to provide an answer is relatively simple, the real answer to the question is actually a little more complicated than simply applying the “Rule of 72”.

### A quick review of the Rule of 72.

Many people have heard of the “Rule of 72” when it comes to investing, but if you haven’t I will bring you up to speed. The Rule of 72 is not necessarily a rule, but rather a rule-of-thumb estimate that helps you approximate how long it would take you to double your money given some constant annual rate of return. To do this you take the number 72 and divide it by the rate of return presented as a whole number (e.g. 3% would be 3 and not 0.03). For example, if you wanted to estimate how long it would take you to double your money if you were able to receive a 10% annual return, all you need to do is divide 72 by 10. In this example, your answer would be 7.2 years, which is very close to the actual answer of 7.27 years (see the yellow highlight in Table 1). The Rule of 72 can provide a reasonable estimate of how long it would take to double an original investment amount so long as the assumed rate of return is greater than zero and within a realistic range of annualized returns. Table 1 below presents the number of years required to invest to double (2x), triple (3x), 5x, 10x, and 15x an original investment amount over various rates of return. The calculations within Table 1 assume a fixed annualized rate of return that compounds annually. Just like many other rules-of-thumb, the applicability of the Rule of 72 is quite limited and only applies to the doubling of a lump sum of money, invested at the beginning of a time period, and while receiving a constant annual rate of return over the entire period. Outside of those specific parameters, the Rule of 72 cannot be used beyond doubling a lump sum – say, 5x your investment at a 12.5% annual rate of return, for instance. Thus we find ourselves back to the original question, “how long will it take to double (triple, etc.) my money?” As you may have guessed, the answer to that question requires a little more exploration before an answer, or more appropriately an estimate, can be rendered. ### What do we really mean when asking how long it will take to double our money?

To start, more clarification is needed to know what is truly meant by “doubling, tripling, etc.” your money. Taking it at face value, the answer seems to be very obvious – turn $100 into$200. But is that really what it means? It is likely that when we say that we want to “double, triple, etc.” our investment, we mean it in the explicit sense (e.g. the amount of money doubles, triples, etc.) as well as the implicit sense (e.g. we also expect to gain the ability to aquire 2x, 3x, etc. more goods and services with the funds in the future). In other words, while most of us think of doubling our money in literal terms ($100 to$200), we also think of the things that they can buy in the future in today’s dollars.

In economic terms, these concepts are known as ‘nominal’ and ‘real’. To keep matters relatively simple we can think of ‘nominal’ as the amount of money (today and in the future), and ‘real’ as the amount of things (goods and services) that money can buy. In a world with no inflation or deflation and stable supply and demand, there would be no difference in nominal and real prices. In such a world one-hundred dollars would purchase the same amount of goods and services today as it did 10 years ago (or 20 years in the future).

However, such a world only exists in the minds of economists, realistically an appropriate inflation estimate would need to be accounted for over the investment period to account for the loss of purchasing power over time due to inflation when attempting to answer our question. Given this, it becomes clear that the original question should not be “how long will it take,” but rather “given the time that I have to invest, and the anticipated inflation rate in the future, what is the rate of return that I need to achieve to double, triple, etc. my investment in real terms?”

### A better question. Not how much time, but what rate given time and inflation.

In investments there are innumerable uncertainties but there are a couple things that we know for sure (1) nobody can accurately predict the future and (2) that our lives are finite. Thus, it is likely more productive to focus on what we can do with the time we believe we have, as opposed to focusing on how long something will take. The question, “given the time that I have to invest, and the anticipated inflation rate in the future, what is the rate of return that I need to achieve to double, triple, etc. my investment in real terms,” forces you to contemplate, not only the finite amount of time that you are affording an investment, but also the anticipated inflation rate over that period of time. So rather than solving for “how long” you are solving for what return is required given time and inflation. This leads you to the question that actually needs to ultimately be answered: “given the time I have to invest, my assumptions about inflation, and my desired goal of doubling (tripling, etc.) my investment in real terms, is the resulting rate of return realistically possible to achieve?”

### Two part question (1 of 2): What is the required rate of return given the time horizon and inflation expectation?

While the estimate of time horizon is highly unique to each person’s age, health, and circumstance, your inflation expectations can be calibrated against historical data. Using the Ibbotson’s Stocks, Bonds, Bills, and Inflation (SBBI) US Inflation annual data, the median annual inflation rate in the United States was 2.68% and the average rate of inflation was 2.98% from 1926 to 2021. As we can observe from the graph below, over the last 96 years the inflation rate in the US has largely remained within a range of 0% to 7% (approximately 80% of the time). Thus, using inflation rates within the historic norms when trying to determine a required rate of return would be a reasonable approach. In Tables 2a-2d below we calculate the required rate of return necessary to achieve a certain multiple of an investment in real terms (i.e. adjusted for inflation) within a particular period of time (i.e. Investment Time Horizon). Each one of the tables below are the same in all regards except the rate of inflation. The inflation rate is stepped up in 2.5% increments from 0% to 7.5% inflation. What you should note when reviewing and comparing the tables below is that: (1) the longer the time horizon the lower the required rate of return and thus a less need for risk in the portfolio, (2) the higher the inflation rate the higher the required rate of return, (3) the larger the desired multiple (2x, 3x, etc) the higher the required rate of return. As an example, let’s assume that you currently have $10,000 and that you would like to triple (3x) the investment in real terms within 15 years. You assume the rate of inflation will be 5% over the period and would like to know what annualized rate of return you would need to meet your objective (see highlighted cell above in Table 2c). What does “triple in real terms” mean? In short it means that you would like the ability to buy$30,000 worth of goods and services in future dollars. Using this example specifically, $30,000 in purchasing power in today’s dollars would be equivalent to$62,368 fifteen years in the future given a 5% inflation rate. Thus, your investment of $10,000 will need to grow to$62,368 within 15 years to provide you with your desired result of tripling the purchasing power of your original investment. The estimated required return to provide such a result would be 12.98% given the inputs.

### Two part question (2 of 2): Is there a reasonable chance of achieving the required rate of return?

Up to this point our exploration of these questions have largely been thought exercises, but ultimately to achieve some desired result we will need to earn the required return through investment, saving, or a combination of the two. Now that we have an understanding of what required rate of return would be needed, we can look to historic investment returns to gain an understanding of the likelihood of achieving a particular required return. In Tables 2a-2d, the annualized required returns ranged from 2.34% (see Table 2a – 2x with 0% Inflation and a 30 year time horizon) to 84.77% (see Table 2d – 15x with 7.5% Inflation and a 5 year time horizon). It is important to remember that thus far we have only been dealing with a single lump sum investment made at the beginning of a time period. We will relax that assumption in the final example, but for now we will continue with the lump sum investment assumption.

In Table 3 below a number of historic trailing return statistics have been calculated using the annual Ibbotson® SBBI® US Large-Cap Stocks (Total Return) data from 1926 to 2021. We will use these statistics to provide us with the historic perspective needed to answer the second part of our question, “is there a reasonable chance of achieving the required rate of return?” As you will notice when comparing the annual required returns presented in Table 2a-2d and the historic returns in Table 3, there are many required return levels that are well outside the historic norms. For example, tripling (3x) your investment in real terms in a 5-year period with a 2.5% inflation rate requires an annual return of 27.69 percent. Given our historic data we could determine that this high return is nearly the maximum and only occurred once (in the 5-year period from 1995 to 1999) out of the 92 trailing 5-year returns within our data. So what can you do if it is not reasonable to assume that you can receive the required rate of return over a time horizon?

### A Final Example: Reduce the required rate of return with investment contributions.

We’ve covered a good deal of information within this article and have provided some general framework around how we can think of investment goals. Without going into a much more detailed analysis, I would like to close out by giving one final example. Let’s assume you have a 20-year time horizon and you want to grow an investment of $50,000 by 10x in real terms over the next two decades. You believe that over the period we will experience an inflation rate of 5% which would require that you earn an annualized return of 17.81% over the period (see Table 2c). Unfortunately, based on the historic trailing 20-year returns, the chances of achieving a 17.81% rate of return within Large Cap US Stocks is only about one percent. If you would like to improve your chances of achieving your goal to at least 75% using Large Cap US Stocks you would need to adjust your expected annualized rate of return down to 8.12 percent. Referring to Table 3, this represents the 25th percentile in the 20-year trailing return statistics (highlighted in yellow). Thus, to meet your objective you will need to contribute additional funds either by (1) increasing the initial funding amount from$50,000 to $278,378, or (2) make annual contributions of$23,469 per year (or ~\$1,956/month) over the investment period. Obviously, there are many other configurations that could be applied to achieve the objective.

The overarching purpose of this article was to provide a broader insight into some of the most important considerations when attempting to answer the deceptively complex question “how long will it take to…” I hope that you found it to be worth your time and if you have any questions do not hesitate to contact me at brian@canyoncreekinvestment.com or visit our website at canyoncreekinvestment.com. Thank you.

Author: Brian Fraser, CFA, CAIA 