Parametric portfolio selection: evaluating and comparing to Markowitz portfolios/Selecao parametrica de portfolios: avaliacao e comparacao com portfolios de Markowitz.
| Autor | Medeiros, Marcelo C. |
| Cargo | Articulo en ingles |
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Introduction
The research of portfolio selection is motivated by two factors. First, the increase of market complexity and computing power pose both challenges and opportunities to the investor's decision problem. Second, the traditional mean-variance approach of Markowitz (1952), despite several recent improvements, deals with a big number of arguments to be optimized given that we need to estimate all portfolio weights directly as arguments in the objective function. The Markowitz approach can be viewed as the minimization of the portfolio expected conditional volatility given a target conditional expected return and asset's covariances. The application of this approach goes beyond the simple minimization of the expected conditional volatility, it considers several other restrictions and fixes such as constraining portfolio weights, imposing transaction costs, shrinkage of the estimates and other adjustments that ensures the model is as close as possible to the reality of the market and the estimates are adequate. However, the complexity of the model increases significantly when the number of restrictions grows bigger. A survey of these studies is found in Brandt (2010).
Our objective in this paper is to evaluate the parametric portfolio optimization approach proposed by Brandt et al. (2009) using monthly data of Brazilian stocks. We used as characteristics (explanatory variables) the book-to-market ratio (BTM), the market equity (ME), defined as the number of stocks of a given company times its price, and the one year momentum (MOM). We compared the results of the parametric portfolio to the value-weighted portfolio, equal-weighted portfolio, and the traditional Markowitz-based portfolio. The base parametric model refers to the model with a simple linear restriction on weights. The parametric model is extended to include short sale constraints (1), maximum absolute weight on individual stocks, transaction costs, and the inclusion of a risk-free asset in the investable set.
The optimized portfolios consistently show risk-adjusted returns above all other portfolios. These results remain after imposing weight constraints and market costs. The optimized parametric portfolios are also superior to Markowitz based portfolios. We find the parametric approach to yield better results, be computationally simpler, it makes easier to handle changes in the investable set through time, and market costs are easy to implement. We used the constant relative risk aversion (CRRA) utility function to estimate the optimal portfolios, we tested the model for several levels of risk aversion, and even for extreme risk aversions [gamma] = 100, the investor chooses to keep some of his wealth in stocks when the risk free asset is available. Nevertheless, the certainty equivalent, in this case, converges to the risk free rate as the relative aversion grows bigger. Furthermore, we tested the parametric portfolio optimization out of sample performance and it yielded higher returns than the market and the out of sample Markowitz based, even when transaction costs are included. Although the parametric optimization is described by Brandt et al. (2009) as a method of moments estimator from Hansen (1982), the estimation of the covariance matrix of the parameters may be troublesome when facing nonlinear constraints, to solve this issue we used bootstrap techniques and the parameters were estimated through nonlinear optimization methods. Additionally, the risk-adjusted returns above markets obtained using a few publicly available data suggest the Brazilian stock market is still inefficient. Finally, the sample period we used involves several changes in the economical and financial environment in Brazil, which are captured in all portfolios. The fact the parametric portfolios are formed by constant coefficients across the entire period is a strong argument in favor of (i) the parametric approach and (ii) the inefficiency of Brazilian stock market.
This paper is organized as follows: section 1 describes the parametric portfolio optimization and its extensions. In section 2 we discuss the bootstrap for statistical inference, section 3 describes the data and an empirical application. Our final remarks are in section 4.
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Methodology
2.1 Parametric approach
This section aims to present the basic parametric portfolio optimization, originally and better described in Brandt et al. (2009).
Let [N.sub.t] be the number of stocks in the investable set at each date t. Each stock i has a return [r.sub.i,t + 1] from period t to period t + 1 and is associated to a vector of characteristics (2) observed at period t.
A portfolio is a vector of weights [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] which has a return [r.sub.i,t + 1]. The investor's problem is to choose the portfolio that maximizes his conditional expected utility,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)
The parametric approach has such name because it parametrizes the optimal portfolio weights as a functions of stocks' characteristics. The parametrization may be very general, but in this paper we used the weights as a linear function of the characteristics from Brandt et al. (2009), such
that:
[W.sub.i,t] = [[bar.W].sub.i, t] + 1/[N.sup.t][[theta].sup.[??]] [[??].sub.i, t] (2)
where [[bar.W].sub.i, t] is the weight of stock i at time t on a benchmark (3) portfolio, [theta] is the vector of coefficients to be estimated, [[??].sub.i, t] t are the observed characteristics of stock t, they are standardized cross-sectionally to have zero mean and unit standard deviation across all stocks at date t. The symbol [??] indicates that the vector is transposed.
The zero mean cross-section ensures the weights always sum one, regardless the values of [theta]. The unit standard deviation makes sure [[??].sub.i, t] is stationary trough time, while the raw [x.sub.i, t] may be non-stationary. Finally, the term 1/[N.sup.t] is a normalization that allows the portfolio weight function to be applied to any time varying number of stocks. Without this normalization, an increase in the number of stocks would result in increased leverage with larger long and short positions.
The idea of the linear parametrization of equation (2) is that of a portfolio management relative to a performance benchmark. Since the characteristics are cross-sectionaly standardized, the term [[theta].sup.[??]][[??].sub.i, t] will have a zero cross-sectional mean, and consequently, the deviations from the optimal portfolio weights from the benchmark will sum to zero, and this ensures the optimal portfolio weights sum one.
In the traditional Markowitz approach, one would have to estimate portfolio weights for each stock for each period of time. In the parametric portfolio optimization we estimate weights as a single function of characteristics, the parameters are the same for all stocks and for every period of time.
The key to understand Brandt et al. (2009) parametrization is the constant coefficients of [theta] for all stocks and for the entire period. Constant coefficients across stocks imply that the portfolio weights depends only on stocks' characteristics, i.e. stocks with similar characteristics must have similar weights even if their past returns are different. The implications of the constant [theta] through time are even stronger, the coefficients that maximize the investor's conditional expected utility at a given date are the same for all dates, and therefore they also maximize the investor's unconditional expected utility. If a set of stocks is described by five parameters, they will be the only five parameters for all assets in the entire period. Thus, the coefficient of time can be excluded from the expectation of equation (1):
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)
The coefficients that maximizes the investor's unconditional utility are:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)
Thus, the sample analogue of (4) for some given utility function and the portfolio police of equation (2) is:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)
Due to the relatively low dimensionality of the parameter vector, the suggested approach is computationally simple. The complexity of the problem increases only if we choose to add more characteristics to the model, the increase of the number of assets has nearly no effect in estimating [??]. Besides, the use of few parameters increases numerical robustness and may reduce the risk of in-sample overfitting.
Regarding the utility function, we chose to use the constant relative risk aversion (4) (CRRA), with a base relative risk aversion [gamma] of five. This function has three desirable characteristics. First, they incorporates preferences towards higher-order moments without introducing additional preference parameters. Second, the function is twice continuous differentiable, which helps the optimization process. Finally, the CRRA function is optimal under a partial myopic behavior (e.g. Mossin (1968)).
The optimal coefficients [??] defined in equation (5) can be interpreted as a method of moments estimator. However, the constraints imposed to the problem in the next section prevent us from using the asymptotic properties of the covariance matrix [[[conjunction].summation over].sub.[theta]]. Instead, [[conjunction].summation over].sub.[theta]] is estimated by bootstrapping techniques.
2.2 Extensions
This section aims to find the adequate extensions of the base model described in 2.1 for the Brazilian stock market. First, the absence of a liquid market for short positions during most of the evaluated period requires limits on negative weights. Second, local regulations imposes maximum weights on single stocks. Third, we discuss the inclusion of a risk free asset in the investable set. Last, the parametric approach...
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