## Time Weighted vs. Money Weighted Returns

### Market Updates

02/08/18Many people assume calculating performance returns on investment accounts to be a very simple process. In its simplest form, you can get your percentage investment return by using this simple equation: (Ending Value – Beginning Value) / Beginning Value. With that in mind, it can get more complicated when you add in factors such as external cash flows, dividends/distributions, timing of when to value the portfolio, amounts of cash held in the account and expenses/fees. To add a little more complication, there are two types of return calculation methodologies that are used: Time-Weighted Return (“TWR”) and Money-Weighted Return (“MWR”).

__Time-Weighted Return:__

There is actually more than one TWR calculation and they include: the Original Dietz method, the Modified Dietz method and the Daily Valuation method. The best method of these three is the Daily Valuation method, which gives you a “true” TWR. TWR breaks the total performance for a desired period into sub-periods that are defined by any occurrence of an external cash flow. As the name of the return indicates, the return is weighted on the amount of time in each period. The basic characteristics of each of these time-weighted return calculations are the following:

- Total returns must be used.
- TWR adjusts for external cash flows. This is done by subtracting contributions and adding withdrawals to the ending value before calculating each sub-period return.
- Each sub-period is geometrically linked (they each are multiplied together).

__Money-Weighted Return:__

The MWR sets the terminal value (ending value) and the present value of all cash flows in the desired period equal to the initial investment. Simply put, it is the internal rate of return. As the name implies, the periods with the most money will be weighted higher in the return calculation. Large external cash flows can have a massive influence on the final return figure. After you have entered the numbers correctly in a spreadsheet, Excel’s “=XIRR” formula makes the calculation quite easy. What you need in order to calculate the MWR are the following:

- Beginning portfolio value and any contributions over the performance period being measured are added as positive values.
- The ending portfolio value (terminal value) and withdrawals from the portfolio are inputted as negative values.

The easiest way to understand the differences between TWR and MWR is by visualizing them in a few examples:

__Example A: __

A client’s starting portfolio value on January 1^{st} is $100,000 and the year ending value is $150,000. During the year, the portfolio returns are quite volatile and there are no external cash flows (contributions or withdrawals). What is the return under each methodology?

Given that there will be no external cash flows; the TWR and MWR are both 50%. Remember, TWR calculates each sub-period by normalizing the external cash flows. With no external cash flows, even if the portfolio has large swings in its value during the year, the only thing that determines the TWR is the ending value of $150,000.

The 1^{st} and 2^{nd} columns in the TWR spreadsheet above are just the beginning date and portfolio value at that date. The 3^{rd} column is the external cash flows throughout the year. The 4^{th} column, “Sub-Period Return + 1” is just a simple monthly return calculation: ((Beginning Value – Ending Value) / Beginning Value) + 1. The reason we add “1” to each return in the 3^{rd} column is because we use these monthly sub-periods to multiply together to get our final TWR. The 4^{th} column is the product of the previous geometrically linked returns and the current sub-period return. Putting it all together you get a TWR of 50%. It sounds complicated, but try it for yourself. Multiply the 12 monthly sub-period returns in column 4 and you will end with a value of 1.5 which is a 50% return.

Using the same example on the right for the MWR is an even easier calculation with the help of Excel. After entering the $100,000 for beginning value in the 1^{st} row in column 2, $0 in each month of the year representing no external cash flows, and the ending value of the portfolio at the end of year as a negative number, we can use the XIRR excel formula to calculate the MWR. The formula is displayed above the actual MWR calculation for your reference. That all seems pretty simple and logical but let’s run through one example which includes an external cash flow.

__Example B:__

The client’s portfolio value is $100,000 to start but the client decides to contribute $900,000 on June 1^{st} after the portfolio performed very well for the first 5 months of the year. Sadly, the portfolio ends the year with only $800,000 even after accounting for the $900,000 contribution. What is the TWR and MWR for the year?

Even though the portfolio value is $800,000 at the end of the year, which is less than the total amount of money deposited into the account, the TWR was actually a positive 146.15%. Over that same time period, the MWR was a negative 30.02%.

__TWR vs. MWR:__

You’re probably asking yourself a few questions.

- What is the “right” way of calculating returns?
- How can I have a dollar amount that is less than my total contributions and at the same time show a positive TWR?
- Why are those return figures between the TWR and the MWR so different and what can this tell an investor?

The first question is a trick question because there isn’t an exact “right” way of calculating returns. The Global Investment Performance Standards, also known as GIPS, recommends that money managers use the TWR calculation. The reason they recommend using the TWR is because it backs out any external cash flows which doesn’t penalize or reward the money manager for the amount and timing of those contributions over which they have no control. It also gives investors the ability to properly evaluate money manager’s investment decisions along with giving them a return that can be used across the board to compare with other investments.

The second question is answered by comprehending what TWR calculations actually show you. The easiest explanation is that given the starting point for the calculation, for every dollar invested on that date, your money would have grown by the stated percentage. In example B, if you had started with $100,000 and didn’t contribute the $900,000 on June 1^{st}, the portfolio would have been worth $240,153.80 at the end of year which is a 146.1538% return. If you look at the actual return figures in the example, the portfolio was up 650% over the first 5 months of the year before the contribution and then had a negative return of 55.37% for the remaining 7 months. By using the TWR calculation, you can compare managers over the same time period and thus have a direct comparison to determine which investment performed better.

The answer to the last question can show you if you are good at timing the market. If the MWR is higher than the TWR, you have been successful at timing the market in regards to your contributions into your account. On the other hand, if your TWR is higher than your MWR, like in Example B, you are doing yourself a disservice by trying to time the market. In this case, instead of accumulating a lot of cash over time and investing it all at once, you might want to consider dollar-cost averaging by regularly adding small amounts of money to your account over time rather than in one big chunk.