# TWR vs MWR: calculating the benefits of deposit and withdrawal

Time-Weighted Return (TWR) and Money-Weighted Return (MWR) are excellent metrics to analyze your portfolio’s performance. But they tend to be misunderstood. I want to unravel this today by using my own portfolio and the multiple deposits I did over the course of a couple of months.

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On 3 March I added 7,500 EUR to my portfolio from a tax deduction I received. Then one month later, on 5 April, I added another 7.5K, from that same deduction plus my salary to increase the total deposits to 40,000 EUR.

The main benefit of doing that is being back in the green, in terms of cash. While using margin after the first 7.5K deposit can be beneficial if the market moves up again, it does affect your gains since you have to pay a (relatively) small percentage of interest on the borrowed money.

For Lynx this percentage is:

Benchmark (BM) | BM rate | Credit | Debit |

Source: https://www.lynx.be/tarieven-beleggen/alle-tarieven/#1499945626553-4ab91661-dc67 | |||

EONIA (Euro Overnight Index Average) | -0.550% | 0.0% | BM + 3,50% (=2.95%) |

In other words, I pay 2.95% every year, calculated monthly. In practice, this means that over the last 3 months I paid approximately the following in interest:

Date | Amount |

4 Mar. 2020 | -3,40 EUR |

3 Apr. 2020 | -6,78 USD (~6.28 EUR) |

3 Apr. 2020 | -4,36 EUR |

5 May 2020 | -8,40 USD (~ 7.75 EUR) |

5 May 2020 | -10,20 EUR |

Total | -31.96 EUR |

But as a result of this margin, I made a larger gain. Even though the market recovered losses it incurred, I’m already back above my initial deposit.

## TWR and MWR

It might seem like an easy and obvious case, given the recent bear market and subsequent recovery, but let’s take a look at the true numbers.

I’m going to use both Time-weighted Return (TWR) and Money Weighted Return (MWR). This will help make it clear whether my decision to add the use of margin (i.e., buy shares by borrowing money), and the following deposits were a positive move or not.

Before I continue, I want to state I know that the Corona-crisis is still ongoing. Governments have only recently started to look at exit strategies or only executed the first steps to reopening after a lockdown. Still, since I topped up my account and am no longer in red, I think it’s an excellent time to drill into the returns.

### What does time-weighted return mean?

The time-weighted rate of return, also known as the geometric mean of return, measures the return of each intermediate interval and aggregates it to generate the return of the entire period. Intermediate cash inflows or outflows do not impact this rate. The growth in each period is multiplied to arrive at the rate of return for the actual period.

Sounds complicated so let us use numbers. Assume that we want to determine the 1-year return of the following cash flows:

Time (in months) | Initial Value | Deposit/withdraw | Final Value |

0 | 100 EUR | 0 | |

6 | 110 EUR | 10 EUR | 100 EUR |

12 | 120 EUR | 0 EUR | 120 EUR |

The final value is the initial value adjusted for any deposits or withdrawals. Since there is a withdrawal in our case, the final value is lesser than the original portfolio value after 6 months.

If we ignore the impact of the intermediate cash flows and holding period returns, the 1-year return would simply be 120/100 – 1 = 20%

However, there has been a withdrawal of 10 EUR that needs to be accounted for.

the TWR formula is:

(1+*HPR1*)* (1+*HPR*2)*……… ( (1+*HPR*n) – 1

Here *HPR* stands for holding period return. In our case, there are two periods: 0-6 and 6-12. Therefore the final formula is:

TWR = (1+10%)*(1+20%) – 1 = 32%

By breaking the 1 year into two sub-periods, TWR ensures that the timing of inflows or outflows does not impact the return.

The result would have been the same had the 10 EUR withdrawal been at the end of 3 months with the value at that time being 110 EUR.

#### Is there such a thing as a good or bad TWR?

I think from the examples above, you can see when there would be good or bad times to use TWR to measure how your portfolio has done.

A small portfolio with 10,000 EUR that excelled over 1 year and doubled up to 20,000 EUR at the end of that year, made a great return. If you then deposited 200,000 EUR and lost 50% in the following year, you would have a crippling loss of 110,000 EUR in the final year. However, your TWR is a 0% return – that is a bad TWR and not representative of your portfolio gain or loss.

If there are consistently small additions and subtractions from the portfolio is when you will have a “good” TWR because it more accurately represents the portfolio positioning. Large deposits and withdrawals can easily turn a “good” TWR into a “bad” TWR.

### What is a money-weighted return?

Money weighted return or the Internal Rate of Return is the rate of return that makes the present value of all future cash flows equal to the initial cash outlay. Using the same numbers from earlier, MRR can be calculated as:

100 = 10/(1+MWR) +120/(1+MWR)2

Or MWR = 14.66% (use Trial and error or goal seek option in Excel). Now, this is the 6-month return. The 1-year rate would be 1.14662 – 1 or 31.47%

Since there was a withdrawal just before the time when the growth was higher, MWR had a lower rate when compared to TWR.

## The differences between Time-weighted return (TWR) vs money-weighted return (MWR)

Now we know what both measures are and how to calculate them. Let us have a look at what the exact differences are between TWR and MWR.

Time-Weighted Return (TWR) | Money-Weighted Return (MWR) |

Is not impacted by cash inflows and outflows that take place in between | Timing and value of the cash inflows and outflows can affect the MWR |

Can have only one value sine it simplifies the period into multiple intervals each having a single return | Can have multiple values in case there are multiple deposits and withdrawals within the time period considered. |

Is easy to calculate. | Requires estimation which can be time-consuming especially in the case where there are multiple MWRs. |

Normally used to assess the performance of a fund manager since the intermediate cash flows are not controlled by said managers. | Can underestimate or overestimate performance depending on when the cash flows took place. Hence, it may distort the actual assessment. |

## In practice: my own TWR and MWR results compared

With the theory out of the way, let us take a look at the results here for my portfolio.

As I did the calculations myself I didn’t take the margin used into account as I do not have that data available through my portfolio analyzer. However, since I did a deposit I can still show the impact of making stock purchases:

As you can see, the time-weighted return I have is -1.41% over the 6-month period from December 1 to the end of May, which is -2.80% annualized. However, have I really lost 2.8% in that time period?

While it seems like a big loss, given my initial amount and deposits equal $43,715, and my account is $43,318, there is only a $400 difference. That is less than 1% of a dollar loss, which is where the money-weighted return comes in.

The money-weighted return might make more sense in this situation, as it comes in at only -0.37% over the 6 months, or -0.731% annualized.

### Portfolio Analyzer

Doing these calculations isn’t easy, luckily most brokers provide tools that do it for you. Lynx.be is such a broker. Through their portfolio analyzer, I was able to get a detailed TWR and MWR for any period. Here is what that looks like:

(Hover over the image to see the MWR)

### Higher; lower

Having a higher MWR than TWR shows that the deposits had a positive impact on my performance. since I’m the “manager” it shows that my decision to deposit money was a good one. The reverse happens as well. If I make a withdrawal before a strong bull trend, the MWR will be lower than the TWR.

## Bonus: Value weighted return

Let us say you have a portfolio composed of $25 bonds and $75 stocks. The bonds have a return of 4% while that of the stocks is 10%. Value weighted rate of return determines the rate of return of the portfolio based on the weights of each component in the portfolio.

The value weight for a bond is 25/(25 + 75) or 0.25 and that of stocks is 0.75

Value weighted rate of return = 0.25*4% + 0.75*10% = 8.50%

In contrast to an equally weighted return that places equal weights on bonds and stocks in this case, the value-weighted return is influenced by the asset class that has a higher value in the portfolio. Therefore, the value-weighted return of the portfolio is closer to the equity return of 10%.

**Sources**:

- Thor Wealth Management. (2019, December 9).
*Time weighted vs. money weighted returns*. THOR Wealth Management, Inc. https://www.thorwealthmanagement.com/time-weighted-vs-money-weighted-returns/ - Starlight Capital. (n.d.).
*TWR vs MWR*. https://starlightcapital.com/en/investor-education/mutual-fund-investments/twr-vs-mwr

Thx for doing such a detailed post on this. It is good to understand these concepts

THx Amber Tree. I had this in mind for a while now but it wasn’t easy to get it in writing. I found it interesting to see how TWR and MWR correlate and how it evolved during different time frames. I recommend you look at it in detail if you are interested in learning a bit about your portfolio’s performance.