What Is A Time Weighted Rate of Return?

A Time Weighted Rate of Return (TWRR) is one type of return you can calculate on your investments.
Yes, you can actually calculate different returns, even if you earn one amount on your investments!  For example, lets say you invested $100 into the market and your investments went down by 5%.  Then you invest more money in and the markets go up by 10%. Since you had more money invested when the markets went up, would your return be higher than 7.5%?  Not with TWRR.
TWRR attempts to show you how much you earned in percentage terms regardless of how much money you had invested.
In other words, the TWRR tries to answer the “growth of a dollar” question: if you invested $1 at the beginning, and didn’t add or take out any money since then, how much would that be worth now?
This differs from a money weighted return (IRR) which does take into account how much money was invested.

How Does the Time Weighted Rate of Return Formula Work?

A time weighted return works by first calculating a return for a time period, usually a day or a month, and then “links” these returns together.
For example, let’s say you have a $100 investment that grows to $105 for a 5% return in January.
In February, you earn another $5.  You will calculate another return for February which would take the $5 gain over the starting value of $105, which is a 4.76% return (calculated as $5 / $105).
To get the the time weighted return over the entire time period (Jan 1 – Feb 28), you can either:
1) “Link” these returns together (also known as compounding) to arrive at the total return of 10%:
(1 + 5%) * (1+ 4.76%) – 1 = 10%
Or..
2) Simply take the $10 total that we earned on $100, and get the same 10% return! That is calculated as $10 / $100 = 10%, which seems simple…but, wait!
You can only use option #2 when there are no flows.  This is very important.  Because investments usually have cash flows, option #2 is usually not a valid way to calculate a time weighted rate of return formula.  In fact, when there are no flows, all returns (time weighted and money weighted) will be 10% in the above example.

Introducing Flows

I have another post which dives into the details of flows, but I will just give you a quick overview here.  Cash flows are when an investor adds or takes money out of the portfolio.
The reason why these complicate returns so much, is because we can’t simply look at what we started with and ended with to calculate our return. In other words, we cant simply say we started with $100 and ended with $110 for a 10% return.
Let’s go back to our example from above where we started with $100 in investments and ended at $110 in investments.  But this time, I also tell you that the $5 increase in value during February was due to the fact that you earned $2 and deposited $3.
What is the January return and what is the February return?  Well, in January, the return is still 5% since you gained $5 on $100 and there were no flows.
The February return, however, is not 4.76%.  You actually only earned $2 on the investments, the other $3 increase which got you to the $110 was due to the fact that you deposited money.
Assuming you put the $3 in at the beginning of February, that means that you earned $2 on $108.  The $108 comes from the $105 you ended with in January + $3 you deposited.  The return for February is $2 / $108 = 1.85%.
If you link these returns together, you get 6.94%
(1 + 5%) * (1 + 1.85%) – 1 = 6.94%
Well, this seems close to just taking the $7 in profits ($5 from January + $2 in February) and dividing them by $100.  Can’t we just do that?
No, you cannot!  In this scenario we had only two monthly returns and a relatively small deposit ($3 over $105).  When there are large deposits and withdrawals and you have longer time periods (e.g., years), you need to follow the linking process or your returns will not be close to what you earned at all!

Different Types of TWRR

Is there only one type of TWRR formula?  Absolutely not!  However, all TWRR formulas break down into two concepts:
• Profit or loss, or Gain/Loss (the numerator)
• How Much Money You Invested, aka “Investment Basis” (the denominator)
If you look up the TWRR in a book or on the internet, you will commonly find a formula that looks something like this:

Don’t be scared if you don’t know what that means!  I’ll interpret it for you:
The numerator is simply how much you made on your investments, in other words your profit or loss.
The denominator is simply how much you had invested, or what we will call the “investment basis.”

The only difference between the different TWRR formulas is the denominator.  The numerator is always the profit or loss.  The denominator is what changes.
1) How Frequent They Value Your Assets: some versions of the formulas calculate the amount of money you had invested on a monthly basis ,while others measure how much money you had invested on a daily basis.  These daily vs monthly valuations is called the “valuation frequency” in the industry.
2) Money Being Contributed or Taken Out Of The Portfolio: some versions assume that when you put money into the portfolio, it could be invested right away and therefore, any gains/losses should be divided by a number that includes contributions.  This is called “beginning of day” (BOD) cash flow assumption.  Other versions assume the money can’t be invested right away so therefore, any gains/losses should be divided by a number that does not include these contributions.  This is called the “end of day” (EOD) cash flow assumption.  Whether you use a BOD or EOD cash flow assumption should be stated in your cash flow policy.
Here is a grid of the 8 most popular combinations, and I will go through the details below using an example.

Example
A portfolio starts the month with $100 and ends the month with $128.  That’s a change in value of  $28.   We are also told that during that time the client contributed $20.  This means that the profit on the investments is only $8. OK, so we have the first piece – the profit or loss.
The other piece of the puzzle (or formula) that is needed is what you should divide the $8 by, also called the investment basis.  It could be that:
  • The $8 was earned before the additional $20 came in, and therefore you earned $8 / $100.
  • The $8 was earned on the full $120 and therefore the return is $8 / full $120
  • The $20 came sometime in the middle of the measurement period, either as one contribution of $20 or in multiple increments.
In the first two cases, you can use the simplified formula (either $8 / $100 or $8 / $120).  However, in the last case  – think about this – what should you divide the $8 by?  If I told you that 1/3 of the way in you got $10 and then another 1/3 of the way in you got another $10.  You can see how that would affect what you divide the  $8 profit over.   Some formulas require you to know the date of the flow, while other just require you to make an assumption of when the additional funds came in.
Another complexity is what profits were earned in each of the time periods when additional money came in.  Let’s say you earned $2 in the first period, $10 in the second period, and -$4 in the 3rd period). Some formulas require you to know what the profits were in between each cash flow (to try to be more exact), while others just use the full $8 in the numerator.
I would also like to note that daily-valued mutual funds, which use a NAV/share to calculate a return, are using the daily “true” TWR.  See the following proof.

History of Time-Weighted Rate of Return

I would like to reference a great post on the where the name time-weighted rate of return came from.
For more details on daily “true” time weighted return, see the page dedicated to it here.

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