An interest in investing, and an understanding of percentages.
This blog post explains the basics of measuring the performance of two things: investment vehicles such as stocks and mutual funds, and an investor’s account. The emphasis is on explaining the meaning and purpose that underlie the calculation methods.
At the very core of things, measuring the performance of an investment is dead simple. Suppose you invest an amount of money. You let it do its thing for some period of time, and then you look at at it again. The performance, or rate of return, of your investment for that period of time is the change in value as a percentage of the beginning value. For example, if you invest $10,000 and find that after 5 years, your investment has turned into $12,500, then the performance, or rate of return, of your investment for those 5 years is $2,500 as a percentage of $10,000, which is 25%. Of course we can also talk about the performance of a stock or a mutual fund without actually investing in it. In that case, the performance is the percent change of a hypothetical initial investment in the stock or mutual fund over the respective time period. We’ll refer to things that can be invested in, such as stocks and bonds, as investment vehicles.
The performance, or rate of return, of an investment vehicle for a period of time is the percent value change over that time period of a hypothetical amount of money invested at the beginning of the time period.
This is the basic definition, and it never changes. However, for the purpose of measuring investments in real life, some refinement is needed.
Suppose you are researching mutual funds on your brokerage’s web site. For each mutual fund, there will be a tab labeled “Performance,” and it will have a table that looks something like this:
Is the number in each column the rate of return as defined above for the respective period of time? No. That wouldn’t make much sense, because the numbers would not be comparable. For example, if the performance of the mutual fund were dead consistent, say, like a CD with a 5% annual interest rate, then the 1-year number would be 5% and the 2-year number would be 10.25%. You wouldn’t even be able to see that the performance was actually consistent. The only meaningful way to fix that is to use the annualized rate of return instead.
So what’s the annualized return? One thing I could do would be to show you the mathematical formula for calculating it, but that would not be very enlightening. Or, I could say, “The annualized rate of return is the annual geometric mean of the rate of return.” That’s better, assuming that you are sufficiently mathematically inclined. But performance reports are not for the mathematically inclined; they are for investors.
Here’s an equivalent definition of annualized return that assumes only the most basic knowledge and understanding of finance. As before, suppose you invest an amount of money, let it do its thing for some period of time, and then look at at it again. Now imagine that instead of making that investment, you would have put your money in an account with a fixed annual interest rate, like a somewhat idealized savings account. Now you can ask, what is the annual interest rate that would have resulted in the same ending balance as my actual investment? In other words, which annual fixed interest rate would have replicated my investment? That is the annualized rate of return of your investment. Applying this to investment vehicles, we get:
The annualized rate of return, or annualized return for short, of an investment vehicle for a period of time is the annual interest rate of the fixed rate account that replicates the performance of the investment vehicle for that period of time. 
The annualized rate of return is what’s used to report the performance of investment vehicles such as mutual funds. That’s what the numbers in a table like the one shown above actually are. 
Now let us assume that you own a mutual fund in your portfolio, and you want to measure the performance of your investment in that mutual fund. Or perhaps, you want to know what the performance of your portfolio as a whole is. How is that different from what we’ve discussed so far? It is different insofar as, in all likelihood, we are not talking about a sum of money that was invested and then left to grow (or to decline, for that matter). It is very likely that there were deposits and/or withdrawals along the way. That’s a whole different ball game. Consider, for example,
a mutual fund that loses half its value and then recovers to where it started, like this:
Suppose that you had $100,000 in this mutual fund at the beginning of the time period, and you bought another $50,000 worth halfway through, when the fund was down 50%. At the end of the time period, you’ll have $200,000: the initial investment is back to where it was, and the additional $50,000 have doubled. The performance of the fund as an investment vehicle is 0.00% for this time period. But you turned $150,000 into $200,000. That is not a 0.00% performance. $50,000 is not zero, by no stretch of the imagination.
So how do we measure the performance of an investment in the presence of deposits and withdrawals? Well, we already know how to measure performance: it is the annual interest rate of the fixed rate account that replicates the performance of the investment. That concept applies to investments with deposits and withdrawals just as easily as it does to a single initial investment that is then left alone. We call this rate of return the fixed rate equivalent, or FREQ
for short .
The fixed rate equivalent, or FREQ
for short, of an investment with cash flows (deposits and/or withdrawals) is the annual interest rate of the fixed rate account which, when subjected to the same cash flows as the actual investment, results in the same ending balance as the actual investment. In other words, it is the fixed rate that replicates the investment with all its cash flows. 
An alternate, perhaps more intuitive description of the FREQ is as follows: picture your investments as a black box, then imagine there’s a savings account inside the black box. What was the annual interest rate of that savings account? That rate is the account’s fixed rate equivalent. In other words, a FREQ of x% means you were effectively running an x% savings account.
Note that the annualized return as defined earlier is merely a special case of the FREQ: it’s the FREQ in the special case where there is only one cash flow, namely, the initial investment. That is reassuring as it provides evidence that we are looking at a meaningful and consistent definition. However, it certainly makes sense to consider the annualized return as its own thing with its own name. That’s because it describes the performance of investments where a sum of money is invested and then left alone, with no further cash flows. The annualized rate of return is thus suitable for describing the performance of investment vehicles such as stocks and mutual funds. The FREQ, by contrast, is something that aims to measure an investor’s specific performance, taking into account the investor’s deposits and withdrawals.
So far, so good. “But wait,” you might exclaim, “isn’t it true that the performance of an investment with deposits and withdrawals should be measured by means of the internal rate of return, sometimes also referred to as the personalized rate of return? What gives? When you tell me to use the FREQ, are you reinventing the wheel, or are you serving old wine in new bottles, or what?” Good question. Since we’ve already covered a lot of ground here, I’ll defer the answer to a separate blog post.
The use of the term “annualized return” for the annualized return of an investment vehicle, or an investment with no deposits and withdrawals, is less than perfect. That’s because every rate of return can be annualized. Other terms have been suggested. We believe that the rate of return of an investment vehicle, or an investment in the absence of deposits and withdrawals, should be called the return on initial investment, or ROII for short, because that’s what it is.
When I say that the performance numbers for a mutual fund are annualized rates of return, there is, strictly speaking, an adjective missing. These numbers are annualized total
rates of return. The qualification “total” means that the performance is measured under the assumption that payouts such as dividends and capital gains are reinvested, that is, as soon as such a payout occurs, additional shares are bought with the proceeds of the payout. If that is not the case, performance measurement becomes a different beast. This blog entry
explains how to measure performance in that situation.
I said earlier that it was ok to define the annualized rate of return as the annual geometric mean of the rate of return. That is indeed a meaningful and informative definition. However, it still focuses on the way that the annualized return is calculated rather than the underlying meaning and purpose. When we move on to the case of multiple cash flows, the definition of performance as a geometric mean breaks down, while the definition as the replicating annual fixed rate carries over seamlessly, giving us the FREQ. In the presence of multiple cash flows, the FREQ is no longer a simple geometric mean, because different portions of the money were at work for different periods of time. You can read more about the mathematics of the FREQ here
Here’s a high-level interpretation of the rationale behind the FREQ as a measure of performance. Ultimately, every measurement is like measuring the length of an object. Measuring length means to hold a yardstick to the object whose length is to be measured, then reading the value on the scale of the yardstick where the object matches. For the sake of measuring the performance of an investment, the “yardstick” is the fixed-rate account, and the scale on the “yardstick” is the annual interest rate.