## Wednesday, January 4, 2012

### Time and Money Weighting: making sense of the differences

When teaching our Fundamentals of Investment Performance course, when writing my books, and often when simply having conversations with clients, I am often faced with the task of explaining, in as clear a manner as possible, the differences between time and money weighting. This topic is one of the most confusing in our industry. I've heard, on several occasions, performance measurement veterans misspeak when it comes to these matters.

At the core it all boils down to cash flows: whether to include them in the process, or eliminate (or at least reduce) their impact on the resulting return. And while a few folks suggest that their implementation has nothing to do with who controls the cash flows, the reality is that this is definitely the main reason behind deciding upon which to use (though there are times when we actually ignore this, in favor of the insights provided).

And it also boils down to linking. That is, the geometric linking of returns.

Time weighting comes in two forms: exact and approximate. Exact methods revalue the portfolio for all cash flows, and calculate returns between each of these revaluations. Approximation methods may revalue for large flows, but not all flows (or they'd be exact). And linking occurs at any point when the portfolio is revalued (either when large flows occur, or at month-ends).

We typically use either the Modified Dietz or Internal Rate of Return (IRR) formula in our approximation methods. Both of these formulas, by themselves, are actually money-weighted methods. We transform them into time-weighting when we employ geometric linking!

The following graphic contrasts money and time weighting:

As you can see, we are calculating returns in two ways: by time and money weighting. The essential difference is that with time-weighting, we value the portfolio multiple times during the period, and link the intermediate results, while for money weighting, we only value at the end points.

Can more be said on this topic? Yes! And more will be, so stay tuned.

1. Stephen Campisi, CFAJanuary 4, 2012 at 9:52 AM

To quote your last blog entry, I say "Down with obfuscation." The truth is that linking money weighted returns does not convert them into time weighted returns, regardless of how often this error is repeated. Any errors or biases embedded in a short-period money weighted return with a significant cash flow are then buried by the linking process; these errors are not eliminated. This was the guidance we received from the great investment writer Bruno Solnik when he was questioned on this process. Unfortunately, he made no headway with the CFA Institute when he voiced his objections to this misstatement.

2. Steve, they convert them into "approximations" of TWRR. This, I believe, is indisputable.

3. I think it's an obfuscation to use the word "approximation" for a significant return difference caused by a material cash flow. If the question concerns a manager's performance, then the impact of a significant cash flow must be eliminated using sub period returns before any linking occurs. That is indisputable. What is disputable is whether a money weighted approach that produces a significant deviation from the manager's true return can be used to generate a time weighted return. I believe that it cannot, since the biased return is not part of an inaccurate linked return. One should only use the word "approximate" when the difference is immaterial.

4. I like your emphasis on linking. I spent many hours explaning to people why calculating Modified Dietz over a somthing like a ten year period is not really a good idea if the goal is to calculate a time-weighted return, while calculating and MWR over ten years produces an exact result.
Instead of "cash flows", I prefer to speak of contributions and withdrawals: "It all boils down to how we treat variations in the capital invested, i.e. contributions and withdrawals."

5. I don't believe it's an "obfuscation," though you can, of course, take exception to my wording (which even Dietz would concur with). You may also re-read the above post to see my reference to revaluing for "large flows." You are, of course, entitled to your opinion, though it runs in conflict with Peter Dietz, the AIMR-PPS, GIPS, the BAI, the ICAA (now IAA), and countless other sources that consider these measure to be "time-weighted" returns.

6. Andreas, thanks for your note. As for the use of the term, "cash flows," this is a broader term, which also works at the subportfolio level, where contributions and withdrawals don't apply (as you well know). Granted, I argue for only MWRR at this level, but also recognize that I haven't won everyone over yet to this idea.

7. If these discussions are to produce anything productive, then we must first agree on what question we are answering. In this case, there are actually two questions, both of which are seeking an answer in a rate of return. That rate of return itself only has meaning in the context of an amount of capital on which the return is earned. As noted, the time weighted return (TWR) only has meaning when it is applied to a static investment amount. A money weighted return (MWR) produces a return on a variable (or average) amount of capital. TWR is relevant to a manager while MWR is relevant to a portfolio, or in the case of attribution within a portfolio, to its sectors, over which the varying amounts of capital are the essence of the active investment process.

To link shorter period MWR where the cash flows are immaterial is not substantially different from simply running TWR on beginning-of-period capital amounts. If the differences were significant, then this demonstrates that sub-period returns should have been used, thereby producing a true time weighted analysis. That said, the end of a true time weighted analysis is nothing more than the ending capital value divided by the beginning value and then annualized. Obviously, the only thing that is changing the capital value is the return on the investment. All of the periodic returns simply inform on the volatility of the return stream. They are not essential to answering the question: "What was the return on the static capital amount invested?"

In the end, this ongoing discussion is a demonstration that performance practice tends to rush to a calculation before understanding the question to be answered, and as such is "much ado about nothing." That may seem like a thumb in one's eye, but it is rather "accurate." And the "fact" that several (sorry, not "countless") sources believe the TWR story is not proof of its correctness; the masses once believed the earth to be flat. For my money, I'm going with Solnik. MWR and TWR are fish and fowl, oil and water - they just don't mix.

8. Steve, it appears that you wish to change 46 years of performance measurement history, beginning with Dietz's '66 paper. The BAI in '68 published the first standard on performance measurement. They recognized that the "exact" method (i.e., where one revalues for each flow) is the ideal approach. However, they recognized that this isn't always possible, and so they introduced the "Linked IRR" (today called "Modified BAI"), which links subperiod IRRs to produce a TWRR (they didn't even bother to refer to it as an approximation method!). I recognize that the exact method is superior, but for various reasons, not everyone is able to calculate the return this way, and so they use an approximation method (e.g., Modified Dietz). Even GIPS doesn't refer to it as an "approximation method," but does permit its use, and since January 2010, requires revaluing for large flows.

Your issue seems to be associating the term "time weighting" with anything but the "exact method." That's fine. That's your view. And I'm fine with it. But I, and many, many others disagree. Sorry.