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June-15-15

How to Calculate Your Approximate Time-Weighted Rate of Return (ATWRR)

In my recent blog post, I explained that the Modified Dietz rate of return gives a decent estimate of the money-weighted rate of return (MWRR). However, both types of returns are not ideal choices for investors who are interested in benchmarking their performance against appropriate indices.  

Some investment firms mitigate this issue by approximating a time-weighted rate of return for their clients. This can be accomplished by calculating the Modified Dietz rate of return over monthly time periods, and then geometrically linking the results (going forward, I will refer to this methodology as the “approximate time-weighted rate of return”). This requires month-end portfolio values, but avoids having to value the portfolio whenever an external cash flow occurs (which is required when calculating the time-weighted rate of return).

The approximate time-weighted rate of return (ATWRR) can differ substantially from the time-weighted rate of return (TWRR) when large cash flows occur during months of significantly fluctuating portfolio values. This makes the ATWRR less ideal for benchmarking portfolio managers or strategies than the TWRR (but generally does a better job than the MWRR or the Modified Dietz rate of return). For example:

  • When a large contribution is made prior to a sub-period month of relatively good (bad) performance, the ATWRR will overstate (understate) a portfolio’s performance, relative to the TWRR.
  • When a large withdrawal is made prior to a sub-period month of relatively good (bad) performance, the ATWRR will understate (overstate) a portfolio’s performance, relative to the TWRR.

To make the ATWRR calculation easier for investors, I’ve created a user-friendly Modified Dietz annual rate of return calculator (with geometric linking of the monthly returns), available for free download on the Canadian Portfolio Manager Blog. Simply input the month-end portfolio values for the year in column E, and any contributions (+), withdrawals (-), and the day of the month that each cash flow took place in the columns to the right of column E (the calculator allows for up to five cash flows per month). I’ve included the calculator screen snapshots below for both investors, based on values from our original example.

Approximate Time-Weighted Rate of Return: Investor 1

Source: Canadian Portfolio Manager blog

Approximate Time-Weighted Rate of Return: Investor 2

Source: Canadian Portfolio Manager blog

In order to help us better understand the differences between the ATWRR and the TWRR, we must first calculate the sub-period returns during the months when any external cash flows occurred (using the TWRR methodology). In our example, the only cash flows that occurred were during the month of September.

Example: Time-Weighted Rate of Return for the Month of September – Investor 1

Example: Time-Weighted Rate of Return for the Month of September – Investor 2

For both investors, their sub-period rate of return before the cash flow occurred was -0.85%. After the cash flow occurred, the portfolio returned -3.42% for the remainder of the month (a relatively worse return than the first half of September). Over the entire month of September, the portfolio returned -4.24%.

Investor 1 had a relatively worse ATWRR in September of -4.35% (when compared to the TWRR of -4.24%). This was because Investor 1 contributed $25,000 before a monthly sub-period of relatively bad performance (-3.42% versus -0.85%). They also had a lower ATWRR of 9.67% during the 2014 calendar year, when compared to the TWRR of 9.79%.

Investor 2 had a relatively better ATWRR in September of -4.13% (when compared to the TWRR of -4.24%). This was because Investor 2 withdrew $25,000 before a monthly sub-period of relatively bad performance (-3.42% versus -0.85%). They also had a higher ATWRR of 9.92% during the 2014 calendar year, when compared to the TWRR of 9.79%.

Although the ATWRR can differ from the TWRR when large external cash flows are made during a volatile month, it is still a decent choice for investors who are looking for an approximate method of calculating a rate of return that can arguably be benchmarked against index returns. As most investors will not be provided with this type of return on their account statements going forward (and collecting daily portfolio valuations when external cash flows occur is not realistic), the approximate time-weighted rate of return would be our recommended choice for investors who are interested in benchmarking their portfolio returns.

Performance Results

By: Justin Bender | 1 comments
June-12-15

How to Calculate your Modified Dietz Rate of Return

The Modified Dietz rate of return attempts to estimate a money-weighted rate of return (MWRR) by weighting each cash flow by the proportion of the measurement period it is present or absent from the portfolio.

Similar to the money-weighted rate of return, the calculation requires the investor to know the portfolio values at the start and end of the measurement period, as well as the cash flow amounts and dates when each cash flow occurs. Unlike the MWRR, the calculation does not require an exhaustive trial and error procedure, or sophisticated computing power.

Source: CFA Institute

The Modified Dietz rate of return can differ substantially from the time-weighted rate of return (TWRR) when large cash flows occur during periods of significantly fluctuating portfolio values (just like the money-weighted rate of return). This makes the Modified Dietz rate of return less ideal for benchmarking portfolio managers or strategies than the TWRR. For example:

  • When a large contribution is made prior to a period of relatively good (bad) performance, the Modified Dietz rate of return (ModDietz) will overstate (understate) a portfolio’s performance, relative to the time-weighted rate of return (TWRR).
  • When a large withdrawal is made prior to a period of relatively good (bad) performance, the Modified Dietz rate of return (ModDietz) will understate (overstate) a portfolio’s performance, relative to the time-weighted rate of return (TWRR).

Using the values from our original example, we would plug in the appropriate numbers and calculate the rate of return for each investor.

Example: Calculation of wi

Example: Modified Dietz Rate of Return (MDRR) – Investor 1

Example: Modified Dietz Rate of Return (MDRR) – Investor 2

Performance Results

As we can see in the chart above, the Modified Dietz rate of return is nearly identical to the money-weighted rate of return. In my final blog post of the series, we will examine how calculating the Modified Dietz rate of return over monthly time periods can help an investor better estimate the time-weighted rate of return.

By: Justin Bender | 0 comments
June-05-15

How to Calculate your Money-Weighted Rate of Return (MWRR)

Starting in July 2016, dealers and portfolio advisors will be required to report investment performance to their clients. The money-weighted rate of return (MWRR) was chosen by the Canadian Securities Administrators (CSA) as the industry standard for these performance calculations. Although the MWRR is arguably more relevant to the individual investor (as it can reward or penalize investors for the timing of their cash flows), it is considered by most advisors to be inadequate for benchmarking purposes. This is because the timing of the investor's cash flows (which most advisors have little to no control over) can cause the performance to be over or understated, relative to the time-weighted rate of return (TWRR).

The money-weighted rate of return can be thought of as the rate of return, r, which equates the right hand side of the following equation to the ending portfolio value, V1.

Source: CFA Institute

This method can be useful for calculating the rate of return when there have been only small external cash flows during the measurement period, relative to the size of the portfolio. It may also be the only available option for investors who do not have access to daily or month-end portfolio values (I often come across investors who receive quarterly statements, as opposed to monthly).

As the MWRR assumes all cash flows receive the same rate of return while invested, its return can differ substantially from the time-weighted rate of return (TWRR) when large cash flows occur during periods of significantly fluctuating portfolio values. This makes the MWRR less ideal for benchmarking portfolio managers or strategies than the TWRR. For example:

  • When a large contribution is made prior to a period of relatively good (bad) performance, the money-weighted rate of return (MWRR) will overstate (understate) a portfolio’s performance, relative to the time-weighted rate of return (TWRR).
  • When a large withdrawal is made prior to a period of relatively good (bad) performance, the money-weighted rate of return (MWRR) will understate (overstate) a portfolio’s performance, relative to the time-weighted rate of return (TWRR).

Without the help of computers, the calculation is just a series of trials and errors. Using the above equation and the values from our original example, Investor 1 would begin by plugging in return "guesses" for r until the right-hand side of the equation equals the ending portfolio value, V1. They would eventually stumble across 8.98% as the plug return that equates the right-hand side of the equation to 298,082 (or as close as possible).

Example: Manual MWRR calculation for Investor 1

An easier way for investors to calculate their MWRR would be to download the Money-Weighted Rate of Return Calculator, available in the Calculators section of the Canadian Portfolio Manager Blog. This calculator requires minimal inputs and is fairly intuitive to use. It also annualizes (averages) returns over periods longer than a year.

After downloading the Excel spreadsheet, select the start and end dates for your measurement period, entering the total portfolio value to the right of each date. Next, enter the dates and amounts of any portfolio contributions (+) or withdrawals (-) during the measurement period. I’ve included examples for both investors below.

Money-Weighted Rate of Return (MWRR): Investor 1

Money-Weighted Rate of Return (MWRR): Investor 2

The MWRR results are noticeably different than the TWRR results from our first example. Investor 1 contributed $25,000 to their portfolio before a period of underperformance (-5.56% versus +16.25%) and ended up with a significantly lower MWRR of 8.98%. On the other hand, Investor 2 withdrew $25,000 from their portfolio before a period of underperformance, which resulted in a significantly higher MWRR of 10.64%. This makes intuitive sense; Investor 1 made a bad timing decision by adding funds right before the markets went down, while Investor 2 made a good timing decision by withdrawing funds right before the markets went down.

Performance Results

Each investor’s cash flow decision resulted in a higher or lower MWRR, relative to the TWRR. Their investment strategy was exactly the same in each case (i.e. to track the MSCI Canada IMI Index). By comparing their MWRR to an index return, both investors may incorrectly conclude that their portfolio manager has underperformed or outperformed the benchmark (which is why a money-weighted rate of return should not be used for benchmarking purposes).

Next up, we will examine the Modified Dietz Rate of Return.

By: Justin Bender | 0 comments