The multiple faces of return I

It sounds like a joke, but the return of a portfolio can have multiple faces.

In reality, it is not that there are multiple faces, but rather formulas that are more or less convenient depending on the situation. Let’s start with the simplest.

Simple calculation of return

When there are no new contributions or withdrawals in an investment account, the simple calculation of return is more than enough to know the return on investment.

In these cases, the calculation is very simple: the ratio between the initial capital and the final capital gives us an accurate view of our accumulated yield.

The complete formula expressed as a percentage is:

Simple return = (Final Capital/Initial Capital)-1 x100

Example 1:

Initial Capital = €10.000

First year 10% return

Final capital (year 1) =10,000+ 10%*(10,000) =10,000+1,000 = 11,000

Second year 10% return

Final capital (year 2) = 11.000+10%*(11.000) = 11.000+1.100 = 12.100

Simple return = (12.100/10.000) -1 x100 = (1.21-1) x 100 = 21%

But what if our investment is somewhat more complex. Let’s imagine that starting from the previous case, the investor at the end of the first year contributes €10,000 more. It is usual in an investment account to add more capital or withdraw money over time. It is the usual fact of adding or withdrawing capital in a portfolio that complicates the evaluation of the return of a portfolio.

Example 2:

Initial Capital = 10.000

First year 10% return

Final capital (year 1) = 10.000+ 10%*(10.000) = 10.000+1.000 = 11.000

New initial capital for the second year 11.000 + 10.000 = € 21.000

Second year 10% return

Final capital (year 2) = 21.000+10%*21.000= 23.100

Simple return = (23.100/20.000)-1 * 100= 15.5%

We see that by the mere fact of having added capital in the second year, the simple calculation of return converts 10% (year 1) plus 10% (year 2) into 15.5% instead of the 21% that we obtained with the simple calculation.

In this case, the simple return formula gives an incorrect view of profitability. To solve this, the financial industry has implemented alternative ways of calculating return, the most commonly used being time-weighted return.

Time-weighted return

The time-weighted return formula helps us to correct this problem. It is very common to see this formula expressed in English as “Time Weighted Return” or TWR.

The time-weighted return formula would give the same result for both examples:

((1+Performance Year1) * (1+Performance Year2))-1×100

((1 + 0.10) * (1 + 0.10)) – 1 = 21.0%

Using the yield for each period (10%) and isolating the effect of the new cash at the end of year 1 for example 2, the time-weighted return formula gives the same result as the simple calculation.

The time-weighted return is independent of cash flows. This provides us with a target return for each portfolio in any given situation. Neither the amount nor the timing of payments has an impact on the return calculation. It is therefore suitable for comparing all types of investment products and investment strategies with each other. This allows us to directly compare the time-weighted return of any individual ETF or any other asset class with the time-weighted return of an entire portfolio.

The time-weighted rate of return is the preferred industry standard and is therefore widely used by investment professionals, and is the one we use by default at inbestMe as well.

Mathematically, the time-weighted return geometrically links the return factors of each period, in our example the two years at 10%. This would be valid for any period, even with daily contributions. To calculate the TWR, the analysis period is divided into as many sub-periods as there have been cash movements. Once divided, the yields of the sub-periods are calculated and then calculated as if they were a geometric progression to obtain the corrected yield (TWR) for the period analyzed.

In our example each year would be a sub-period:

(Final capital/(initial capital + additional capital))-1

Year 1: 11.000/ (10.000+0)-1= 0.10 = 10%

Year 2: 23100/ (11.000+10.000)-1= 0.10 = 10%

We see again how the 10% annual return appears, which is nothing more than the % taken as a starting point in our example. If we wanted to convert the accumulated return to annual return, we would only have to correct the formula initially used and raise it to ½, since the investment contemplates two years.

For our example:((1 + 0.10) * (1 + 0.10))^1/2 – 1 = 10%.

In this case, we again obtain 10%, since the years have been equal in return. In a normal situation (non-regular returns) this figure would be different. We will see in another post other alternative formulas to calculate the return and how the result of the calculation varies in each formula.

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