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«Equity Style Investing RONG, WU How to cite: Equity Style Investing, Durham theses, Durham University. RONG, WU (2013) Available at Durham E-Theses ...»

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Because of the difficulty in modelling conditional distributions of stock returns, academia has been exploring different approaches to simply the investment process. Recently Brandt (1999) develops a framework to directly estimate the optimal portfolio weights based on the state variables. This approach is intuitively appealing since it bypasses the auxiliary yet difficult procedure of estimating the joint conditional distributions of stock returns. Ait- Sahalia and Brandt (2001) argue that the predictability of the first (expected returns) and second moments (covariance) of stock returns is difficult to be translated into portfolio selection advice because the two moments may be predicted by different variables. In addition, a variable may be both significant for predicting the variations of expected return and variance but such variations offset hence it is essentially useless for determining optimal portfolio weights. Indeed, the investor’s ultimate interest is to obtain optimal portfolio weights while the moments of returns serve as inputs to the underlying problem and are therefore endogenous to investor’s preference. Interestingly, within this framework, Brandt and SantaClara (2006) propose a method to solve a dynamic portfolio selection problem for a mean-variance investor who optimises the expected endof-period wealth. By introducing managed and timing portfolios in the asset space, they provide an approximation to the problem that is easy to apply by investors in the traditional static Markowitz paradigm.

5.2 Motivation and research questions

Chapter 5 is motivated by the identified gap in the literature regarding the optimal multi-asset investing (style timing) over business cycles.

First, as mentioned previously, while the extant literature provides perspective on the benefits of considering business cycle predictors on the asset allocation process, in most previous studies the investable equity instrument is designed as market portfolio only, which is not realistic. To offer a fresh insight, this chapter contributes to the extant literature by allowing the investors to have access to different market segments of equity stocks and invest different equity style portfolios with no restrictions of long or short. Such investors can be regarded as hypothesised “fund of hedge funds” investors.

Second, existing literature on the portfolio choice implications of business cycle effect often focus more on the time-varying nature of return distributions driven by different business cycle predictors.

However the role such predictors play on determining optimal multi style allocation is less directly explored. The transmission mechanism of business cycle volatility to asset return dynamics and consequently the optimal style allocation is not extensively studied. If a multi-style investor believes that business cycle variables predict the conditional distributions of equity style returns, the moments of style returns to be predicted are endogenous to the investor’s preference due to model specification. Hence, in order to capture the changing investment opportunities related to the business cycle fluctuation, the investor should focus primarily on identifying how the same exogenous state variable directly predicts her ultimate style investing choices (i.e.

optimal style timing weights).

Based on the methodology of Brandt and Santa-Clara (2006), Chapter 5 contributes to the literature by applying an optimisation framework to test several equity style investing strategies based on business cycle information and examine their ex ante in-sample and ex post out-ofsample performance in the U.K. stock market. The aim of this chapter is to give multi-style investors an intuitive manner to understand their asset allocation process of incorporating business cycle predictability.

This chapter will answer some key questions such as if business cycle predictor variable X increases, should the investor move to or move away from Y style? Formally, the major objective of this chapter is to

investigate:

 Can a mean-variance multi-style investor benefit from using business cycle information to optimally implement multi-style investing strategies according to the time-varying investment opportunity set?

 If business cycle predictors affect the distribution of equity style returns, how such economic exogenous forces could affect the investor’s style choices in the context of style level asset allocation? Specifically, which economic variable or a set of such variables should be tracked when investors implementing equity investing based on market segments?

 How investor’s style investing policy differs when following the traditional two-step Markowitz approach and with that directly predicts optimal style allocation weights based on the state variables as suggested by Brandt and Santa-Clara (2006)?

5.3 Testable Hypothesis Based on the research questions, there are some hypotheses that can

be examined:

 If business cycle information affects style allocation process, multi-style investing on the basis of- business cycle predictors (i.e. conditional on the state variables) should yield better performance, both in-sample and out-of-sample, as compared to the same strategies disregarding business cycle information (unconditional investing). Such multi-style trading strategy should also outperform single-fixed passive style investing due to its nature of active style timing as suggested by business cycle predictors to capture changing investment opportunities;





 The optimal style allocation weights conditional on business cycle information should exhibit dynamic and large variations in style tilts. Predictability should induce investors to aggressively take extreme positions on specific styles because they can reduce the exposures in bad times given their prior beliefs regarding the conditional distributions of style returns predicted by state variables;

 Business cycle variables should exert different influence on different equity styles in the allocation process. For example, if one variable could positively predict the optimal weight of one specific style (e.g. value stocks), it should also negatively predict the optimal weight of its counterpart style (e.g. growth stocks);

 The optimal style tilts suggested by following the traditional twostep econometric approach and that of Brandt and Santa-Clara (2006) should exhibit significant difference. Since Brandt and Santa-Clara (2006) directly predicts the optimal style weights with business cycle predictors, it can arguably capture higher moments beyond the first and second moments of stock returns that affect asset allocation decision and therefore could yield more extreme weights but better in- and out-of-sample performance.

5.4 Methodology and econometric framework Suppose that at each date t there are N equity styles in the financial market. Each style i has an excess return of ri,t 1 from time t to t  1, and rt 1 is the vector of excess returns for all N styles. The dynamics of rt 1 is associated with a vector of state variables zt that is observable at time t. Consider an investor who implements a multi-style timing strategy. The investor’s problem is to choose the optimal style weights wt  (w1,t, w2,t,, wN,t ) to maximise a utility function of the trade-off between the expected style investing performance and the underlying investing risk. Formally, this unconstrained single-period optimal style selection problem can be conventionally described as (c.f. Brandt

and Santa-Clara (2006)):

–  –  –

The solution to (1), called the style timing policy, maps the preference parameter set  that is ex-ante, the state vector zt and the parameters

of the data generating process  to the optimal style weights wt :

–  –  –

Parameter  can be estimated from a given sample research data set rT  {rt 1}T0, and typically it is unbiased or at least assume consistent t estimates ˆ can be obtained. Thus the estimates of the optimal style

weights are:

–  –  –

Where bt 0, and bt is small enough to ensure that the marginal utility of wealth remains positive. Assume that the state vector observed at time t is zt  ( zt1, zt2, zt3, zt4 )  (divt, spreadt, yldt, termt ) 25.Let rp,t 1 be the excess returns of investor’s style timing portfolio from time t to t+1. After

simple manipulation, (4) can be rewritten as:

–  –  –

In the unrealistic case when excess returns are iid and optimal style weights are constant over time (i.e. wt  w ), the conditional expectation

–  –  –

Alternatively we can assume Z is Fama-French factors (SMB and HML) and/or momentum factor Carhart (1997). These state variables are available for investors in the lagged values. The use of the 4 macroeconomic variables used in this chapter are default risk premium (def), dividend yield (div), the term spread (term) and shortterm interest rate (yld), they are also used in chapter 3.

Given sample research data rT  {rt 1}T0, the moments in (7) can be t estimated using sample analogues.

While not so straightforward, the analytical expression of (7) suggests a link between predictability of state variables and style timing policy.

Theoretically, if state vector zt captures the first and second moments of style returns, one can identify which state variable is important in the style timing policy by first modelling the conditional means, variance and covariance of style returns as a function of zt and then derive the optimal style weights as a function of state variables (e.g.

Ferson and Siegel, 2001). This approach suffers from the difficulty in modelling the conditional covariance with state variables. It is also not parsimonies because there are too many parameters to be estimated.

Brandt and Clara (2006) present an interesting methodology that focuses directly on the portfolio weights, rather on the underlying styles’ conditional return distributions. They argue that this approach is an approximation of the traditional solution provided by Ferson and Siegel (2001). In this framework, the optimal portfolio weights are a linear function of the observed state variables, i.e. wt   zt. Thus the optimization problem (6) becomes

–  –  –

Where vec( ) is a vector that stacks all the columns in, and  is the Kronecker product of two matrices. Now let w  vec( ) and rt 1  zt  rt 1, problem (8) becomes

–  –  –

Since style weight matrix w maximises the conditional expected utility at all time t, it should also maximises the unconditional utility, hence the optimization problem is

–  –  –

Correspondingly, this is to find the optimal unconditional portfolio weights of w for the expanded risky asset set of N  K (i.e. number of styles  number of state variables) with returns of rt 1. Therefore,

–  –  –

Based on the solution (12), the investor can retrieve the weight investing in each of the styles by adding the corresponding products of w and zt.

Now consider an economy with 4 investable equity styles, S1, S2, S3 and S4, corresponding to Small-Value (SV), Small-Growth (SG), LargeValue (LV) and Large-Growth (LG) stock groups, respectively. While one can always use 9 styles to fill the entire equity universe, it is more efficient to choose only 4 highlighted styles to capture the interaction of two basic style dimensions that have shown to have wider return spreads in Chapter 3. The selection of these 4 styles is also justified by recent empirical findings. For example, Horowitz et al. (2000a) find that the observed size premium is not linear across all stocks but is concentrated only in smaller firms. Likewise, Fama and French (2008) observe that the size premium is the strongest among U.S. tiny stock groups based on data from 1963-2005. Fama and French (2012) also find that both value premiums and momentum effect differ across size dimension, specifically, value premiums and momentum returns decrease from smaller to large stocks.

These 4 styles (s1, s2, s3, s4 ) act as basis assets and are obtained by sorting stocks according to company characteristics of PC, BM and DY, respectively. This process is consistent with previous Chapter 3 and 4 in the research. Consider the time series of 60 months historical

observations of excess returns for these 4 styles:

–  –  –

Equation (7) directly gives the Markowitz solution of optimal static weights for these 4 styles, namely w  (w1, w2, w3, w4 ). This solution takes into account the sample covariance matrix of style excess returns and the vector of sample mean excess returns.

Suppose now the conditional distribution of style excess returns is affected by the business cycle effect, and the investor can observe a set of economic variables that relate to the business cycle. The state variables are zt  ( zt1, zt2, zt3, zt4 )  (divt, spreadt, yldt, termt ). It should be noted that these variables are only known at the beginning of each return period hence are one month lagged behind. The matrix of the time

series of state variables is:

–  –  –

In the spirit of Brandt and Clara (2006) approach, the basis style

assets return matrix (13) can be expanded in the following manner:

(15) The optimal static portfolio of this expanded set of assets can be computed by equation (12) using sample analogues. The static solution is w  (w1, w2, w3,, w20 ), corresponding to each of the 4 basis styles and 16 managed portfolios in matrix (15). Based on these

results, the optimal weights invested in the 4 styles are:

–  –  –



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