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Measuring the Performance of Accounts with Deposits and Withdrawals

## Enter Deposits and Withdrawals

Building upon the example of the previous section, let us now assume that instead of investing \$10,000 at the beginning of the 5-year period, you would have started with \$5,000 at the beginning of the five years and then added another \$5,000 after one year. Assuming further that the portfolio holdings were the same throughout, your portfolio value over time would have looked like this:

Figure 2

The end value is now \$15,417.89. Remember that in the original example, where there were no deposits and withdrawals, it didn't really matter whether we viewed the account as a mutual fund or as an individual investor's personal account. The issues of measuring performance and investment skill were the same. In the presence of deposits and withdrawals (or cash flows, as one would say in the world of mutual funds), this is no longer the case. We will now discuss those two points of view separately.

## The Mutual Fund Point of View

So let us first assume that you actually are a mutual fund manager, and the chart above shows the total asset value of your fund over time. Then it is quite likely that the two investments of \$5,000 came from two different people who do not know each other. In fact, the first investor does not know and does not need to know about any cash flows other than his own. In view of this, there can be no doubt as to what return the fund must publish for the 5-year time period in question. The investor who put in the first \$5,000 will base his claim after 5 years on the reported return. Since the underlying portfolio was the same as in our original example, the return must be the same as well: it must be 35.0252%.

And that's pretty much it for the mutual fund point of view: the return for a given reporting period describes the performance of an initial investment made at the beginning of that period, regardless of any further cash flows during that period. This is very real insofar as investors may demand to be paid based on that return. This metric is also known as the time-weighted return, or TWR for short. It is almost always given in its annualized form (see the appendix), in which case it is also called the time-weighted rate of return. We prefer the term return on initial investment, or ROII for short, because that's exactly what it is. Since the term time-weighted return is widely used, we will stay with it in this article.

The only thing about the TWR that is somewhat less than trivial is its calculation. In the absence of cash flows inside the reporting period, as in our original example, the time-weighted return is simply the percentage change of the account balance (total asset value). With the additional cash flow after one year, as shown in the chart above, the 35.0252% change is nowhere visible. In order to recover it, one must, in a manner of speaking, remove the effect of the additional cash flows from the ending balance. If you are mathematically inclined, you can read about it here.