This is the second part , here are the link for the first and third part:
The hidden cost of borrowing
As mentioned previously the return of a LETF is :
One can notice that we have an almost quadratic form in L, so we will plot the gain Mg as a function of L for different values of R thanks to the previous formula.
This graph summarizes the performance of different leveraged on the S&P 500 according to the borrowing interest rate. On the graph, all the curves pass through the same point at leverage 1, which represents the unleveraged index and therefore has no borrowing costs. Moreover, for R=0 , I plotted both the actual and estimated returns to once again prove that the assumption of a normal distribution for the S&P 500 is valid. We can see that if there were no borrowing costs during the period, a leverage of 3 would have been the most beneficial. However, with a borrowing interest rate of 4% per year, a leverage of 2 performs almost the same as without leverage (today the €STR is roughly 4%). Finally, a leverage of 3 turned out to be a bad idea under many different interest rates.
However, we do not have the detailed performance of a LETF over time. The next section will therefore focus on constructing the performance of such a LETF.
CL2 return from 1928 to 2024
Introduction
First, I will focus on a LETF available in France (CL2) and its underlying index MSCI US Leveraged of 2 in euros. Therefore, the borrowing cost of both the index and the LETF is driven by the European central bank rate and not the FED rate. Thus in the following section I will consider the European Central bank rate. Moreover, the data I work with are in euros and not dollars. Even if this LETF is available in France the idea behind the reconstruction of the CL2 is still the same with other LETF.
In this section, I will focus on reproducing the performance of CL2 between 1928 and 2024 and explaining the assumptions made to model this performance. First, CL2 is an ETF from Amundi that attempts to replicate the MSCI USA index with a leverage of 2 in euros. The MSCI USA index includes the 610 largest US market capitalizations, making it very similar to the S&P 500. In the performance of the MSCI USA leveraged index borrowing cost (€STR) has been already taken into account.
From 2014 to 2024
For this time interval, reconstructing the Amundi LETF is easy because we have the values of the ETF for this period provided on the Amundi website.
Here is the graph for the concerned period:
From this graph, we can actually extract the interest costs that Amundi pays to achieve its leverage. Indeed, according to the equality:
With µLETF being the daily return of the CL2 and µ the daily return of the MSCI USA index (without leverage).
Therefore, R is:
We can plot R:
Which can be compared to the €STR and EONIA (which are linked to the European central bank rate) over the same period. By comparing the two graphs we can clearly see the impact of €STR on the interest rates paid by the LETF.
From 2001 to 2014
During this period, we do not have data for the MSCI USA leveraged x2 index or for the Amundi LETF. However, we do have data for the MSCI USA index and historical rates for €STR and EONIA. Therefore, we will reconstruct the performance of the CL2 LETF using these two data sources.
To reconstruct the Amundi LETF, we first use the performance of the MSCI USA index in euros for each trading day 𝑖 written µi .
Then, we will estimate the daily performance of the MSCI USA Leveraged 2 index, written
, using the formula:
With
Ri The €STR or EONIA rate of the day i
Ti The number of days between two consecutive trading days
Finally, we will apply annual management fees, written as 𝑓. We then denote µLETFi the estimated performance of the LETF on day 𝑖, which is calculated as follows:
We take the 252nd root because there are approximately 252 trading days per year
Thus, if an amount 𝑒 was invested in the Amundi LETF (CL2) on the first day, the amount on the 𝑁th day is:
The equation above is purely theoretical, so we will investigate how valid it is in practice. To check the validity of the formula we will plot the ratio between the actual performance of the LETF and the simulated performance. We will then adjust the management fee 𝑓 so that the ratio is as close to 1 as possible. In other words, we will plot the ratio of the true performance of the LETF to the estimated performance given by the equation above.
With management fees of 0.35% per year (𝑓=0.0035), the deviation between the model and the Amundi LETF does not exceed 0.3% over a 10-year period. This might seem surprising because the management fees listed in the CL2's KIID are 0.5% per year, suggesting that the CL2 manages to outperform its benchmark index in the long term, resulting in lower actual management fees. This outperformance can be attributed to the synthetic construction of the ETF.
There is two major conclusions from the previous graph:
1-It seems that there daily leverage reset cost for the CL2, in fact it seems to have additional gain.
2-The equation given above deviates only slightly (0.3% over 10 years) from the actual value of the Amundi CL2.
We can then extend the performance of the CL2 using the above equation back to 2001 with very good accuracy (around 0.5% accuracy in 2001).
From 1999 to 2001
During this period, the euro still exists, but I no longer have values for the MSCI USA index. We will then use the following formula with data from the S&P 500 Total Return in euros.
We will then plot the ratio, as we did in the previous section, between 2001 and 2024. Given that the previous estimate for the period 2001-2014 was very reliable, we will assess the accuracy of the next estimations using the reconstructed Amundi LETF up to 2001. We obtain the following graph with f=1.4% per year if we take the S&P500 total return in euros:
We observe that the ratio fluctuates within a margin of approximately 5%. The fees are higher because the MSCI USA index is calculated with net dividends reinvested, while the S&P 500 Total Return is calculated with gross dividends reinvested. This tax difference is reflected in the higher f fees. We can then extend the CL2 back to early January 1999.
From 1988 to 1999
During this period, the problem is that the euro did not exist, so I don’t have data for either the €STR or the euro-dollar exchange rate. Therefore, the data used for the simulation will be the S&P 500 Total Return in dollars and the effective rate of the U.S. Federal Reserve (FED). I will maintain the fees at 1.4% per year for this period.
I fully acknowledge that the approximations are significant, which means the CL2 graph deviates considerably from what it might have been. Therefore, the following graph should be used for indicative purposes only.
Here is the simulated graph of the CL2 back to 1988.
From 1928 to 2024
For this last period, I no longer have data for the S&P 500 Total Return, but only for the S&P 500 Price Return. Therefore, we will plot the ratio between the model using the S&P 500 Price Return data in euros and the Amundi LETF between 2001 and 2024 in order to estimate the value of the management fee 𝑓 that compensates for the difference between the MSCI USA Net Return and the S&P 500 Price Return.
We find that 𝑓=−2.4% per year provides the best calibration of the curve. The fees are negative to offset the absence of dividends in the S&P 500 Price Return.
We can therefore extend the CL2 back to 1928 by using the S&P 500 Price Return in dollars with a management fee of -2.4% per year.
For the U.S. central bank rates, I used the values published up until 1954. For the period from 1928 to 1954, I utilized the FED rate values published daily by the Wall Street Journal and the New York Herald Tribune.
Here is the graph of the CL2, taking into account the period from 1928 to 1988, calculated as specified above.
The values from 1928 to 1999 can significantly deviate from reality because the data used is based on the dollar and the FED rate.
We will compare the graph above with the S&P 500 Price Return in dollars (dividends not included) over the period 1928-2024.
The long-term return outperforms only slightly the S&P 500 Price Return (without reinvested dividends).
In the following graph, we compare the S&P 500 Total Return (with gross dividends reinvested) and the reconstructed CL2 between 1988 and 2024.
There is a third part because I can no longer add graph
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