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Alternative alpha represents risk-adjust absolute return of an alternative investing instrument regressed on alternative risk factors. Over the years, the definition is extended to the absolute return generated from alternative asset speculation-long-only or long-short strategy on alternative assets to generate additional return on top of existence alpha. In this article, we examine and propose a model with state-dependent stochastic differential equations based on Gaussian mixture model and multi-class Gaussian-kernel support vector machine to analyze hedge fund ETF alpha. We provide a new type of long-short speculation which trades on hedge fund strategies. This long-short alternative portfolio is build based on a Sharpe-ratio-like alpha ratio optimization program, and the historical performance from the portfolio shows statistically significant improvement adding to existing alphas. For passive investors, the portfolio also yields a simple portable alpha strategy which outperforms the S&P 500 benchmark return by 7.8% since 2012.

On March 25, 2009, Index IQ issued a hedge multi-strategy tracker exchange-traded fund (ETF) (ticker: QAI) to replicate the risk-adjusted returns of hedge funds covering various strategies including long-short equities, global macro, market neutral, event-driven, fixed-income arbitrage, and emerging markets. With a structure of fund of primary assets, QAI aims to statistically track the characteristics of IQ Hedge Multi-Strategy Index which is a multi-strategy hedge fund index generated by publicly available hedge fund performance. In the same year, Index IQ launched hedge fund replication products covering specific strategies: a macro tracker ETF (ticker: MCRO), a merger arbitrage ETF (ticker: MNA), and a real return ETF (ticker: CPI). Compared to hedge funds or hedge fund-style mutual funds, ETFs provide lower cost and higher liquidity for investors and the benefit for short selling.

The question is: can a standardized exchange-traded fund with pre-defined trading model generates stable and positive alpha comparing to a hedge fund or a fund of hedge funds? A bigger challenge, which is the key focus for this article, is whether speculating on hedge fund ETFs can generate alternative alphas additional risk- adjusted absolute returns on top of existing alphas. In Fung and Hsieh [^{1}. Hence, with the ability to short sale ETFs, we aim to provide a long-short alternative alpha portfolio which is designed to short sell bearish hedge fund strategies. This long- short hedge fund ETF portfolio will provide an opportunity to speculate on investing strategies, not just on assets.

Fung and Hsieh [^{2}, the excess kurtosis of the alphas reduces significantly, and the alternative alpha dynamics can be captured by a AR(1)-GARCH(1,1) model. Furthermore, the two-factor Fama-French model^{3} on these indexes has R-square of at least 81%, which indicates that the equity long-short strategy can be captured adequately by this model for two reasons: the small cap minus big cap (SMB) factor is essentially an equity long-short strategy, and the market portfolio is assumed to be an equity portfolio. Consequently, this particular hedge fund strategy might be replicated statistically. Despite the promising results for quantifying alternative alphas and the possibility of applying these results to portfolio diversification, these hedge fund indexes and the Fame-French model merely provide an indirect guideline for alternative investments. The actual performance of the portfolio depends on the actual investable vehicles and the investable alternative investing strategies.

With the focus on hedge fund ETFs, we first examine the possibility of modeling hedge fund ETF alphas. We show that alpha distributions have multiple peak and can be fitted with Gaussian mixture models [

Based on the alpha state-dependent model framework, a long-short hedge fund ETF portfolio is constructed based on an optimization program to maximize Sharpe-ratio-like alpha ratio. The alpha ratio computes the expected increment/decrement of future portfolio alpha and adjusts it by ex ante alpha volatility. We show that the optimum weight of each hedge fund ETF should be proportional to alpha increment and inverse proportional to the square of alpha volatility. Since the optimization dependents on the SVM-OU model simulation, we refer the portfolio constructed by the optimization program to SVM-OU long-short hedge fund ETF portfolio (SVM-OU portfolio). Followed by monthly portfolio construction, we compare the portfolio performance to another three strategies. While HFRI Fund Weighed Composite Index and an equality weighted long-only hedge fund ETF portfolio show negative alphas from Jan 2012 to Aug 2014, our SVM-OU portfolio provides positive alpha regressed on S&P 500 Index portfolio. Furthermore, SVM-OU portfolio shows positive alternative alpha regressed on HRFI Fund Weighted Composite Index. The historical portfolio performance from 2012 to 2014 has Sharpe ratio 1.22.

With stable and positive risk-adjusted absolute returns, a passive benchmark portfolio manager might invest in the strategy after gaining more capital flexibility by switching out cash positions to futures positions. This alpha generating and transporting process is called portable alpha. The portable alpha strategy works based on the lower margin requirement for investing in futures rather than underlying assets. Fung and Hsieh [

The returns of hedge fund ETFs are positively correlated with S&P 500. In Panel A of

We first examine a simple one-factor dynamic model by rolling linear regressing hedge fund ETFs on S&P 500 Index ETF. We conduct the linear regression on daily returns for a 252-trading-day rolling window. Equa- tion (1) represents the Hedge-Fund-on-Market alternative alpha:

Panel B in

QAI | MNA | CPI | MCRO | |
---|---|---|---|---|

Mean | 1.3456 | 0.9933 | 0.6021 | 0.4507 |

Standard deviation | 42.1649 | 59.8359 | 23.6789 | 46.8892 |

Skewness | −0.7638 | −0.2748 | −0.0589 | −0.5607 |

Kurtosis | 35.4299 | 9.7479 | 6.5084 | 13.1548 |

Correlation | 0.5313 | 0.4230 | 0.2146 | 0.5438 |

QAI | MNA | CPI | MCRO | |
---|---|---|---|---|

Mean | 1.7765 | 1.6372 | 0.5036 | 1.0515 |

Standard deviation | 0.7495 | 1.3820 | 0.6065 | 1.4254 |

Skewness | 0.3531 | −0.3376 | −0.4209 | −0.0084 |

Kurtosis | 3.4275 | 2.0417 | 2.3686 | 2.2310 |

Correlation | 0.0426 | 0.0563 | 0.0265 | 0.0112 |

Gaussian mixture model with probability density function is shown in Equation (2):

where f is the density function of a normal random variable with mean

Please refer to Bishop [

With a fixed number N, the parameters for each Gaussian component can be determined by expectation- maximization algorithm. If N is unknown for the case of an unsupervised classification case, the number of Gaussian components needs be determined by cross-model evaluation. There are two common methods for model selection. Akaike Information Criterion (AIC) [

We can now label each component from a fitted Gaussian mixture distribution to the alpha time series.

For a Gaussian mixture model with only two components, a two-class SVM suffices to choose the boundary based on the kernel function which maximizes the margin between the two classes. Gaussian kernel is adapted in our two-class SVM to incorporate the cases that the classes cannot be separated by a linear function, for example, one class might be surrounded by the other class. With

class with highest score. The fact that the simple one-versus-the-rest approach works for our study is because each class is on the same scale and no symmetry is required in the classified data. We shall also see in the following that we can embed the calibrated multi-class SVM model into a continuous-time stochastic process.

After identifying the labeled alpha time series from Gaussian mixture models, the patterns further reveal mean-reverting characteristic. If we want to build a mean-reverting stochastic differential equation with mixed Gaussian distribution characteristic, a state-dependent Ornstein-Uhlenbeck process is an ideal candidate. With labeled states determined from a multi-class Gaussian-kernel SVM, we can identify the regimes of Ornstein- Uhlenbeck processes and their coefficients. The alternative alpha dynamics

where

and

where N is the number of Gaussian mixtures labels, and

MCRO for example, we have N = 4, so the Ornstein-Uhlenbeck coefficient set is

and j is determined by the multi-class Gaussian-kernel SVM based on the rolling alpha at time t, which is

For an active alternative portfolio management, alternative alpha represents the absolute return from speculating alternative instruments. Unlike a typical fund of hedge funds, we can short sale hedge fund with hedge fund ETF; hence, our goal is to construct an alpha long-short strategy under SVM-OU framework and compare it to a HFRI Fund Weighted Composite Index and a simple equally weighted hedge fund ETF portfolio which are both alpha long-only.

Since we expect to construct an alternative portfolio to generate stable alternative alpha, we propose a long- short hedge fund ETF portfolio to maximize the Sharpe-ratio-like alpha ratio:

where

Maximizing the alpha ratio is equivalent to solving a constrained optimization problem that

subject to the volatility constraint

Let

The first-order conditions for optimality yield

where

and maintain a portfolio with no leverage that

To construct the described portfolio, we simulate under SVM-OU model for each hedge fund ETF to derive the mean and variance for a future time T. The optimization is under a typical mean-variance paradigm, but it conveys three important messages. First, instead of maximizing return which might be highly correlated to market return, we maximize portfolio alpha and expect a state-dependent momentum on future alpha to generate absolute return. Second, a Sharpe-ratio-like alpha ratio is adjusted by the alpha volatility. Lower alpha volatility shows better hedge fund manager’s skill and better hedge fund strategy. Hence, this portfolio is built to capture a steady alpha under various market conditions. Third, we look for a long-short alternative portfolio to short sale alpha if needed. An alternative alpha therefore represents both the absolute return generated from a portfolio of alternative instruments and the absolute alpha by long-short traditional alphas. Since the optimization de- pendents on the SVM-OU model simulation, we refer the portfolio constructed by the optimization program to SVM-OU long-short hedge fund ETF portfolio (SVM-OU portfolio).

Alternative Alpha PortfolioSVM-OU portfolio speculates on future alphas of hedge fund ETFs and creates long-short positions. By running a rolling window of 252 days, we calibrate the parameters for the SVM-OU processes and buy and hold the portfolio for a month. The calibration process includes adjusting the Gaussian mixture model for data labeling, SVM model training for supervised classification, and coefficients for Ornstein-Uhlenbeck processes. Using the alpha data generated starting from 2011, we compare the monthly performance of our SVM-OU alternative alpha portfolio to two other portfolios: 1) HFRI Fund Weighted Composite Index, and 2) a simple averaging portfolio by assigning equally positive weight to each hedge fund ETF.

The results are shown in

Despite the short history, the result shows the possibility of alternative alpha by speculating on alternative instruments. As more hedge-fund-like products become available to general public, we can expect a further decline on hedge fund long-only alphas. However, based on our result, a long-short strategy on hedge-fund strategies may have proven to be a good alternative investing strategy. With the increasing popularity of hedge- fund replicate products in mutual funds, ETFs, and ETNs^{4}, we can expect a trend of speculation on this asset class which sells various investing strategies, and the new alternative alphas might eventually replace existing alphas.

A portable alpha strategy is not new to portfolio managers. The idea is to replace cash positions with futures contracts, which only needs a lower capital requirement if trading with margin. Subsequently, portfolio mana- gers can invest the additional capital in alpha-generating assets or strategies. As a cash position can be tracked by a futures position (for instance, S&P 500 ETF SPY tracked by S&P 500 mini futures) for a preset time horizon, the portable alpha strategy is expected to steam roll out additional returns while maintaining a futures portfolio with the same asset weights as the cash portfolio at the end of the time horizon. However, the implementation relies on the identification an effective alpha-generating strategy. The insight provided by Kung and Pohlman [

Though hedge funds or funds of funds are the typical choice for generating alphas, they are not suitable for a portable alpha implementation because of their long redemption periods which are too illiquid for a short-term cash-futures interchange. Hedge fund ETFs can therefore serve a better purpose without incurring similar liquidity risk. In previous sections, we show that alternative alphas generated by hedge fund ETF speculation can provide stable additional returns on S&P 500 Index portfolio and on HFRI Fund Weighted Composite Index comparing to a long-only fund of funds multi-strategy. Combining the result of low correlation between SVM- OU and the market portfolio, SVM-OU portfolio will then serve as a good alpha generating component of a portable alpha implementation.

We look at the daily returns of a portable alpha strategy by incorporating our long-short hedge fund ETF

HFR Index | Long-Only | SVM-OU | |
---|---|---|---|

a | −0.0185 | −0.1483 | 0.0267 |

b | 0.3631^{*} | 0.2478^{*} | 0.1141^{*} |

Long-Only | SVM-OU | |
---|---|---|

a | −0.1261 | 0.0738 |

b | 0.6650^{*} | 0.2438^{*} |

R-Squared | 0.679 | 0.207 |

Sharpe Ratio | 0.99 | 1.22 |

portfolio to S&P 500 Index portfolio.

This paper introduces four hedge fund ETFs and the alternative alphas generated by speculating on hedge fund ETF alphas. Data justification starts by fitting historical rolling alpha distributions to Gaussian mixture models, and then we impose a supervised multi-class Gaussian-kernel support vector machine (SVM) classification and integrate it into state-dependent Ornstein-Uhlenbeck processes. This new framework is referred as SVM-OU model. Then, a long-short hedge fund ETF portfolio is constructed to maximize Sharpe-ratio-like alpha ratio based on the simulated ex ante alpha mean and volatility under SVM-OU model framework. From Jan 2011 to Aug 2014, the SVM-OU portfolio return shows low beta and sustainable positive alpha while HFRI Fund Weighted Composite Index and a simple equally-weighted long-only hedge fund ETF portfolio show negative alpha with much higher beta. This article provides an additional use case by implementing the SVM-OU alternative alpha in a portable alpha strategy. Comparing to S&P 500 Index portfolio, the portable alpha strategy generates additional 7.8% return from January 2012 to August 2014. A generalization of this long-short alpha speculation to other alternative assets is planned for future investigation.

Peter C. L.Lin,11, (2016) Alternative Alphas from Hedge Fund ETF Speculation. Journal of Mathematical Finance,06,34-42. doi: 10.4236/jmf.2016.61004