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 w_{i}

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

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

Performance Results

Methodology

Investor 1

Investor 2

Time-Weighted Rate of Return (TWRR)

9.79%

9.79%

Money-Weighted Rate of Return (MWRR)

8.98%

10.64%

Modified Dietz Rate of Return (ModDietz)

8.97%

10.66%

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.

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.

I completely agree, in that I believe the Modified Dietz method is a suitable MWRR estimation method. As opposed to a TWRR method.

My only question to you would be…in the context of CRM2 regulatory requirements, and the need to provide MWR returns to clients…do you think Modified Dietz is a “fair” ROR to provide to them, that reflects their own deposit/withdrawal timing and amount decisions?

@Tom – the Modified Dietz method (without monthly geometric linking of returns) would be expected to approximate the MWRR. I would consider it to be fair, but the Canadian regulators have not made a similar claim (so they would likely disallow this type of return).

The Modified Dietz method (with monthly geometric linking of returns) would be expected to approximate the TWRR.

I cannot seem to get the MWRR calculator to work correctly for me. I have only 8 entries for the year with a start date of Dec 31, 2014. In the return percentage I only get ####. Even if I move my end date to 2016. Any suggestions?

@kulvir – the MWRR calculator requires the period to be at least one year in length. Please email me your spreadsheet and I’ll take a look at it: jbender@pwlcapital.com

In this example, if you have more cashflows , will we need to do the procced for each cashflow? or we can sum all the cash flow (CF*W)+(CF*W) (…)
Thank you.

@Rui Costa – for the numerator section, you would add up all of your monthly cash flows to equal “CF”. For the denominator, you would weight each cash flow as you mentioned. I would recommend using my Modified Dietz calculator – it’s much easier: http://www.canadianportfoliomanagerblog.com/calculators/

I have a question that I cannot–for the life of me–figure out the answer to. With Modified-Dietz, are dividends considered a cash flow to be weighted in the denominator? I am not sure because, they are not contributions/withdrawals, but inflows due to the investment and part of the overall period gain. For example, if I start the month with $100, end the month with $150, have a dividend of $2 paid on the 14th (31 day convention), have a cash inflow of $20 on the 3rd and have a cash outflow of $5 on the 17th:

Numerator: (150-100+2)-20+5); the total gain, in this case, is 37.
Denominator (With Dividend): (100+(2*(17/31))+(20*(28/31))+(5*(17/31))); the weighted average amount at risk over the month, in this case is, 121.9032
Return: 0.3035194

Numerator: (150-100+2)-20+5); the total gain, in this case, is 37.
Denominator (Without Dividend): (100+(20*(28/31))+(5*(17/31))); the weighted average amount at risk over the month, in this case is, 120.8065
Return: 0.3062750

A 28 bps difference is too large for me to ignore. Which way is more precise/correct

@Daniel – dividends that are paid will automatically be included in the month-end portfolio market value, so they do not need to be included as cash flow inputs.

Thanks for your quick response. If a dividend is received in the month (like in my example on the 14th), that “new” capital ($2) is assumed to not get reinvested for the percentage of the month it was in the account (17/31, or 0.5484)? Is this why it doesn’t get added to the denominator? Technically, if a dividend is assumed to be reinvested it should contribute to the denominator just like any other cash inflow, correct?

@Daniel – no, the dividend will be included in the numerator (as part of “V1). If it is paid and sits in cash, then it’s contribution to the return for the rest of the month will be zero. If it is reinvested, then it will affect the remainder of the month return by either increasing the value of V1 in the numerator or decreasing it.

“If it sits in cash” is the key assumption here. Aren’t dividends in indices assumed to be reinvested (for benchmarking)? I understand if the dividend becomes reinvested it will add to the ending market value in the numerator, but if it is also assumed to be reinvested, I still think it should also be placed in the denominator for the days it was invested in order to capture that fact that the dividend was deployed inter-month.

For example, under what you’re suggesting, if I start the 31-day month with $100, receive a dividend of $2 on the 16th, and have $105 at the end of the month my return would be:

(105-100+2)/(100) = 7%

However, this does not capture the fact that the dividend was redeployed into other investments and at risk for 15 days of the month, meaning that I think the return should be:

@Daniel – perhaps the following three examples will help you better understand the concepts:

1. A $100 stock increases to $107 after 15 days (it pays no dividend at this time). At the end of the month, the stock has increased further to $110. For the first half of the month, the return is 7% [$107/$100 – 1]. For the second half of the month, the return is 2.80373% [110/107 – 1]. If we link these returns together, the monthly return is 10% [(1.07) x (1.0280373) – 1]. You could also have calculated the monthly return as $110/$100 – 1.

2. A $100 stock increases to $107 after 15 days, at which point it pays a $2 dividend (the price of the stock immediately drops by the same amount, to $105). The investor immediately reinvests the $2 back into the stock (so they again have a stock investment worth $107). For the first half of the month, the return is 7% [(105 + 2) / 100 – 1]. Since we know from example #1 that the return of the stock in the second half of the month is 2.80373%, our $107 investment increases to $110 by month-end [107 x (1.0280373)]. The monthly return would also be 10% [$110/$100 – 1].

3. A $100 stock increases to $107 after 15 days, at which point it pays a $2 dividend (the price of the stock immediately drops by the same amount, to $105). The dividend is left in cash (so does not benefit from the stock’s return in the remainder of the month. The existing $105 stock increases by 2.80373% in the second half of the month to $107.94 [105 x (1.0280373)]. There is also the $2 dividend sitting in cash at the end of the month, for a total portfolio value of $109.94. The monthly return is therefore 9.94% [$109.94/$100 – 1].

The first two examples show that it makes no difference whether a dividend is paid by the stock or not (as long as the dividend is immediately reinvested). If the dividend is not immediately reinvested and sits in cash (third example), it can affect the rate of return (positively or negatively), but this is already accounted for without weighting the dividend in the denominator.

So, is the idea to use monthly statements to manually record the total value at the end of each month?

Also, I was thinking cash flows were simply (large) contributions (e.g. a TFSA/RRSP contribution). You mention above that dividends would be included in the month end value and thus no need to single out. Does that also apply to other cash flows like account fees and withholding taxes?

@skube: In order to calculate the Modified Dietz return using my downloadable rate of return calculator, you will require month-end statements to obtain the total value at the end of each month (or you’ll have to login to your account at the end of each month after the closing values have been updated).

You are correct – account fees and withholding taxes can be ignored, as these will all reduce the value of the month-end portfolio value (and lead to a lower return).

Thanks! I’m now a little confused regarding your downloadable calculator which is titled Modified Dietz Rate or Return (MDRR) but your white paper seems to conclude that the Monthly Modified Dietz Rate or Return (MMDRR) is the most preferable method.

You state:

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.

I completely agree, in that I believe the Modified Dietz method is a suitable MWRR estimation method. As opposed to a TWRR method.

My only question to you would be…in the context of CRM2 regulatory requirements, and the need to provide MWR returns to clients…do you think Modified Dietz is a “fair” ROR to provide to them, that reflects their own deposit/withdrawal timing and amount decisions?

Cheers,

@Tom – the Modified Dietz method (without monthly geometric linking of returns) would be expected to approximate the MWRR. I would consider it to be fair, but the Canadian regulators have not made a similar claim (so they would likely disallow this type of return).

The Modified Dietz method (with monthly geometric linking of returns) would be expected to approximate the TWRR.

Hello Justin,

I cannot seem to get the MWRR calculator to work correctly for me. I have only 8 entries for the year with a start date of Dec 31, 2014. In the return percentage I only get ####. Even if I move my end date to 2016. Any suggestions?

@kulvir – the MWRR calculator requires the period to be at least one year in length. Please email me your spreadsheet and I’ll take a look at it: jbender@pwlcapital.com

Hello Justin,

In this example, if you have more cashflows , will we need to do the procced for each cashflow? or we can sum all the cash flow (CF*W)+(CF*W) (…)

Thank you.

@Rui Costa – for the numerator section, you would add up all of your monthly cash flows to equal “CF”. For the denominator, you would weight each cash flow as you mentioned. I would recommend using my Modified Dietz calculator – it’s much easier: http://www.canadianportfoliomanagerblog.com/calculators/

I have a question that I cannot–for the life of me–figure out the answer to. With Modified-Dietz, are dividends considered a cash flow to be weighted in the denominator? I am not sure because, they are not contributions/withdrawals, but inflows due to the investment and part of the overall period gain. For example, if I start the month with $100, end the month with $150, have a dividend of $2 paid on the 14th (31 day convention), have a cash inflow of $20 on the 3rd and have a cash outflow of $5 on the 17th:

Numerator: (150-100+2)-20+5); the total gain, in this case, is 37.

Denominator (With Dividend): (100+(2*(17/31))+(20*(28/31))+(5*(17/31))); the weighted average amount at risk over the month, in this case is, 121.9032

Return: 0.3035194

Numerator: (150-100+2)-20+5); the total gain, in this case, is 37.

Denominator (Without Dividend): (100+(20*(28/31))+(5*(17/31))); the weighted average amount at risk over the month, in this case is, 120.8065

Return: 0.3062750

A 28 bps difference is too large for me to ignore. Which way is more precise/correct

@Daniel – dividends that are paid will automatically be included in the month-end portfolio market value, so they do not need to be included as cash flow inputs.

Justin,

Thanks for your quick response. If a dividend is received in the month (like in my example on the 14th), that “new” capital ($2) is assumed to not get reinvested for the percentage of the month it was in the account (17/31, or 0.5484)? Is this why it doesn’t get added to the denominator? Technically, if a dividend is assumed to be reinvested it should contribute to the denominator just like any other cash inflow, correct?

@Daniel – no, the dividend will be included in the numerator (as part of “V1). If it is paid and sits in cash, then it’s contribution to the return for the rest of the month will be zero. If it is reinvested, then it will affect the remainder of the month return by either increasing the value of V1 in the numerator or decreasing it.

“If it sits in cash” is the key assumption here. Aren’t dividends in indices assumed to be reinvested (for benchmarking)? I understand if the dividend becomes reinvested it will add to the ending market value in the numerator, but if it is also assumed to be reinvested, I still think it should also be placed in the denominator for the days it was invested in order to capture that fact that the dividend was deployed inter-month.

For example, under what you’re suggesting, if I start the 31-day month with $100, receive a dividend of $2 on the 16th, and have $105 at the end of the month my return would be:

(105-100+2)/(100) = 7%

However, this does not capture the fact that the dividend was redeployed into other investments and at risk for 15 days of the month, meaning that I think the return should be:

(105-100+2)/(100 + ((15/31)*2) = 6.93%

Hope this brings clarity.

@Daniel – perhaps the following three examples will help you better understand the concepts:

1. A $100 stock increases to $107 after 15 days (it pays no dividend at this time). At the end of the month, the stock has increased further to $110. For the first half of the month, the return is 7% [$107/$100 – 1]. For the second half of the month, the return is 2.80373% [110/107 – 1]. If we link these returns together, the monthly return is 10% [(1.07) x (1.0280373) – 1]. You could also have calculated the monthly return as $110/$100 – 1.

2. A $100 stock increases to $107 after 15 days, at which point it pays a $2 dividend (the price of the stock immediately drops by the same amount, to $105). The investor immediately reinvests the $2 back into the stock (so they again have a stock investment worth $107). For the first half of the month, the return is 7% [(105 + 2) / 100 – 1]. Since we know from example #1 that the return of the stock in the second half of the month is 2.80373%, our $107 investment increases to $110 by month-end [107 x (1.0280373)]. The monthly return would also be 10% [$110/$100 – 1].

3. A $100 stock increases to $107 after 15 days, at which point it pays a $2 dividend (the price of the stock immediately drops by the same amount, to $105). The dividend is left in cash (so does not benefit from the stock’s return in the remainder of the month. The existing $105 stock increases by 2.80373% in the second half of the month to $107.94 [105 x (1.0280373)]. There is also the $2 dividend sitting in cash at the end of the month, for a total portfolio value of $109.94. The monthly return is therefore 9.94% [$109.94/$100 – 1].

The first two examples show that it makes no difference whether a dividend is paid by the stock or not (as long as the dividend is immediately reinvested). If the dividend is not immediately reinvested and sits in cash (third example), it can affect the rate of return (positively or negatively), but this is already accounted for without weighting the dividend in the denominator.

So, is the idea to use monthly statements to manually record the total value at the end of each month?

Also, I was thinking cash flows were simply (large) contributions (e.g. a TFSA/RRSP contribution). You mention above that dividends would be included in the month end value and thus no need to single out. Does that also apply to other cash flows like account fees and withholding taxes?

@skube: In order to calculate the Modified Dietz return using my downloadable rate of return calculator, you will require month-end statements to obtain the total value at the end of each month (or you’ll have to login to your account at the end of each month after the closing values have been updated).

You are correct – account fees and withholding taxes can be ignored, as these will all reduce the value of the month-end portfolio value (and lead to a lower return).

Thanks! I’m now a little confused regarding your downloadable calculator which is titled

Modified Dietz Rate or Return(MDRR) but your white paper seems to conclude that the(MMDRR) is the most preferable method.MonthlyModified Dietz Rate or Return@skube: My downloadable calculator uses monthly returns as well.