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The Basics of Return, Compounding, and Annualization
Suppose you invest some amount of money, say, in a mutual fund. After some time, you look at the current value of your investment. The return of the investment for that period of time is the balance change divided by the initial investment. As a measure of performance, this metric is less than ideal because investments with different time periods are not comparable. The solution is the annualized return. The annualized return is the annual interest rate of the fixed rate account which, when seeded with the same amount of money as the real investment, results in the same ending balance as the real investment. In other words, it is the annual interest rate of the fixed rate investment that replicates the actual investment.

When looking at an investment that periodically pays a fixed interest rate, say, an annual
interest, the reverse question arises: if we add the interest to the principal at the end of
each year for compounding, then what is the balance after several years? The answer is: the
balance after n years
is B × (1 + a)^{n},
where B is the beginning balance
and a is the annual interest rate. The principal doubles
roughly every .7/a years.

In this article, we have covered the basics of return, compounding, and annualization. Here is an outlook on further topics that are important in the world of investing and performance measurement.

When you evaluate different fixed rate investment options such as certificates of deposit (CDs), savings bonds, and the like, you will often encounter this situation: an annual interest rate is given, and it is qualified with an expression like "compounded quarterly," "compounded daily," or "continuously compounded." When that is the case, the terms APY (annual percentage yield) and APR (annual percentage rate) are not far off. Understanding all this is important for the purpose of comparing different investment options. It is explained in a white paper that's coming soon to GreaterThanZero.

In this article, we talked about the return of a single investment: an amount of money is invested, and its performance is observed over time. The reality of individual investing is more complicated than that. Individual investors have accounts (such as 401k accounts, brokerage accounts, and accounts with financial advisors) that undergo deposits and withdrawals. What matters to the investor is the performance of these accounts. And even if an investor wants to know about the performance of an investment in a specific stock or mutual fund, chances are that there wasn't just a single initial purchase of shares of that stock or mutual fund. It is quite likely that shares were bought and sold at various points in time; so again, we are looking at the equivalent of an account with deposits and withdrawals.

Measuring performance in a manner that takes into account the effect of deposits and
withdrawals is also called *money-weighted performance measurement*.
The appropriate way of doing this is by calculating the
*fixed rate equivalent*, or *FREQ* for short. It is the main focus of
GreaterThanZero. Here is a list of white papers where
you can read about it:

- “Quick Introduction to ANP and FREQ” (no mathematics required)
- The FREQ vs. the Internal Rate of Return (no mathematics required)
- “The Math of the Fixed Rate Equivalent” (for the mathematically inclined)
- “The Math of the Fixed Rate Equivalent, Extended Version” (for the mathematically trained, PDF Format Only)