Risk-controlled leveraged ETFs

It is interesting to note that the expected growth rate of the leverage fund is a quadratic polynomial function with respect to the leverage factor L. Consequently, it implies the existence of an optimal leverage factor L^* that maximizes the growth rate. We get: (see Giese, 2009)

$L^{*}=\frac{1}{2}+\frac{\mu -r}{\sigma ²}\mathrm{ }\left(10\right)$

Relation (10) shows the arbitrage that an investor faces when choosing $L$ between pushing up the growth rate and assuming rebalancing losses due to rising volatilities. In our case, L^* =1.07. We replace L^* from (10) in the function of the growth rate (7) to find the optimal growth rate of the leveraged strategy:

$g^{*}=\frac{\left[\mu -r\right]²}{2\sigma ²}+\frac{\mu +r}{2}+\frac{\sigma ²}{8}\left(11\right)$

The maximum expected growth rate g^*$$in our case study is 18.77%.

In what follows, using financial data, we examine the variation of the optimal leverage and its risk control introduced by Giese (2010). We denote by L_σ$$this time varying optimal degree of leverage:

$L_{\sigma }=\frac{1}{2}+\frac{\mu -r}{{\sigma }^2}\left(12\right)$

According to equation (12), the optimal way of executing leveraged ETFs is to implement the risk-controlled leverage factor L_σ. The degree of optimal leverage is higher when the volatility is low and the risk premium (u - r) is high, which is typically the case for bull markets. On the other hand, in the case of bear markets (which are typically characterized by high levels of volatility and low risk premium values) the optimal leverage factor is small. Therefore, for investors using leveraged ETFs for a long-term investment strategy, a risk-controlled methodology is proposed that involves adjusting the leverage of the ETFs according to equation (12).

In this section, we analyze the performance of a leveraged fund over a time period [0, T], which we divide into small intervals $\mathrm{\Delta }t=\frac{T}{N}$ and the rebalancing takes place at discrete moments$t_i=i \raisebox{1ex}{$T$}\!\left/ \!\raisebox{-1ex}{$N$}\right.$. In addition, for each subinterval [t_i , t_i+1], we have independent standard random variables $x_i$ which represent the randomness of the returns of the underlying index in the respective time slots. The underlying index return between ${ t}_{i }$to $t_{i+1}$ is equal to $u\mathrm{\Delta }t+{\sigma }_i{x}_{i+1}\sqrt{\mathrm{\Delta }t}$, where the variable σ_i represents the volatility in the respective time interval ${[t }_i{, t }_{i+1}].$In addition, we use an interest rate r_i in each time interval. We get:

$F_T=F_0\prod _{i=0}^{N-1}\left(1+L\left(u\mathrm{\Delta }t+{\sigma }_i{x}_{i+1}\sqrt{\mathrm{\Delta }t}\right)-(L-1\right)r_i\mathrm{\Delta }t)\left(13\right)$

As a result, using Taylor expansion of order 2, the expected growth rate of the fund is given by:

$g_{\mathrm{\Delta }t}=\frac{1}{T}E\left[ln\left(\prod _{i=0}^{N-1}\left(1+L\left(u\mathrm{\Delta }t+{\sigma }_i{x}_{i+1}\sqrt{\mathrm{\Delta }t}\right)-(L-1\right)r_i\mathrm{\Delta }t)\right)\right]$


$= \frac{\mathrm{\Delta }t}{T}\sum _{i=0}^{N-1}\left[Lu-\left(L-1\right)r_i-\frac{1}{2}{L}^2{\sigma }_i^2+\frac{1}{3}{L}^3\sqrt{\mathrm{\Delta }t }E\left[{\sigma }_i^3{x}_{i+1}^3\right]\right]+ O\left({\mathrm{\Delta }t}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)$


$=\begin{array}{c}L(u-\stackrel{-}{\sigma }^2/2)-(L-1) \stackrel{-}{r} -1/2 L(L-1)\stackrel{-}{\sigma }^2\\ \downarrow \\ Continuous term\end{array} + \begin{array}{c}1/3 L^3 \stackrel{-}{\sigma }^3 \kappa \_\Delta \surd \left(\Delta t \right)\\ \downarrow \\ Correction term\end{array} + O\left({\mathrm{\Delta }t}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)\left(14\right)$

where E represents the expectation operator, $\stackrel{-}{\sigma }={\left(\frac{1}{N}\sum _{i=0}^{N-1}{\sigma }_i^2\right)}^{\frac{1}{2}}$, $\stackrel{-}{r}=\frac{1}{N}{\sum }_{i=0}^{N-1}{r}_i$ which represent respectively the average values of volatility and interest rate and$K_{∆}=\frac{1}{N}{\sum }_{i=0}^{N-1}\left(\raisebox{1ex}{${\sigma }_i^3$}\!\left/ \!\raisebox{-1ex}{${\stackrel{-}{\sigma }}^3$}\right.\right)E\left[x_i^3\right]$ is the skewness of the random variables x_i. The key question is whether the correction term has a significant influence on the average performance of leveraged ETFs, where Δt is each daily, weekly or monthly rebalancing period. As the correction term increases in the rebalancing period, we limit our analysis to monthly rebalancing. We can see that, for a leverage factor equal to four, the correction term is clearly less than one basis point per year. This implies that the rebalancing frequency has a negligible influence on leveraged ETFs in the real world. We conclude that the characteristics of the long-term performance of Leveraged ETFs are accurately described with daily, weekly and monthly rebalancing.

Considering volatility on a regular basis, we illustrate the advantages of this methodology by simulating a leveraged ETF based on the returns of the EUROSTOXX 50 stock index. In order to avoid transaction costs related to high turnover rates, we proceed to the adjustment only once a month. More precisely, on the first trading day of each month, we evaluate the optimal degree of leverage according to formula (3). Our database used in this study is composed of the monthly returns of the EUROSTOXX 50 index for ten years from January 31, 2007 to January 31, 2017. For volatility, we use the monthly historical volatility of this index for the same period. Then regarding the interest rate, we use the monthly Overnight USD Libor rates to calculate the leveraged versions of the underlying index. Then, the growth rate $\mu$ achieved by the underlying index EUROSTOXX 50 in the long run is supposed to be constant but in the medium term it is interesting to detect possible bear markets through this rate.

In what follows, we round up the optimal leverage to integers 0,1,2...

- If L_σ is equal to zero, then during the whole month the investment amount is invested in the USD Libor market without any exposure to risk.

- If L_σ$$is equal to one, then the investment amount is invested in an unleveraged ETF using the underlying EUROSTOXX 50 asset.

- If L_σ is equal to two or three, then the investment strategy will be invested in a double or triple leveraged ETF using the underlying EUROSTOXX 50 asset.

According to Figure (4), we can see that the leverage factor L varies in the opposite direction of the volatility. For example, the lowest leverage level of 0.62 that was realized on October 01, 2008 corresponds to the highest volatility during the 10 years of the study, which is equal to 60%. That is quite logical because this period corresponds to the financial crisis of 2008 during which the markets became volatile and bearish. On the other hand, the higher leverage level of 3.75$\cong$4 achieved in October 2013 corresponds to the lower volatility 14% which is consistent with the interpretations of Equation (12). Then Figure (5) compares the performance of the leveraged risk-controlled strategy for the underlying Eurostoxx 50 index and a constant leverage ETF of 2. In this figure, we note that the two strategies with constant and conditional leverage almost match. Knowing that the average of monthly leverage levels - applied monthly depending on monthly volatility - is equal to 1.7 which can be rounded to 2 which is leveraged strategy with a constant leverage factor. From the end of 2007, which corresponds to the beginning of the subprime crisis, constant and time-varying leveraged strategies begin to underperform the underlying index up to the year 2017.

In this case, the most interesting leverage to apply is as it was found with equation (10) in the single-active model equal to 1.07 corresponding to fixed L^* that is close to 1. In other words, it is more interesting to replicate the underlying without taking risks.

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Figure 4: The evolution of the leverage according to the volatility of the risky asset.

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Figure 5: The historical performance of the risk-controlled strategy relative to the returns of the underlying index and those of the leveraged fund.

Conclusion

Leveraged or inverse leveraged ETF are becoming increasingly popular in the financial industry. Leveraged ETF offer an advantage and disadvantage for the investor. The main benefit of the daily rebalancing strategy is that, in the case of positive market movements, exposure to risk through investment in an underlying asset increases on a daily basis to improve profits. When markets change in the adverse direction, exposure is reduced on a daily basis to protect investors from serious losses. On the other hand, the main disadvantage is the occurrence of losses due to the volatility of daily rebalancing. Therefore, the long-term performance of these products depends on the trade-off between leveraged returns, which are linear in the chosen leverage factor and the loss due to volatility, which is quadratic in the leverage factor. Note that usually the leveraged ETFs use a constant degree of leverage that does not respond to the market environment. In this paper, we have discussed the benefit of introducing a time varying leverage factor which is adjusted according to the concept of optimal leverage in order to limit the losses investors face in bear markets. Our empirical results are rather disappointing showing that the optimal leverage introduced by Giese (2009) is quite close to 2, even during the financial crisis. Such feature suggests the introduction of another time-varying leverage more based on risk measures such as quintile or expected shortfalls in the line of Ben Ameur and Prigent (2014).

References

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