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Measuring the Performance of Accounts with Deposits and Withdrawals
Measuring Performance in the Presence of Cash Flows
Mutual Fund:
Return on Initial Investment (ROII)
(time-weighted)
Individual Investor's Account:
Fixed Rate Equivalent (FREQ)
(money-weighted)
Cash flows are purchases and redemptions made by independent investors. Cash flows are deposits and withdrawals made by the investor.
Investors who have purchased shares at the beginning of the reporting period base their claims at the end of the reporting period on the reported performance. This claim is independent of the cash flows in between. What matters to the individual investor is the performance of the account as a whole. The performance measure must reflect the amounts and the timings of all deposits and withdrawals.
The ROII (time-weighted rate of return) describes the performance of a single initial investment made at the beginning of the reporting period: it is the percentage change of such an investment over the entire period. The FREQ is not the percentage change of any specific part of the invested money. It is the fixed annual interest rate that replicates the performance of the account: had the deposits and withdrawals been made to a fixed rate account, this is the annual interest rate that would have resulted in the same ending balance.
The ROII is almost always given as an annual rate. However, it makes sense both as an annualized and unannualized return. (See the appendix below for a brief discussion of annualization.) The FREQ is inherently an annual rate. Its purpose is to specify a certain (imaginary) fixed rate account. While that specification could just as easily be made using a rate other than an annual one, not many people would be able to gain much intuition from a non-annual number.
Data needed for calculating the ROII:
  • Beginning Date and Balance
  • Ending Date and Balance
  • Amounts of Cash Flows
  • Account Balances on the Dates of the Cash Flows
Note: The begin and end date are needed for annualization only.
Data needed for calculating the FREQ:
  • Beginning Date and Balance
  • Ending Date and Balance
  • Dates and Amounts of Cash Flows

Appendix: Annualization

In our original example, where there were no deposits and withdrawals, we were looking at a return of 35.0252% for the full five year period: the initial investment of $10,000 grew by 35.0252%. Now suppose we're also interested in the performance for the two year period beginning at the start of 2005. The beginning balance is $11,115.00, and the ending balance is $13,502.52. That's a percentage change of 21.48%. How does that compare to the 35.0252% for the full five year period? Well, not at all: the returns for a five year period and a two year period are incomparable. The number for the five year period should be greater just because the money is at work for a longer period of time. It is not clear how much greater exactly it would have to be so we could say that the performance was better over the five year period as opposed to the two year period.