The prospect of a better method had me following the link to Justin Bender’s post called “Rate of Return Calculator – Modified Dietz Method.” He offers spreadsheets to help you calculate your returns and says that “By using an approximate time-weighted rate of return (such as the Modified Dietz method), investors will be better able to gauge their performance relative to index benchmarks.”

**Measuring investment skill**

Justin’s calculators don’t actually “track your portfolio returns more accurately.” Your money-weighted return is a good measure of the return you got. But comparing it to a benchmark mixes the results of your active decisions with cash-flow luck. Justin is trying to reduce this luck factor to isolate whether your active portfolio decisions made or lost money when compared to a benchmark. If you happen to get a bonus at work that you invest when the markets are low, then the boost to your return is just luck. You can get unlucky as well with when you have new money to add at a market high or need to make a withdrawal at a market low.

At first I thought Justin was claiming that the Modified Dietz method of calculating returns is time-weighted (but I’m guessing that’s not what he meant). It is clearly money-weighted. Modified Dietz is just a way to approximate the Internal Rate of Return (IRR) method, which is also money-weighted. Modified Dietz’s only virtue is that it’s easier to calculate than an IRR.

Justin’s calculator tries to remove the cash-flow luck factor by calculating a time-weighted return rather than a money-weighted return. One catch in calculating a time-weighted return is that you need to know your portfolio value each time you add or remove money from your portfolio. But, account statements usually only have month-end portfolio values. This is where the Modified Dietz method comes into play.

What Justin is actually doing with his calculators is using month end portfolio values together with Modified Dietz to approximate the time-weighted return for each month of the year. He could just as easily have used the IRR method for each month. Then he compounds the monthly returns to get an estimate of the time-weighted return for the whole year.

All this doesn’t mean that the return Justin’s calculator gives is somehow more accurate than the money-weighted return that will appear in your account statements under the new CRM2 rules. When you compare your return to a benchmark, Justin’s method of calculating your return does a better job of isolating the value of your active portfolio decisions. If you just compare your money-weighted return to the benchmark return, you might be mixing in a significant cash flow luck factor. Although, as I’ll show, Justin’s calculator doesn’t eliminate this luck factor completely.

If you’re going to use Justin’s calculator, you have to make sure to apply cash flows correctly. If you have new money available to invest, but try to time your entry, that’s an active decision. So, you should treat the new money as entering the portfolio immediately, but sitting in cash. So, the date of cash entry in Justin’s calculators should be the date the money was available to invest, not the date it was used to buy equities.

**Another way to measure skill**

Justin’s method of measuring the value of your active portfolio decisions is just one method. He does it by adjusting your return to remove most of the cash-flow luck. Another method is to compute a benchmark return that takes into account your cash flows. With this method, you compute a return as though your portfolio was on auto-pilot being invested in a benchmark. Each cash flow leads to a trade in the benchmark index. This gives an adjusted benchmark index that you can compare directly to your money-weighted portfolio return.

One virtue of this method is that it’s usually easy to find benchmark values for any day of the year and not just month-ends. So, you can more accurately remove cash-flow luck when you compare this adjusted benchmark return to your actual money-weighted return.

**Example**

To explain all this in more detail, I’ll make use of a simple example:

XYZ index fund had a decent year, but it wasn’t steady. By June 30 it had lost 20%, but later rallied. It finished both August and September down 10%, but briefly hit break-even on September 15. An end-of-year rally left the fund up 10% for the year.

Tim had $15,000 in XYZ fund to start the year. He had another $8000 available to invest on June 30 but with XYZ falling, he got nervous and waited until September 15 to invest the $8000 in XYZ.

**Analysis**

We’ll treat XYZ index fund’s 2016 results as Tim’s benchmark.

On September 15, Tim had $15,000 in XYZ fund and added another $8000. Then the fund went up 10%, leaving Tim with $25,300. Tim’s IRR for the year is 13.31%, and his Modified Dietz return is 13.27%, not much different. They’re both much higher than XYZ fund’s 10% return, so Tim looks like a genius, but this comparison mixes in cash-flow timing luck.

If Tim had just invested the $8000 right away instead of waiting for “things to calm down,” he could have ended up with $27,500. That’s $2200 more than he actually got. His IRR could have been 23.9%, and by the Modified Dietz method, it would have been 23.7%. Either way, that’s a much higher return than Tim actually achieved. This is an example of adjusting the benchmark return for Tim’s cash flows to isolate his skill, such as it was.

Any reasonable analysis would conclude that Tim’s market timing effort was a failure. Even though his return is higher than XYZ fund’s 10% return, that’s just because of the lucky timing of his added $8000. In fact, he squandered most of his luck by waiting 3 months to invest it. The comparisons that make Tim look smart just show that his good luck was greater than the poor skill he showed.

**Time-weighted returns**

The idea of time-weighted returns is to eliminate the effect of cash flows in and out of a portfolio. The idea is that if portfolio managers can’t control when you put money in or take it out, they shouldn’t get credited or penalized by the effect of cash flows.

We measure time-weighted returns by breaking up the year based on when the cash flows occur, calculating the return for each interval, and compounding the returns. In Tim’s case, if we ignore his market timing mistake for the moment, his return up to September 15 was 0%, and for the rest of the year was 10%. This compounds to 10%, the same as XYZ fund’s return for the year.

It’s no coincidence that the time-weighted return matches the fund’s return. After all, the goal was to eliminate the effect of cash flows. If Tim had invested the $8000 right away, his return after 6 months would have been -20%. The return from then to September 15 would have been 25%, and 10% for the rest of the year. Once again, these returns compound to 10%.

Let’s return to the case where we recorded the cash flow on September 15 and ignored the poor market timing. The actual time-weighted return is 10%. However, Justin’s calculator computes it as 5.0%. This is because the calculator did not have the portfolio value on September 15 and had to try to estimate September’s time-weighted return.

The calculator essentially treats the cash flow as happening with the market down 10%, when it was actually at break-even. And when the final portfolio value doesn’t reflect this apparent good luck of buying low, the calculator gives a low return.

This is what I mean when I say that the calculator doesn’t completely eliminate cash-flow timing luck. Ordinarily, the error is smaller than it is in this example of wildly fluctuating returns, but the error will still be there in most cases.

To get Tim’s true time-weighted return, we should really be taking into account the $8000 Tim let sit for 3 months gathering no interest. Looked at this way, Tim’s return for the first 6 months was -20%, at which point the new $8000 made his total $20,000. This grew to $23,000 by September 15 for a 15% gain. Finally, he made 10% in the rest of the year. These returns compound to 1.2%, much less than the benchmark return of 10%. So, the time-weighted return method agrees that Tim’s market timing hurt his returns, but it has to be calculated with the cash flows at the correct times.

In this case, Justin’s calculator correctly calculates the time-weighted return as 1.2%. That is because the only cash flow during the year happened at the end of a month. So, there was no error due to not having the correct portfolio value on the date of a cash flow.

**Confusing definitions of “time-weighted”**

There is some possible confusion when it comes to the definition of a time-weighted return. To understand this, we have to go back to the Simple Dietz method of calculating investment returns. This method just treats all cash flows during the year as though they happened at mid-year. The idea is that if you are making regular contributions, the average contribution happened roughly at mid-year.

To make this calculation more accurate, the Modified Dietz was born. This method weights each cash flow by how much of the year is left when the cash flow happens. So, Tim’s contribution on September 15 with 3.5 months left in 2016 gets a weight of 3.5/12. With Simple Dietz, all cash flows get weight 1/2.

This time-weighting used in Modified Dietz is not the same as “time-weighted returns.” The fact that they use the same words is a potential source of confusion. Both Simple Dietz and Modified Dietz are money-weighted methods of calculating returns.

**More about each method of calculating returns**

Regardless of which method you use to calculate your investment return, the idea is to take your starting portfolio value, your cash flows, and your final portfolio value and figure out what steady investment return would have produced the same outcome. The differences among calculation methods come down to what we mean by “steady.”

**Internal Rate of Return**

In the case of the internal rate of return (IRR), we define “steady” as a return at a fixed compounding rate throughout the year. Getting back to Tim’s example above, if he had invested in a fund that produced a steady compounding return of 13.31% per year, he would have ended up with the same amount of money as investing in XYZ fund with his cash flows.

One of the criticisms of the IRR is that it sometimes doesn’t give a single answer. But this only happens in wild situations where any measure of portfolio return is meaningless. An example is a portfolio that starts with $10,000, grows so wildly that it’s possible to withdraw $36,000 after 4 months, and after adding $43,100 at the 8-month mark, drops crazily to $17,160 at year-end. Three different IRR returns fit this data: 33.1%, 72.8%, and 119.7%. In almost all portfolio situations that happen in real life, IRR gives just one answer.

The following figure shows the steady IRR compounding return model. It might look like a straight line, but it’s actually increasing exponentially. If we extended it for a few years, this would be easier to see.

**Modified Dietz**

For the Modified Dietz method of calculating returns, we define “steady” as a kind of reverse simple interest. Instead of the return to date being proportional to time elapsed, the return to the end of the year is proportional to the time remaining. This creates a hyperbolic curve. If Tim had invested in a fund whose returns were of this form that ended the year at 13.27%, he would have ended up with the same amount of money as investing in XYZ fund with his cash flows.

The following figure overlays this reverse simple-interest pattern on top of the earlier compounding pattern for the IRR calculation. The difference between them is tiny, and it’s not hard to see why IRR and Modified Dietz gave very close to the same calculated return.

If Tim had invested the $8000 as soon as it was available, his IRR would have been 23.9%, and the Modified Dietz return would have been 23.7%. The following chart shows the compound interest and reverse simple-interest patterns. Now we see a slightly larger difference between the curves. This is because the difference between simple interest and compound interest is greater as interest rates rise. But it’s still not a big difference, and it’s not hard to see why they give close to the same results.

**Simple Dietz**

The Simple Dietz method has a more basic definition of “steady.” It has close to half the year’s return appear in the first instant of the year, and the other half at the last instant of the year. This method says Tim earned 12.1%. The following chart shows this return pattern.

**Beardstown Ladies**

An investment club called the Beardstown Ladies had their own method of calculating returns. They just ignored new contributions, effectively treating them like part of the investment return. By this flawed method, Tim earned a 69% return!

Their implied definition of “steady” has the investment return leap to infinity in the first instant of the year, and then drop back down to earth in the last instant of the year (see the chart below). This way, any cash flows have no impact because the portfolio value is infinite. If the $8000 cash flow has no impact, then the only way Tim could grow his $15,000 to $25,300 is with a 69% return.

**Conclusion**

There is nothing wrong with using the internal rate of return method to calculate your personal returns. This is what the new CRM2 rules will give you. The Modified Dietz method and time-weighted methods of calculating returns are not more accurate if you just want to know how your portfolio did.

However, just comparing your portfolio’s return to a benchmark can be misleading, depending on what you’re trying to measure. Being higher or lower than the benchmark can be just a matter of luck in the timing of when you had money available to invest or when you needed to make withdrawals.

To isolate the value of your active investment decisions, you need to factor out the luck of cash-flow timing. One way to do this is to compute your time-weighted return and compare it to a benchmark return. Another is to compare your actual return to what you return would have been if you had invested on auto-pilot in a benchmark.

Investing on auto-pilot won’t give you exactly the benchmark return if your portfolio had cash flows. Your personal benchmark requires a return calculation based on what your cash flows would have been if you hadn’t used any of your own discretion. You have to remove the luck factor for things you didn’t control to get at a measure of the results of the things you did control.

So why not just take the value of your portfolio on the day you make a contribution or withdrawal and calculate the actual time weighted return? This should also be available to any DIY investor.

ReplyDelete@Anonymous: That would work, but for most investors, it would mean recording the portfolio value each time you have a cash flow. Note that some cash flows are automatic, such as for regular contributions. If you fail to record the value, as most of us would do at least some of the time, you may find it a lot of work to reconstruct what the portfolio value was on a particular day. I assume that is why Justin created his spreadsheets to work from month-end values.

DeleteNice post! I spent a lot of time researching how to calculate and compare returns several years ago, it would have been a lot easier if I had this post then.

ReplyDelete@Greg: Glad you like it. I've had some of these ideas bouncing around in my head for a while, but this is the first time they solidified into something useful I could write down.

Delete