INTRODUCTION

Hedge funds engage in dynamic trading strategies, use leverage opportunistically, take positions in non-linear instruments and take concentrated bets. These techniques result in non-linear payoffs at the single Hedge Fund and at the Fund of Hedge Fund level. Q.M.S Advisors' approach is therefore to assess Hedge Fund strategies' risks, to forecast Hedge Fund style returns and to construct optimal strategic and tactical hedge fund allocations that reflect the option-like and nonlinear features of those strategies.

DESCRIPTION

Hedge funds usually exhibit non-normal payoffs for multiple reasons such as the use of derivatives, structured products, and dynamic trading strategies. Further, hedge funds take state-contingent opportunistic bets that account for a significant part of their returns and risks. Despite ever growing research on the topic, most studies on hedge funds' performance so far focused on classical linear factor models, non-parametric models or linear factor models with option like factors.

Q.M.S Advisors' framework departs from those approaches and utilizes factor models based on the regime switching theory, where non-linearity in the exposure is captured by factor loadings that are state dependent. The regime switching approach first identifies the current and future likely states of the markets - a stochastic process based on numerous risk factors with forecasting power that identifies and translates the current and future states of the markets in quintiles "VL, L, M, H, VH" - after which Q.M.S Advisors' state dependent factor loading is able to capture the exposure of the hedge funds to the market risk factor in these different states or market conditions.

Q.M.S Advisors' empirical results show that switching regime factor models can explain a larger proportion of the variation in returns of hedge funds, as opposed to classical linear factor models, non-parametric models and linear factor models with option like factors. The justification for this greater explanatory power can be linked to both the dynamic nature of hedge fund strategies and to the tendency of market factors' to synchronize or become more dependent in periods of stress.

For instance our research reflects the empirical evidence that hedge funds' market exposures or betas are usually very low in normal market conditions, but that those sensitivities can rapidly become significant in abnormal market conditions. In classical models, hedge funds' sensitivities to market factors will usually remain close to zero (as they are "averaged" across all states); unable to capture the small yet critical probability that hedge funds strategies' seemingly uncoordinated returns become co-dependent across market factors and across hedge fund strategies in periods of market stress.

By utilizing a regime switching factor model framework, Q.M.S Advisors addresses most of the shortcomings linked to classical models. The cyclical, non-Gaussian and unstable nature of asset distributions' moments is imbedded in our structure by appropriately integrating markets' and strategies' dynamics:

• The regime switching factor model reflects the empirical evidence that markets and hedge fund strategies display time-variations in investment opportunities -or cyclicality- and therefore exhibit non-normal return probability distribution functions
• It captures the stylized fact that hedge fund returns exhibit non-stable co-dependence parameters with market factors and across hedge fund strategies. Regime dependent correlation structures adequately capture those phenomena and are key in devising robust strategic and tactical asset allocations in multi-step optimization frameworks.
• Volatility clustering phenomena are also appropriately represented as the model is based on both:
• State contingent covariance matrices
• A stochastic process driving both current and future states of the markets (based on underlying risk factors that identify the current and future unobservable states of the markets)

Those aspects of the framework are key as not only do they provide a clear map in terms of dynamic risk exposures, but also invaluable inputs for strategic and tactical asset allocation purposes in simulation frameworks.

One of the main feature of the model is the regime switching models' ability to capture and forecast the evolution of the underlying market state in terms of changes from one regime to another. In that sense, this "state factor" would lead the tactical asset allocation aspect of the portfolio construction process by acting as an "opportunistic" risk aversion parameter.

OVERVIEW OF THE THEORETICAL FRAMEWORK:

To provide a simplified yet valid overview of our framework, we present the dual state Markov chain based regime-switching model:

Q.M.S Advisors’ switching-regime model is a structure where systematic and un-systematic events may change from the presence of discontinuous shifts in average returns and volatilities. We utilize a Markov chain as it allows for conditional information to be used in the forecasting process. (The change of regime in Q.M.S Advisors’ model is ultimately partially predictable, yet should be regarded as purely stochastic in this paragraph, or when implemented for Strategic Asset Allocation purposes). More specifically, a formal representation of our approach is such that:

R_t = a + b(S_t)I_t + wu_t

I_t = m(S_t) + s(S_t) e _t

Where S_t is a Markov chain with n states and transition probability matrix P. Each state of the market index I has its own mean and variance and the same applies to the returns of the hedge fund index. More specifically, the hedge fund mean returns and volatility are related to the states of the market index and are given by the parameter a plus a factor loading, b, on the conditional mean of the factor, where b could be different conditional on a state of a factor risk.

When n = 2 (two separate states) the model is such that:

R_t = a + b₀I_t + wu_t   if S_t = 0

R_t = a + b₁I_t + wu_t   if S_t = 1

Where the state variable S depends on time t, and b depends on the state variable

b(S_t) = b₀   if S_t = 0

b(S_t) = b₁   if S_t = 1

And the Markov chain S_t is described by the following transition probabilities:

Pr(S_t = 0| S_t-1= 0) = p          Pr(S_t = 1| S_t-1= 0) = 1-p

Pr(S_t = 1| S_t-1= 1) = q          Pr(S_t = 0| S_t-1= 1) = 1-q

The parameters p and q determine the probability to remain in the observed regime at t-1. This model allows for a change in the variance of returns only in response to occasional, discrete events. Despite the fact that the state S_t is unobservable, it can be estimated statistically (Hamilton).

Q.M.S Advisors’ model specification is similar to the well-known "mixture of distributions" model. However, unlike standard mixture models, the regime-switching model is not independently distributed over time unless p and q are equiprobable. Indeed, one key aspect of the switching regime model is that if the volatility has been efficiently characterized with different parameters for different periods of data, it will be probable that in the future the same pattern will apply. As the switching regime approach accounts for brief and sporadic events, it provides a precise representation of the left tail of asset return distributions.

The main improvement coming from utilizing a Markov chain is that the former allows for conditional information to be used in the forecasting process. The predictability of the future states conditional on our knowledge of the current state helps us exploit those regime shifts via our "opportunistic" risk aversion parameter,

Q.M.S Advisors’ regime-switching model's main advantages resides in its capacity to

1. Properly fit Hedge Fund strategies time series, providing more stable and therefore reliable parameters
2. Capture the well known cluster effect, under which high volatility is usually followed by high volatility (in presence of persistent regimes)
3. Generate better forecasts since regime-switching models generate a time conditional forecast distribution rather than an unconditional forecasted distribution.

Q.M.S Advisors’ approach is superior to non-parametric and semi-parametric models as it relies on dynamic mixtures of multiple normal distributions that evolve dynamically through time. Q.M.S Advisors’ parametric model separates the states of the world and makes inferences on time-varying risk exposures of hedge fund strategies, derives forecasts and calculates conditional expectations.

More specifically, Q.M.S Advisors’ approach allows for dynamic factor loadings with different betas. Q.M.S Advisors’ approach captures and separates factors into different quintiles based on historical performance, and assesses the exposure of hedge fund strategies' returns to factors in each of the quintiles. The use of quintiles implies the exogenous definition of states which we let the model determine. Further, Q.M.S Advisors’ model forecasts the future states of the world via our stochastic risk aversion parameter; a transition probability.

Formally once the current state estimated, forecasts of changes in regime can be readily obtained, as well as forecasts of the hedge fund strategies' sensitivities b_t. Formally, k-step transition matrix in Markov chain models is given by P^k, the conditional probability of the regime S_t+k given date-t data R_t = (R_t;R_t-1; : : : ;R₁) takes on a particularly simple form:

Prob (S_t+k = 0 I R_t) = p₁ + (p- (1-q)) ^k [Prob (S_t = 0 | R_t) - p₁]

where Prob (S_t = 0 | R_t) is the probability that the date-t regime is 0 given the historical data up to and including date t. Using similar recursions of the Markov chain, the conditional expectation of b_t+k can be readily derived as:

E[ b_t+k | R_t ] = a'_t P^k b

a_t = [ Prob (S_t+k = 0 I R_t)   Prob (S_t+k = 1I R_t) ]'

b = [ b₀ b₁ ]'

 Time-varying betas can be determined via E[ b_t+k | R_t ] = a'_t and assuming that k = 0. That in turn provides us with a robust framework to analyze the time-varying risk exposures of hedge funds strategies. Moreover, this framework can be used to calculate expected time varying risk exposures for hedge funds for various factors, by setting k to be more than 0. For example, if k=1, we can calculate the evolution of expected one-week beta exposures to different factors.

Q.M.S Advisors’ model is built on those principles, yet is extended in several ways as it incorporates a wide array of factors (multifactor beta switching model).

CONCLUSION:

Q.M.S Advisors’ model detects the exposure of hedge fund strategies to multiple factors conditional on the state that characterizes the stochastic yet predictable market index factor. It effectively characterizes the exposure of hedge fund strategies to risk factors using a regime switching beta approach.

This approach allows our team to analyze the time varying risk exposure of hedge funds, and in particular, the changes in hedge fund exposure conditional on different states of various risk factors. Further, it provides an invaluable input for strategic asset allocation purposes in simulation frameworks and a powerful tactical asset allocation tool via the "opportunistic" risk aversion parameter; a predictable transition probability upon which conditional expectations can be formulated

Main results:

1. First, we find that it is important to analyze hedge fund exposure conditional on the different states of the market index as exposures may change when market factors switch from states to states.

2. Second, the use of switching beta models allows for a non-linear relationship among hedge fund returns and risk factors, highlighting that these hedge fund strategies also exhibit significant non-linear exposure to numerous factors: bonds, currencies, commodities, volatility, etc. In particular, we show that exposures can be strongly different in the VL market regimes compared to normal states suggesting that risk exposures of hedge funds in the VL and L-market regimes are quite different than those faced during normal regimes. Therefore, investigation of risk exposure of hedge funds requires a deep analysis of the evolution of time-varying risk exposure, and more specifically of hedge fund tail-event behavior.
   

3. Third, it provides a powerful framework for strategic asset allocation purposes in simulation frameworks

4. Fourth, it offers a potent tactical asset allocation tool via the "opportunistic" risk aversion parameter; a predictable transition probability upon which conditional expectations can be formulated
   

5. Fifth, it offers incomparable flexibility as multiple extensions can be implemented in the regime switching model framework.