6

The Basics of Return, Compounding, and Annualization
Just like an individual investor's 401k account or brokerage account, a mutual fund is
nothing but an account with deposits and withdrawals. The only difference is that the
deposits and withdrawals in a mutual fund are purchases and redemptions made by independent
investors who don't know about each other. Now assume that 5 years ago, you invested $10,000
in a certain mutual fund. Today, the mutual fund reports an annual return of 7% for the
trailing 5 year period. Then you can demand to be paid
$10,000 × (1 + .07)^{5} = $11,402.55 upon redeeming your
shares. When calculating its return, the mutual fund is thus constrained by the
requirement that the total value of its investors' shares add up to what the fund actually
has in the pot. This return is called the
*time-weighted return*, or *TWR* for short. It is almost always given as an
annualized return, in which case it is more appropriately called the *time-weighted rate of return*.

The time-weighted rate of return is different from the fixed rate equivalent (FREQ), which is the appropriate measure of performance for an individual investor's account. Speaking very generally, the difference is this: an individual investor wants all the deposits and withdrawals and their timings to be taken into account when judging the performance of his or her account (money-weighted performance). The mutual fund, on the other hand, must figure out how one investor's single investment at the beginning of the reporting period has performed, independently of what other cash flows have occurred since. These white papers have more on the subject:

- “Measuring the Performance of Accounts with Deposits and Withdrawals” (no mathematics required)
- “The Math of the Time-Weighted Return” (for the mathematically inclined)

In this appendix, we prove that when interest compounds at a constant rate, the principal doubles every x number of years, where x is a number that depends only on the interest rate. Moreover, the number x, which is also called the doubling time, is roughly equal to .7/r periods, where r is the period interest rate as a fraction.

Suppose that an amount of money B is invested at a fixed
period rate of r.
Equation
(5) tells us that the value E_{x} of the principal
after x number of periods equals

Since we are looking for the number of periods it takes for the principal to double, we are looking for the value of x that satisfies

Cancelling B out of the equation, we see that x depends only on r:

Taking the natural logarithm on both sides of the equation, we obtain

and thus

For a rough estimate, we can use .7 for ln(2). Also, for small values of r, such as rates between 0 and .1 (meaning between 0% and 10%), the value of ln(1 + r) is close to r, and we obtain