When you mention the word “finance”, probably one of the concepts that jumps into someone’s mind are returns (percentage figures); something in the line of *17.8% performance*. Indeed, returns sit at the very core of most financial calculations.

Today, we are going to look at two different numbers attached to the performance of a portfolio: time-weighted and money-weighted returns.

### Sources driving investment value

Before we get to the them, let’s lay out a bit of context to understand why we need returns in the first place. In the context of portfolio management, there are two sources of money that can make the size of the portfolio vary:

- internal (due to a change in value of the underlying assets);
- external (due to investors inflows and outflows).

To explain how these two sources impact performance measurement, we will follow Giovanni and Joanna in their investments. Joanna is the strategy chief officer of a large fund; Giovanni is an engineering student, investing the money he makes out from tutoring.

Joanna has an investment strategy set to beat the performance of the Eurostoxx 50. To do so, she receives large amounts of money from investors around the continent; and she returns back money to those investors who claim it, including their gains or losses. **She has no control over the size nor timing of these inflows and outflows.**

Giovanni chips in some money every month to his broker account, and invests in companies from the Eurostoxx 50 too. Additionally, every now and then, when he wants to go on holidays, he withdraws some money out.

Now, both Giovanni and Joanna face the same problem: **they need to find a calculation to measure how well or bad they are performing. **Because the size of their portfolio changes constantly over time due to internal and external sources; **they cannot use as a performance metric the amount of money in the bank account.** Doing so could be tremendously misleading.

For instance, if Joanna’s fund is invested in declining (rising) assets, but a very large investor comes in (drops out), it would seem as if she is doing great (poorly). Additionally, her benchmark is an index, that is, a manufactured number whose value is computed based on the evolution of the underlying companies it represents.

### Time-Weighted Returns: Removing the External Sources

To overcome the aforementioned problems, we define the Time-Weighted Rate of Return (TWRR). We need two ingredients to perform the calculation:

- Portfolio valuations for each day (at market closing price): \(A_{t_i}\);
- Cash flows (inflows / outflows) for each sub-period: \(CF_{t_i}\).

With these two elements, we define the daily time-weighted return \(r_{t_i}\) as

This formula assumes that the inflows (outflows) are invested (divested) at the beginning of the sub-period that goes from \([t_{i-1}, t_i]\).

What makes this number great is that it provides a dimensionless figure **to represent the change in value of the investment without the effects of inflows and outflows. **

It measures the change in value 1€ invested at the beginning of the sub-period would experience. Therefore, to know how much 1€ would have changed in a large period, we perform what is known as *geometrical linking* of the associated TWRR sub-period returns:

Performing these calculations, Giovanni and Joanna can attribute a performance number to their portfolios, and hence compare between each other or the index they are trying to beat.

### Beyond Definition

So far, we have covered the usual definition and application of the TWRR. However, there is more to it. By recursion of the daily return definition, one can arrive to the following expression, which relates all the cash flows in the period to the end-of-period portfolio value \(A_{t_N}\), provided the daily returns are known:

This new expression has an elegant interpretation: the end-of-day portfolio value is equal to the sum of each cash flow in the period, accrued by the total return *since its inception in the portfolio.*

### Money-Weighted Returns: Including Timing

Now, the TWRR is not sufficient to give an accurate view of an investors’ portfolio. You see, its calculation contains a flaw:

An investor could find his account has incurred in losses during the period, and yet compute a positive TWRR.

(We will show how this is possible in the numerical example at the end).

To overcome this issue, another performance rate, the Money-Weighted Rate of Return (MWRR), can be computed. The equation for its calculation can be easily derived from the previous expression for the TWRR. **To compute a rate that is sensitive to the actual gain or loss**, we level set the rate at which all cash flows are accrued, and still demand that their sum is equal to the end-of-day portfolio value. This leads to

which is a polynomial equation in \(r\). This expression is also known as the Internal Rate of Return (IRR), and you can see where its name comes from: the same accruing return (albeit for a different exponent) is *weighted* by the size of the cash flow.

### Numerical Example

Finally, in this numerical example we can see both returns in action.

We build a simple setting with two sub-periods. In the first sub-period the assets decline, in the second sub-period the assets rise again, more than they declined in the previous sub-period. There are two cash flow patterns,

- N-WD: No-Withdrawal (an amount is invested and no withdrawal takes place);
- WD: Withdrawal (half of the initial investment is withdrawn after the first sub-period).

You see, the TWRR of the total period is positive, but the WD pattern incurs in losses at the end of it. This is so because the portfolio size was smaller when the assets went up, so those gains were not sufficient to cancel out the previous losses. This fact is correctly captured by the MWRR, whose calculation I leave to you 😉

Tip: the number referring to *sub-periods* is the one that comes out from solving the MWRR equation. The number referring to *total period* is the compounding to the whole period, which in this simplified setting is equal to the power of two.

We can see how in the absence of additional inflows or outflows, both the MWRR and the TWRR coincide.

### Conclusions

Today we have looked at two alternative rates of return for the same stream of cash flows: time-weighted and money-weighted.

The main conclusion is that different percentage figures can be attached to the same amounts. The main benefit of expressing performance in percentage terms is that it allows comparison between mandates with different investment size and cash flow timings.

However, we must be careful in the selection of which number to give, for they do not answer the same question.

Thanks for reading! What’s your experience with return calculations? Let us know in the comments below, see you around!