Long-Term Growth Rate of Expected Utility for Leveraged ETFs: Martingale Extraction Approach
This paper studies the long-term growth rate of expected utility from holding a leveraged exchanged-traded fund (LETF), which is a constant proportion portfolio of the reference asset. Working with the power utility function, we develop an analytical approach that employs martingale extraction and involves finding the eigenpair associated with the infinitesimal generator of a Markovian time-homogeneous diffusion. We derive explicitly the long-term growth rates under a number of models for the reference asset, including the geometric Brownian motion model, GARCH model, inverse GARCH model, extended CIR model, 3/2 model, quadratic model, as well as the Heston and stochastic volatility models. We also investigate the impact of stochastic interest rate such as the Vasicek model and the inverse GARCH short rate model. We determine the optimal leverage ratio for the long-term investor and examine the effects of model parameters.
- 1 Introduction
- 2 Martingale extraction approach for LETFs
- 3 Univariate processes
- 4 Stochastic volatility models
- 5 LETF with stochastic reference and interest rate
- 6 Quadratic models
- 7 Conclusions
- 8 The extended CIR model
- 9 The inverse GARCH model
- 10 Quadratic models
Exchange-traded funds (ETFs) are popular financial products designed to track the value of a reference asset or index. With over $2 trillion of assets under management, ETFs are traded on major exchanges like stocks, even if the reference itself may not be traded. Within the growing ETF market, leveraged ETFs (LETFs) are created to generate a constant multiple , called leverage ratio, of the daily returns of a reference index. For example, the ProShares Ultra S&P 500 (SSO) offers to generate twice () the daily returns of the S&P 500 index. In the LETF market, the most common leverage ratios are and . In particular, investors can take a bearish position on the reference by taking a long position in an LETF with without the need of borrowing shares or a margin account. For many speculative investors, LETFs are highly accessible and liquid instruments that give a leveraged exposure, and particularly attractive during periods of large momentum.
For LETF holders and potential investors, it is crucial importance to understand the price dynamics and the impacts of leverage ratio on the risk and return of each LETF. A number of market observations suggest that LETFs suffer from the volatility decay effect, which reflects the value erosion proportional to the realized variance of the reference index. Recent studies, including Avellaneda and Zhang (2010), Cheng and Madhavan (2009), Leung and Ward (2015), and Leung and Santoli (2016), present discrete-time and continuous-time stochastic frameworks to illustrate the the path dependence of LETFs on the reference, including the volatility decay effect. In fact, SEC issued in 2009 an alert announcement regarding the riskiness of LETFs, especially when holding them long-term.
In this paper, we investigate the long-term growth rate of the expected utility from holding a LETF. Specifically, we consider different stochastic models for the LETF price, denoted by , and analyze the long-term growth rate represented by the limit:
where is the investor’s utility of the power form: with . As such, the coefficient of relative risk aversion is given by , . When , corresponding to zero relative risk aversion, the limit is the long-term growth rate of expected return of the LETF. Hence, analyzing (1.1) allows us to understand the long-term growth rates of expected utility and expected return useful for risk-averse and risk-neutral investors, respectively.
One of main contributions of this paper is to present a novel approach to determine the above limit analytically. For this purpose, we employ the method of martingale extraction, through which the problem of finding the long-term growth is transformed into the eigenpair (eigenvalue and eigenfunction) problem of a second-order differential operator that is associated with the infinitesimal generator of the reference process.
Our results allow us to determine the optimal leverage ratio for the long-term risk-averse investor. For the -LETF with price denoted by , we find the optimal leverage ratio that maximizes the long-term growth rate, that is,
Furthermore, we examine through our explicit expressions the combined effects of risk aversion and model parameters on the optimal choice of leverage.
There are a number of related studies on the long-term growth rate of expected utility. The seminal work by Fleming and Sheu (1999) investigated the optimal growth rate of expected utility of wealth. The utility was of hyperbolic absolute risk aversion (HARA) type, and dynamic programming scheme was developed for different HARA parameters and policy constraints. Akian et al. (1999) studied the optimal investment strategies with transaction costs with the objective to maximize the long-term average growth rate under logarithmic utility. In another related study, Zhu (2014) also examined the long-term growth rate of expected power utility from a nonleveraged portfolio with a fixed fraction of wealth in the single risky asset, and derived explicit limits under some models. In comparison, we study leveraged portfolios under additional single-factor and multi-factor diffusion models.
Christensen and Wittlinger (2012) considered the growth rate maximization problem based on impulse control strategies with limited number of trades per unit time and proportional transaction costs. Guasoni and Mayerhofer (2016) analyzed the optimal strategy to maximize the long-term return given average volatility under the Black-Scholes model with proportional costs. Hata and Sekine (2006) studied a long-term optimal investment problem with an objective of maximizing the probability that the portfolio value would exceeed a given level in a market with Cox-Ingersoll-Ross interest rate. Applying the theory of large deviation, Pham (2003) derived the optimal long-term investment strategy under the CARA utility, and Pham (2015) examined the long-term asymptotics for optimal portfolios that involved maximizing the probability for a portfolio to outperform a target growth rate.
The martingale extraction method is a relatively new analytical technique that has been used to investigate a number of financial and economic problems. Among our main references, Hansen and Scheinkman (2009) and Hansen (2012) developed the martingale extraction method to study the long-term risk in continuoue-time Markovian markets. Borovicka et al. (2011) utilized the martingale extraction method to examine the shock exposure in terms of shock elasticity that measures the impact of shock. In these studies, the authors decompose a pricing operator into three components: an exponential term, a martingale and a transient term, each of which carries a financial interpretation depending on the context of problem. Park (2016) studied sensitivities of long-term cash flows with respect to perturbations of underlying processes by using the martingale extraction method. Qin and Linetsky (2015) further analyzed the Hansen-Scheinkman factorization (martingale extraction) for positive eigenfunctions of Markovian pricing operators. Our contribution on this front is to be the first to apply the martingale extraction technique to compute explicitly the long-term growth rate of expected utility.
The rest of this paper proceeds as follows. In Section 2, we discuss our martingale extraction approach for LETFs. In Section 3, we solve the long-term growth rate problem when the reference price follows a one-dimensional Markov diffusion. Sections 4, 5 and 6 are dedicated to, respectively, stochastic volatility models, interest rate models, and quadratic models. We compute the long-term growth rates and determine the optimal leverage ratios. Section 7 summarizes this paper.
2 Martingale extraction approach for LETFs
Let be a probability space where is the
subjective probability measure. Denote by the filtration generated by a
-dimensional standard Brownian motion . Consider a reference index, such as the S&P500 index, whose price process is a one-dimensional positive time-homogeneous Markov diffusion process satisfying
where the drift process and vector volatility process are both -adapted. At this point, we do not specify a parametric stochastic drift or volatility model, though many well-known models, such as the Heston model as well as other stochastic or local volatility models, also fit within the above framework. In addition, the risk-free rate process is denoted by which may be constant or stochastic depending on the model.
2.1 LETF price dynamics
A leveraged ETF is a constant proportion portfolio in the reference . A long-LETF based on has a leverage ratio . At any time the cash amount of ( times the fund value) is invested in and the amount is borrowed at the risk-free rate . Strictly speaking, for , the fund is not leveraged since only a fraction of the fund value is invested in the risk asset, and no money is borrowed. For a short-LETF with ratio , a short position of the amount is taken on while the amount is kept in the money market account at the risk-free rate . As a result, the LETF price satisfies
Without loss of generality, we set
The LETF value at time admits the expression
where is the usual -dimensional norm.
The investor’s risk preference is modeled by the power utility function
As such, the coefficient of relative risk aversion is given by , . The expected utility from holding the LETF up to time is given by
where we have defined the stochastic exponential
In particular, when , the risk aversion is zero so that the expectation (2.4) is the expected return from holding the LETF over .
Suppose that a local martingale in (2.7) is a martingale. Then, we can define a new measure via
By Girsanov theorem, the process defined by
To analyze the expected utility, we employ the martingale extraction method, which will be described in Section 2.2. This method allows us to express the expected utility in a form that is more amenable for analysis and computation.
2.2 Martingale extraction
We now discuss the martingale extraction method with a generic multi-dimensional time-homogeneous Markov diffusion process on with drift and volatility In the SDE form, we can write by
where is -dimensional column vector and is a matrix. The components of and are differentiable functions and assume that the SDE has a strong solution.
The -dimensional process may represent multiple components of the model, such as the reference, stochastic volatility, stochastic interest rate, or other stochastic factors. Fix a continuously differentiable multi-variate function . Denote by the infinitesimal generator of with killing rate . Suppose that is an eigenpair corresponding to
where and is a positive continuous twice-differentiable function. It can be shown that
is a local martingale by checking that the -term of is zero. Refer to Hurd and Kuznetsov (2008) for a relevant topic.
In this case, we can express equation (2.12) as
and interpret it as the martingale being extracted from With each admissible eigenpair , one can define a new measure by
This measure is called the transformed measure from with respect to . For convenience, we use notation instead of In turn, we apply a change of measure from to to express the expectation
In many cases, the right-hand side is more amenable to computation and analysis. For instance, the expectation depends on the marginal distribution of at time , whereas depends on the whole path of . This observation is particularly useful for our analysis of LETFs since they are also path-dependent.
The dynamic of is also altered under the transformed measure . To see this, we define the Girsanov kernel associated with by
then the martingale satisfies
According to the Girsanov theorem, the process defined by
is a standard Brownian motion under As a result, given an admissible eigenpair , the process evolves under according to
As expected, the eigenfunction arises in the drift adjustment of , but does not affect the diffusion term.
Furthermore, if the density function of under is also available in closed form, one can compute and analyze the expectation Not all but for many cases, we will choose an admissible eigenpair such that the term converges to a non-zero constant. From this we derive the long-term growth rate of the expected utility of LETFs.
Let be an admissible eigenpair of , and be the corresponding transformed measure. If converges to a nonzero constant as , then the limit
3 Univariate processes
We now demonstrate how the martingale extraction technique can be applied to analyze the growth rate of expected utility for LETFs. In this section, the reference asset is a one-dimensional Markov diffusion process that satisfies
where is a one-dimensional standard Brownian motion under the subjective measure . The coefficients and are continuously differentiable functions such that SDE (3.1) has a strong solution. Throughout this section, the short interest rate is a constant
According to (2.4), the expected utility from holding the LETF is given by
To utilize the martingale extraction method, we can view as playing the role of the process in Section 2.2. Define as the infinitesimal generator of with killing rate . As such, we have
A key step in our approach is to find, as explicitly as possible, an eigenpair of with positive It is noteworthy that there always exists such a solution pair as long as (see (Pinsky, 1995, Theorem 3.3)). This condition is satisfied for all LETFs since their leverage ratios satisfy .
Given that there exists an eigenpair which admits the martingale extraction of , the expected utility can be expressed as
where is the corresponding transformed measure. Since the term depends only on the value at time , rather than its whole path, this significantly simplifies the analysis of , as we present in the following models.
Applying Proposition 2.1, we obtain the long-term growth rate of expected utility from holding the LETF in this univariate framework. Precisely, we have
and if converges to a nonzero constant as then the limit in (3.5) reduces to
In particular, we recover the long-term growth rate of expected return for the LETF by setting corresponding to zero risk aversion. Again, the eigenvalue plays a crucial role in the long-term growth rate, along with the first term that depends explicitly on the interest rate , risk aversion parameter , and the leverage ratio . It is important to note that the eigenvalue also depends on , , and , but not .
3.1 The GBM model
As a warm-up exercise, we present the long-term growth rate of expected utility in the geometric Brownian motion (GBM) model
with The corresponding generator is
To apply martingale extraction, we find the corresponding eigenpair
We obtain the expected utility
This implies the limit
The right-hand side consists of two parts: the factor and the negative eigenvalue . Moreover, the long-term growth rate is quadratic in . Using this result, we maximize the long-term growth rate in equation (3.8) over to obtain the optimal leverage ratio
As we can see, the optimal leverage ratio is wealth independent, proportional to the Sharpe ratio, but inversely proportional to the coefficient of relative risk aversion . The investor should select a positive (resp. negative) if and only if (resp. ). It resembles the optimal strategy in the classical Merton portfolio optimization problem.
3.2 The GARCH model
In this section, we consider a positive mean-reverting model for the reference price process . Specifically, it satisfies the continuous-time GARCH diffusion model (see Lewis (2000)):
with The GARCH diffusion model is sometimes referred to as the inhomogeneous geometric Brownian motion (see e.g. Zhao (2009)). The corresponding generator is
To apply martingale extraction, we solve the eigenpair problem to obtain the eigenpair
To evaluate (3.12), we first deduce that
The proof is as follows. The process converges to the Gamma random variable with parameter that is, the density function of converges to as (Theorem 2.5 in Zhao (2009)). We obtain the above result by considering the density function and the limiting density function above. The asymptotic behaviors of near and are as follows. For fixed and any small
Here, for two positive functions and we denote by
if there exists a positive constant such that Refer to Section 6.5.4 in Linetsky (2004) for the density funtion If then
which is finite. Otherwise,
since near In conclusion, we obtain the following long-term growth rate.
Let be the LETF whose reference price satisfies the GARCH model (3.10). Then,
This result implies two distinct scenarios. When , there is a finite long-term limit of the growth rate. Interestingly the long-term limit is linear in and decreasing in , but does not depend on . When , the long-term limit is infinitely large. The limit also applies when , in which case the condition represents an upper bound on the leverage ratio in order to obtain a finite long-term growth rate of return.
By Proposition 3.1 and direct calculation, we maximize
to obtain the optimal leverage ratio for a long-term investor
Surprisingly, in contrast to the GBM model, the optimal leverage ratio under the GARCH model is independent of , which means that under this model investors with different risk aversion coefficients, including zero risk aversion, will have the same optimal leverage ratio . In fact, only depends on the interest rate and volatility parameter . It is also notable that the GARCH model is reduced to the GBM model as ; however, not only the optimal growth rate but also the optimal leverage ratio do not converge to those of the geometric Brownian motion as It is because the path behaviors and other qualitative features of the GARCH model differ significantly from the GBM model.
3.3 The inverse GARCH model
As an alternative to the GARCH model, suppose now the reference price follows the inverse GARCH diffusion model, which is also referred to as the Pearl-Verhulst logistic process in Tuckwell (1974):
with and Both the GARCH and inverse GARCH models are positive and mean-reverting. The process is called the inverse GARCH model because its inverse process follows the GARCH model:
The infinitesimal generator of is
By direct substitution, we verify that
is an admissible eigenpair to . Since the eigenfunction is a constant, the corresponding transformed measure is identical to the original measure (see (2.15)-(2.17)). Following from (3.4), the expected utility is
Since is the GARCH model, we observe from (3.13) that
This leads to the long-term limit summarized as follows.
Let be the LETF whose reference price follows the inverse GARCH model (3.17). Then,
Therefore, the long-term growth rate of expected utility can be finite or infinite, depending on the leverage ratio , risk aversion parameter , and model parameters but not . Interestingly, the limits in (3.16) and (3.20), respectively, for the GARCH and inverse GARCH models are the same, except for the conditions for the finiteness of the limits.
By direct calculation, the optimal leverage ratio for the long-term investor is when the long-term growth rate is finite. While does not depend explicitly on , but plays a role in determining the finite/infinite growth rate scenario. As , the inverse GARCH model reduces to the GBM model. Nevertheless, the optimal growth rate and optimal leverage ratio , being independent of , do not converge to those in the GBM model as . The same phenomenon was observed in the GARCH model case in Section 3.2.
3.4 The extended CIR model
We now turn to the extended Cox-Ingersoll-Ross (CIR) model proposed by Cox et al. (1985):
with parameters and This process is a transient process diverging to infinity given . The corresponding infinitesimal generator is given by
The first square-root term is real provided that , which holds true for all LETFs. It can be verified by direct substitution that
is an admissible eigenpair of according to equation (2.11). Under the transformed measure with respect to this eigenpair, the process follows
where is a -Brownian motion. We note that this is a standard mean-reverting CIR process and the Feller condition is satisfied, thus is an unattainable boundary.
The expected utility is given by
In turn, we obtain the long-term growth rate of expected utility.
Suppose that the reference price process satisfies the extended CIR model (3.21). Then, we have
This result has a number of implications. First, the long-term growth rate is affine in the leverage ratio and excess return , and linear in , but it does not depend on the model parameters and explicitly other than in the condition separating the two scenarios. In the scenario with , denote the limit as a function of : . When , it follows that the long-term growth rate . This is because the resulting “leveraged” ETF portfolio is simply growing deterministically at rate , and the utility is at time . Second, the function reveals the optimal choice for a static investor. In a bullish market with , a higher leverage ratio is preferred, though in practice the available leverage ratios are capped at . In contrast, in a bearish market with then a more negative leverage ratio is better, and in practice the most negative leverage ratio available among LETFs is .
3.5 The 3/2 model
We now consider the 3/2 model for the reference price of the form:
with This is a positive mean-reverting model that has been used to model interest rates and volatility (see Ahn and Gao (1999), Carr and Sun (2007)), so this model would be appropriate for fixed-income and volatility LETFs with a mean-reverting reference price.
The infinitesimal generator corresponding to (3.25) is
we find that
is an admissible eigenpair of Under the transformed measure , the reference price follows
where is a Brownian motion under Notice that satisfies a re-parametrized model under .
The expected utility from holding an LETF can be expressed under the transformed measure by
We show that
The proof is as follows. Define Then is a CIR process with
By considering the density function of the CIR process, which is given in equation (8.2), we obtained the desired result. In conclusion, we obtain the following long-term growth rate.
Let be a -LETF whose reference price satisfies the model (3.25). Then, we have
In general, the sign of depends on the model parameters , risk aversion coefficient , and leverage ratio . Nevertheless, we find that for , which holds for market-traded LETFs, the condition is satisfied.
Next, we investigate the optimal leverage ratio for a static investor (see (1.2)). In the scenarios with , we define
Next, we determine the critical points of Differentiation yields that
When the equation has no solutions and for all . Therefore, is a decreasing function of In practice, is the optimal strategy. On the other hand, when by considering the equation we conclude that the maximum of is attained at