Parametric portfolio policies: evidence from
the Dutch stock market
J.W. Lutterop
s2226243
Parametric portfolio policies: evidence from
the Dutch stock market
J.W. Lutterop
Abstract
Contents
1 Introduction 2
2 The model 3
2.1 specifications and extensions . . . 3
3 Literature review 7 3.1 Portfolio decision making . . . 7
3.2 Asset pricing . . . 9
3.3 Similar literature . . . 10
4 Empirical results 12 4.1 Data . . . 12
4.2 Results for the base case . . . 15
4.2.1 Sensitivity to sample size . . . 19
4.2.2 Sensitivity to risk aversion . . . 22
4.3 Extensions . . . 26
4.3.1 Weight restrictions . . . 26
4.3.2 Transaction costs . . . 28
5 Conclusion 30
6 References 32
1
Introduction
The field of portfolio management or portfolio selection has seen a surge of renewed interest in the recent years. Increasing market complexity and advances in technol-ogy opened the door to new challenges and opportunities in relation to the investor’s decision problem. The economic theory behind the optimal portfolio decision prob-lem saw its inception in the 1950s with the well known mean-variance approach by Markowitz [1952] [33]. Despite being theoretically useful, comprehensible and ele-gant, in practice this approach is econometrically problematic. In a universe with N stocks, N (N + 3)/2 parameters need to be estimated, the expected returns and the variances and covariances of the returns. This is not problematic if N is small, but if N is large the results will be noisy due to having too many parameters relative to the number of observations.
Brandt et al. [2009] [3] tackle this problem with a general model where portfolio decisions rely on firm characteristics, circumventing the estimation of the expected stock returns and (co)-variances. Dimensionality problems are eliminated, as the method relies on a small set of fundamental characteristics, which entails that very large scale portfolio problems can be solved. Moreover, Brandt et al. argue that the method implicitly captures the relation between firm characteristics and first-, second- and higher-order moments and that it allows for easy testing of joint and individual hypotheses.
The aim of the present study is to evaluate the performance of the parametric portfolio policy model (henceforth PPP model) in the Dutch stock market. This is done by evaluating the performance of the PPP model in general and by comparing the PPP model to several benchmark models in terms of statistical measures such as e.g. return, volatility, turnover and the Sharpe ratio (see section 4). Ultimately, the research question is
How well does the PPP model perform in the Dutch stock market,in general and compared to several benchmark models?
2
The model
The model used in the present study is proposed by Brandt et al. [2009] [3]. Let
t denote the date and let Nt be the number of stocks in the investable universe at
that date. Between dates t and t + 1 each stock has a return ri,t+1(i ∈ {1, . . . , Nt})
and is associated with a vector of firm characteristics xit. The resulting expected
utility maximisation problem is given by max wt E u(w0 trt+1) (1)
where u(·) is the investor’s utility function, rt+1is a Nt×1 vector that stacks the
re-turns ri,t+1and wtis a Nt× 1 vector that stacks the portfolio weights parameterised
as
wi,t = ¯wi,t+
1
Nt
θ0xˆi,t. (2)
Here ¯wi,t is the weight of stock i in a benchmark portfolio, θ is a parameter vector
to be estimated and ˆxi,t is the vector of firm characteristics normalised to have zero
mean and unit variance. Parameterisation (2) is merely one of the possibilities, in
this modelling framework any function f (ˆxi,t; θ) will do, the present study focuses
on (2). Note that the normalisation term 1/Ntensures that the model can be applied
to an arbitrary amount of stocks and the normalisation of the firm characteristics ensures that all portfolio weights sum to unity. For further detail and motivation behind the model the reader can consult Brandt et al. [2009] [3].
Due to parameterisation (2), the expected utility maximisation problem is given by max θ E " u Nt X i=1 ¯ wi,t+ 1 Nt θ0xˆi,t ri,t+1 !# . (3)
Suppose the last date in the sample is T , then the estimator of θ is given by ˆ θ = arg max θ T −1 X t=0 u Nt X i=1 ¯ wi,t+ 1 Nt θ0xˆi,t ri,t+1 ! . (4)
One can see that this amounts to a generalised method of moments (GMM) estima-tor (see e.g. Hansen[1982] [22]). For statistical inference see Brandt et al. [2009][3]
1.
2.1
specifications and extensions
As one can see in (2), a benchmark model is needed to obtain ¯wi,t. In the present
study estimation results using both the value weighted (based on the logarithm of the market value) portfolio and the equally weighted. Also, it seems common
1The reason that statistical inference is not included here is that bootstrapped standard errors are
practice (Brandt et al. [2009] [3], Barberis [2000] [1] or Merton [1969] [36], just to name a few) to use the power utility function defined by
u(r) = (1+r)1−λ 1−λ λ ∈ (0, ∞)\{1} ln(1 + r) λ = 1. (5)
Consequently, the power utility function is used in the present study. Note that λ < 0 would mean that the investor is risk loving. Though theoretically possible, risk loving is not a realistic assumption and throughout the paper it shall always be assumed that the investor is risk averse. λ = 5 Shall be the mainly used pa-rameter choice, though estimation results with different papa-rameters are presented in the sensitivity analysis. The main desirable property of the power utility function is that it exhibits constant relative risk aversion. This entails that the investor’s attitude toward risk does not depend on his or her wealth, meaning wealth need not be included in the optimisation problem. Furthermore, the power utility function
is C2 which is helpful for optimisation algorithms that rely on the gradient vector
and the Hessian of the objective function, which is the case in the present study.
Brandt et al.[2009][3] show that the linear PPP model specification nests the Fama French three factor asset pricing model (Fama and French [1993] [16]) and its ex-tension, the Carhart four factor asset pricing model model (Carhart [1997] [5]). Therefore these two models are obvious candidates for testing. Other candidates are variations on the aforementioned two models; a three factor model by Chen and Novy-Marx [2011] [8] and a five factor model by Fama and French [2015] [17]. Erge-man and Taamouti [2015][13] extend the PPP-model by adding common volatility dynamics. Finally, the model can easily be extended to accommodate more realistic assumptions; weight restrictions (e.g. no short selling) and transaction costs for instance. Due to scarce data availability, the model by Chen and Novy-Marx and the Fama French five factor model are not viable models to apply, as they require too specific balance sheet related data, which is hardly available for Dutch firms. Furthermore, the method by Ergeman and Taamouti [2015] [13] is suitable for daily observations(see subsection 4.1 for further details)The Carhart four factor model does not have this problem, as the required data is widely available. Hence, this study shall revolve around this specification.
as an extension of the base case (subsection 4.3.1). This is easily implemented by truncating the allocations at zero
wit+= max[wit, 0]
PNt
j=1max[wit, 0]
. (6)
This ensures that all portfolio weights are positive and sum to unity. Note that (4) is now no longer optimal. The optimisation problem is augmented by subtracting a penalty term given by
10
Nt X
i=1
|wit|1{wit < 0} (7)
where 10 is an arbitrary choice of constant 2 and 1{·} is the indicator function.
Consequently, using (2) in combination with (7), the estimator with the long only restriction is given by ˆ θLO =arg max θ T −1 X t=0 u Nt X i=1 ¯ wi,t+ 1 Nt θ0xˆi,t ri,t+1 − 10 Nt X i=1 ¯ wi,t+ 1 Nt θ0xˆi,t 1{ ¯ wi,t+ 1 Nt θ0xˆi,t< 0} ! . (8)
Furthermore, the basic model relies on the premise that markets are frictionless (i.e. absense of transaction costs). This assumption is convenient for theoretical models, though in practice, trading is not free. Since the unrestricted model per-mits a very high turnover (trading activity), it is possible that the model changes the allocation in a stock to capitalise on a relatively small profit. In a frictionless market this is not an issue, but if transaction costs are introduced, the costs of changing the wealth allocation might not offset the marginal profit that it yields. Consequently, transaction costs are added as an extension of the base case. Trans-action costs are taken to be proportional to its allocation, consequently transTrans-action costs will be determined by the turnover of the portfolio. Euronext, of which the Amsterdam Stock Exchange is a part, pursues a policy where transaction fees are higher for less liquid stocks and lower for highly liquid stocks. Stock liquidity gen-erally is higher for high cap stocks and vice versa, consequently the characterisation of transaction costs used in Brandt et al. [2009] [3] - which is based on the same premise and argumentation - is implemented in the present study , the specification
is as follows. Firstly, let cit be the proportional transaction costs for stock i at
time t. The costs associated with rebalancing stock i at time t are cit|wit− wi,t−1|
(wi0 ≡ 0) and consequently, the portfolio return between time t and t + 1 is given
by rp,t+1 =PNt=1t witri,t+1−PNi=1t cit|wit− wi,t−1|. Here, cit is assumed to depend
negatively on a firm’s size; specifically, cit = 0.006 − 0.0025 ˜meit, where ˜meit is the
2This choice produced satisfactory results. The constant can be chosen higher or lower depending on
log market equity (see subsection 4.1 for definition) rescaled to the interval [0, 1]. Rescaling is done by applying the transformation
˜
meit =
meit− max meit
max meit− min meit
+ 1. (9)
where meit is the logarithm of the market equity of firm i at time t. With the
3
Literature review
As the present study finds its theoretical basis in both portfolio decision making the-ory and asset pricing thethe-ory, literature on both subjects is reviewed in this section. It concludes with a review on the literature of similar studies.
3.1
Portfolio decision making
The economic theory behind an investor’s optimal portfolio choice was pioneered by Markowitz [1952] [33] when he proposed a mean-variance setting where expected portfolio volatility is minimised while attaining a target expected portfolio return. The model by Markowitz laid groundwork for what is now known as modern portfo-lio theory (MPT). All of his findings were later extensively discussed in Markowitz [1959] [34]. Tobin [1965] [45] extended the one-period problem by Markowitz to a multi-period portfolio problem. This formed a basis for the discrete life-time port-folio selection model using dynamic programming by Samuelson [1969] [41] and its continuous time counterpart developed by Merton [1969] [36], the latter of which was later extended by Merton [1971] [37]. The theoretical groundwork introduced by Markowitz provided two key insights: it illustrates the effects of portfolio diver-sification (i.e.“don’t put all of your eggs in one basket.”) and that fully diversified portfolios can only generate higher expected returns by taking on more risk.
Although Markowitz’s contributions earned him a Nobel prize in economics in 1990, his mean-variance setting has its theoretical limitations. First of all, it is based on the assumption that the return distribution is fully characterised by its first- and second order moments, which is only realistic if asset returns follow an elliptical distribution (see e.g. Fama [1965] [14]). Secondly, it uses the return variance as a correct measure of risk, which does not align with investor preferences if they are merely concerned with downside risk (which is a fair assumption to make, see e.g. Sharpe [1964] [42]). These criticisms motivated refinements and extensions of Markowitz’s work, most predominantly by incorporating higher order moments. These improvements form the field of post-modern portfolio theory (PMPT). Kraus and Litzenberger [1976] [30] and Kane [1982] [29] use skewness in their model, i.e. they include third order moments. Fama [1970] [15] and Elton and Gruber [1974] [12] are examples of papers where alternative distributions are used. Finally, Sortino and Price [1994] [43] apply the concept of downside risk.
number of assets is high and the amount of observations is relatively low 3. More-over, the optimisation results will be inherently noisy, as the portfolio optimisers will put high weights on the returns that are prone to large estimation errors (see Michaud [1989] [38]). Michaud calls this phenomenon error maximisation, of which the magnitude was quantified by Jobson and Korkie [1980] [25] for several cases.
The problems concomitant with the estimation of the first and second order mo-ments have been dealt with in several ways. Michaud [1989] [38] points out that imposing constraints on portfolio weights mitigates the noisiness of the results and outlines alternative methods, though he alludes that “more direct approaches” are preferable. By imposing constraints on the portfolio weights one can incorporate realistic assumptions such as taxation, transaction costs and short sale constraints etc. Literature on constrained portfolio optimisation is extensive and includes e.g. Frost and Savarino [1990] [21], Davis and Norman [1990] [10] and He and Pear-son [1991] [23]. An alternative estimation strategy is shrinkage estimation (Stein [1956] [44]), which is implicit in Bayesian inference ( Copas [1983] [9]) and explicit in James-Stein type shrink estimators (James and Stein [1961] [24]). Jorion [1986] [26] proposes an empirical Bayesian approach and shows that his Bayes-Stein esti-mators outperform the sample mean in the context of portfolio selection problems. Frost and Savarino [1986] [19] convey their success in using an empirical Bayesian
approach where they make use of an informative prior4. Finally, another ‘fix’ of the
aforementioned problems is imposing a factor structure on the covariance matrix, see e.g. Chan et al. [1999] [7].
Though the aforementioned techniques provide fairly good fixes for the estimation problems with the covariance matrix, the easiest way is to circumvent the estimation of covariance matrix altogether. This is done by parameterising portfolio weights as functions of observable quantities such as firm characteristics. This idea was devel-oped by Brandt and Santa-Clara [2006] [2] and Brandt et al. [2009] [3]. The former propose in the context of single and multiperiod market timing problems a method in which portfolio weights are parameterised as linear function of state variables. They extend the Markowitz model by choosing between conditional portfolios that invest in each asset proportional to a conditioning variable and timing portfolios that invest in each asset for a single period and in the risk-free asset for all other periods. They proceed to show that this static approach is equivalent to a dynamic strategy that invests in the basis assets. The model by Brandt et al. [2009][3] is sim-ilar, but more suited for large-scale cross-sectional portfolios. They parameterise the portfolio weights as a function of firm characteristics and argue that their method is “computationally simple, easily modified and extended, produces sensible portfolio weights, and offers robust performance in and out of sample.” The model by Brandt
3It is not uncommon to have a thousand assets and ten years of data (i.e. 120 observations for monthly
data).
et al. [2009] [3] is the bedrock of this study. For a comprehensive overview of the theory of portfolio choice problems the reader can consult Brandt [2009] [4].
3.2
Asset pricing
In this section an overview of the asset pricing theory that is relevant to the present study is provided. Specifically, the focus is on the Capital Asset Pricing model (CAPM hereafter) and its static extensions and refinements; bond pricing and op-tion pricing are not treated in this secop-tion. The reasons for this are that the CAPM is closely related to the theory by Markowitz and that the CAPM and its extensions can be used in conflation with the PPP model by Brandt et al. [2009] [3].
The CAPM is a linear model which explains expected asset returns as a function of the expected market return, originating from the work by Sharpe [1964] [42],
Lintner [1965a,b] [31, 32] and Mossin [1966] [39] 5. The reasoning behind why the
CAPM is closely related to the mean-variance framework by Markowitz [1952][33] is as follows. The idea behind Markowitz’s optimisation problem is that a rational investor picks a portfolio with the least volatility given a specified expected portfolio return, or conversely, a portfolio with the highest expected return given a certain amount of volatility. In this risk-return spectrum, these “mean-variance efficient” portfolios form the efficient frontier. Sharpe and Lintner add two assumptions to the Markowitz model. Firstly, all investors agree upon one true joint return dis-tribution and secondly, everyone can borrow/lend at the risk-free rate. The latter assumption allows an investor to allocate it’s wealth between the risk-free asset and a risky portfolio. The former assumption causes all rational investors to invest in a linear combination of the risk-free asset and one risky portfolio on the efficient frontier (i.e. the market portfolio). This is where the CAPM arises, for further details see e.g. Fama and French [2004] [18].
The role of the CAPM in asset pricing is similar to that of Markowitz’s model: it is considered to be the foundation of asset pricing. Indeed, William Sharpe earned a Nobel prize in economics for his contributions in the same year as Harry Markowitz did. Even though the CAPM is an inevitable part of any course in asset pricing theory, the model has certainly endured its criticisms since its inception. Since the
1970s empirical evidence against the CAPM has been piling up6, see C¸ elik [2012] [6]
for an extensive overview. One explanation is that the CAPM is too simple and that the assumptions upon which it is build are unrealistic. This sparked the creation
of extensions and refinements of the model in the decades thereafter 7. Fama and
French [1993] [16] introduced a three-factor model, an extension of the CAPM by adding two factors: a portfolio of small minus big cap stocks and a portfolio of high
5Although, French [2003] [20] argues that Treynor [1961,1962] [46, 47] should also be regarded as one
of the ”primary architects” of the CAPM.
minus low book-to-market ratio stocks. Carhart [1997] [5] extends this three-factor model by adding a momentum factor based on the stock’s previous performance. Other extensions were the use of higher moments: Kraus and Litzenberger [1976] [30] extend the CAPM by using skewness of the returns in the model and Dittmar [2002] [11] uses both skewness and kurtosis.
Extensions of the CAPM by using additional moments or additional factors (port-folios) exist in abundance in present times, see Vendrame and Tucker [2016] [48] for further discussion. Chen and Novy-Marx [2011] [8] propose an alternative to the three factor model by using a market factor, an investment factor and a return on equity factor. In turn, Fama and French [2015] [17] extend their three factor model by adding robust minus weak factor and a conservative minus aggressive factor to capture profitability and investment patterns. In conclusion, the abundance of as-set pricing models that can be applied in conflation with the PPP-model provides a fruitful basis for a comparative empirical study.
3.3
Similar literature
Some empirical tests of the PPP model have been conducted in the literature. First and foremost, Brandt et al. [2009] [3] estimated the PPP-model in conflation with the Carhart four factor model on US data and conclude it produces robust results that outperform the market portfolio, in sample and out of sample. They demon-strate the importance of the firm’s market capitalisation, book to market ratio and momentum in the investor’s decision problem. Furthermore, they extend the model by incorporating transaction costs and weight restrictions and find that it reduces performance only marginally. Hjalmarsson and Manchev [2012] [28] apply the idea by Brandt et al. to the classic mean-variance setting in order to permit a deeper understanding of the portfolio weights estimator. Their empirical results provide us with three insights: combining several characteristics in the optimisation problem is a substantial improvement compared to using one characteristic and the direct-estimation approach generally outperforms the naive regression approach, but this is not the case for the equal-weighted portfolio. Medeiros et al. [2014] [35] test the PPP-model using a three factor approach on the Brazilian market and show that the parametric approach is very efficient out-of-sample. Furthermore, they conclude that the PPP portfolio’s are superior when compared to the value weighted port-folio, the equally weighted portfolio and a Markowitz based portfolio . Hand and Green [2011] [27] use the PPP model on US data to stress the importance of firm characteristics in portfolio selection. Other than the aforementioned studies, I did not find further empirical evidence.
4
Empirical results
In this section the empirical results of the PPP model (and its extensions) described in section 2 are presented, as well are the results of the value weighted and equally weighted portfolios. The base case is the model described by parametrisation (2), where the used characteristics are based on the Carhart four factor model (Carhart [1997] [5]). Note that the contribution of the present study is mainly an empirical one, focusing on the performance of the model. The data and the specific variables used are described in the first subsection, the other subsections are dedicated to the estimation results and the performance of the PPP model and its extensions. All the analysis was done in R, the programming code of which is provided in appendix A
4.1
Data
The raw dataset contains monthly observations of 294 Dutch firms traded on the Euronext Amsterdam Stock Exchange, ranging from January 1973 to October 2017. The investable universe consists exclusively of equity shares. Exchange traded funds and notes, preference shares and warrants are not included. The dataset contains stock price information and other market dependent variables (e.g. market value), which are measured monthly and balance sheet information, which is measured
an-ually 8.
The model used in the present study and the paper by Brandt et al.[2009][3] applies the Carhart four factor model using variables for market equity (in logs), the book to market ratio (in logs) and momentum. The log market equity is defined as the natural logarithm of the market value, where market value is defined as the share price multiplied by the number of outstanding shares. The log book to market ratio is defined as log(btm) = log 1 + book equity market value
where book equity is defined as total assets minus total liabilities, to which balance sheet deferred income taxes and tax investment credit is added and preferred stock value is subtracted if either or both are available. Furthermore, a six month time gap is allowed for the balance sheet information to become available. Here it is as-sumed that balance sheet information becomes available at the end of the calendar year. Hence, the returns for July till December are explained by the log book to market ratio of the preceding year, whereas the returns of January till June are explained by the log book-to-market ratio of the year before that (i.e. two years ago). Finally, the momentum variable is the compounded return of the preceding twelve months. That is, the momentum at time t is the compounded return between t − 13 and t − 1. Due to the definitions of the latter two variables, the first eighteen
Firms between January 1980 and October 2017 Year Fir ms 1980 1990 2000 2010 0 50 100 150
Figure 1: Firms in the investable universe over time.
months of observations are ‘lost’ .
Stock price anomalies have been eliminated manually. By inspection some stock prices exhibited erratic behaviour, presumably due to lack of liquidity. These stocks have been eliminated from the dataset. Furthermore, some firms had stock price observations for dates before they became public. These were easily recognised by their uncharacteristically high values. Naturally these observations were eliminated from the dataset. Some stocks showed severe price plunges in the months preceding the firm’s bankruptcy, this was dealt with via the returns; returns were truncated
from below at the 0.025% quantile to minimise the effect of extreme price drops.9.
Returns are calculated as the growth of the stock price in between months, these stock prices are adjusted for stock splits. Also note that the market equity and book to market ratio are measured in logarithms, which circumvents possible problems with outliers.
Subsequent to dealing with price anomalies, all firms and dates without usable observations were eliminated. The resulting dataset contains observations for 247 firms between July 1980 and October 2017. The amount of usable observations per monthly observation differs, only firms with complete data at a given date are included in the dataset. A time series plot of the amount of firms in the investable universe throughout the dataset is given in figure 1. The firm count of investable
9It turned out that the quantile level chosen did not influence the parameter estimation results by
mean returns Year 1980 1990 2000 2010 −0.4 −0.2 0.0 0.1 sd returns Year 1980 1990 2000 2010 0.1 0.3 0.5 mean me Year 1980 1990 2000 2010 4.0 5.0 6.0 sd me Year 1980 1990 2000 2010 1.4 1.8 2.2 2.6 mean btm Year 1980 1990 2000 2010 6.0 6.5 7.0 7.5 8.0 sd btm Year 1980 1990 2000 2010 0.6 0.8 1.0 1.2 1.4 mean mom Year 1980 1990 2000 2010 −0.5 0.0 0.5 1.0 sd mom Year 1980 1990 2000 2010 0.2 0.4 0.6 0.8
Figure 2: Time series plots of the means and standard deviations of the returns and (non-normalised)characteristics used the PPP-model.
Figure 2 depicts time series plots of the cross-sectional means and standard de-viations of the returns and the non-normalised characteristics. The mean returns appear to be reasonably stable around zero over time, apart from several downward spikes. The cross-sectional standard deviations of the returns exhibit numerous spikes, most prevalent after the change of the millennium, the largest of which to-ward the end of the dataset. The mean market equity is generally upto-ward trending, as well as the standard deviation, whereas the book to market ratio is tentatively downward trending. Finally, the mean of momentum characteristic appears to be rather erratic over time, which is also the case for its standard deviation.
4.2
Results for the base case
In this section the results of the base case are presented, i.e. parameterisation (2) with characteristics as specified in the previous section and with benchmark weights from either the value weighted portfolio (table 1) or the equally weighted portfolio (table 2). Both the in sample and the out of sample results of the PPP model are presented. The meaning of the former is obvious, the meaning of the latter is that historical data is used for investments in the future. It is important to realise that from a practical point of view, the out of sample results are the practically relevant results and that the in sample results are merely of theoretical relevance.
All results in the present study are presented in a similar format; tables 1 and 2 each have four columns of results with rows divided in three categories. The columns respectively contain results of the value weighted, the equally weighted, the in sample parametric and out of sample parametric portfolio policies. The first category of rows contain the parameter estimates of the PPP model with their boot-strapped standard errors. The second category contains (time) averages pertaining to the portfolio weight distribution; respectively the overall mean absolute weight, the mean maximum weight, the mean minimum weight, the mean sum of the neg-ative weights, the mean fraction of negneg-ative weights and the average turnover. The last category contains the average portfolio returns, the standard deviation of the
portfolio returns, the sharpe ratio and the certainty equivalent return 10. The last
category shall be commonly referred to as ‘performance measures’. All subsequent tables have the same categories of rows.
The definitions of the statistics and estimates are to a large extent obvious, pa-rameter estimates arise from equation (4) and its standard errors are computed
using 1000 bootstrap samples 11. The mean absolute weight is computed over all
10Note that the expressions used for the statistics in tables 1 and 2 (and subsequent tables) are a slight
abuse of notation. The choice of expressions is purely for expositional purposes.
11Standard errors are only offered in sample as bootstrapping out of sample is too computationally
Yield on Dutch 10 year Gvt. bonds year risk free r ate 1990 1995 2000 2005 2010 2015 0.00 0.02 0.04 0.06 0.08
Figure 3: Time series plot of the risk free rate.
weights, whereas the mean minimum (maximum) weight is the time average of the cross-sectional minima (maxima). The sum of the negative weights and the fraction of negative weights are time averages as well and the average turnover is the time
average of the sum of the absolute difference between the successive allocations 12.
The average portfolio return and its standard deviation have obvious definitions. The Sharpe ratio is computed as
SR = mean(rp− rf)
SD(rp− rf)
(11) i.e. the mean difference of the portfolio returns and the risk free rate divided by its standard deviation, where the risk free rate is defined as the return on Dutch 10 year government bonds (see figure 3). The Sharpe ratio measures the average excess return, adjusted for risk. Equivalently, the average excess return per unit of standard deviation or volatility.
Finally, the certainty equivalent return is defined as
rCE= u−1 1 T T X t=1 u(rpt) ! (12)
where u−1(·) denotes the inverse of the utility function, which is well defined for
specification (5). Note that (12) arises from the identity
u(rCE) = E(u(rp)) (13)
where the expectation operator is replaced by its sample equivalent. The certainty equivalent return is the return such that the investor is indifferent between investing
12If a certain stock is in the investable universe at time t, but not at time t − 1, we consider the
using the portfolio policy or taking the riskless certainty equivalent return.
Out of sample results are obtained as follows. The in sample parameter estimates are used to compute the weights for each month of the first out of sample year. All the weights and performance statistics are subsequently computed over that year. The sample is then extended by one year, the new parameters are estimated and the process repeats. The reported parameter estimates are the time averages. By default, a sample size of 10 years is used for the both the in sample results and out of sample results (in sample results with a sample size 20 years are also presented in the next section). There are two reasons to use a default sample size of 10 years for the in sample results, instead of the complete dataset. Firstly, it permits the comparison of the in sample performance of samples of multiple sizes. Secondly, it allows the reader to observe the effects from ‘venturing out of sample’, as the in sample parameters are the starting parameters of the out of sample results. But, most importantly, it is concluded in subsection 4.2.1 that the in sample performance is robust to sample size. Note that from figure 1 it becomes apparent that in the first year of the dataset (1980) there are not enough usable observations. In fact, we see a sharp increase in usable observations in 1988, where the size of the investable universe increases from roughly 20 firms to roughly 60 firms. Consequently the sample is taken from July 1988 untill August 1998, where the first observation -July 1988 - is purely for estimation purposes (in (4) this corresponds to t = 0).
Table 1 shows the results for the base case where the benchmark model in (2) is the value weighted portfolio. The statistics for the value weighted and the equally weighted portfolio (benchmark portfolios) are also included. Note that the weights of the value (equally) weighted portfolio is the same as (2) with θ = 0 and the value (equally) weighted portfolio as benchmark weights. Therefore parameter estimates are assigned a value of zero, even though they are not relevant for interpretation. Parameter estimates should not be interpreted as regression coefficients, as it is not the purpose of the model to investigate the economic relevance of the character-istics in an explanatory sense. The coefficient estimates should be interpreted as the influence of the characteristic on the decision of the investor on how to deviate
from the benchmark portfolio13. The parameter estimates in table 1 tell us that an
investor with a relative risk aversion of 5 finds it optimal to positively deviate from the value weighted portfolio for higher values of all characteristics, i.e. the higher the characteristic, the higher the allocation to that stock compared to the value weighted portfolio. Since the characteristics are all normalised, the magnitudes of the coefficient estimates can be compared. In sample, it is optimal to value high mo-mentum stocks the most, then high book to market and lastly high market equity. This order of magnitude also applies to the out of sample results. It is interesting to note that Brandt et al. [2009] [3] consistently find negative coefficient estimates
13Note that the implications of the parameter estimates are discussed, though it can not be stressed
Statistic VW EW PPP in sample PPP out of sample ˆ θme 0 0 1.0454 2.1925 std. error - - (0.9077) -ˆ θbtm 0 0 5.8908 4.7527 std. error - - (2.3920) -ˆ θmom 0 0 10.7047 6.7656 std. error - - (2.6359) -mean |wi| 0.0089 0.0089 0.0722 0.0489 mean max wi 0.0201 0.0096 0.2937 0.2039 mean min wi 0.0016 0.0096 -0.3533 -0.2625 P wi1{wi < 0} 0.0000 0 -3.5471 -2.6274 P1{w i < 0}/Nt 0.0002 0 0.4584 0.3859 P |wit− wi,t−1| 0.0245 0.01502 2.6636 1.6994 CE 0.0034 0.0023 0.0371 0.011 ¯ rp 0.0069 0.0057 0.0755 0.1068 sd(rp) 0.0363 0.0362 0.1315 0.2276 SR -1.5133 -1.547 0.0387 0.3296
Table 1: Results for the base case, where the benchmark model for the parametric policy is the value weighted portfolio, the sample size is ten years and the relative risk aversion of the investor is λ = 5.
for the market equity characteristic, meaning investors tend to underweight in large firms. This is clearly not the case here, though Brandt et al. argue that under-weighting in large firms is consistent with the literature.
It is not surprising to see that the weight allocations of the PPP model are more
aggressive than those of the benchmark models. Mean absolute weights of the
posi-tions is close to a half on average, which is a bit lower for the out of sample results (0.3859). These observations underline the importance of using weight restrictions (i.e. restrictions on short selling), the results of which are found in subsection 4.3.1. The trading activity observed with the parametric portfolios is also substantial com-pared to the benchmarks. In sample, an average turnover of 2.6635 (266.45 %) and out of sample an average turnover of 2.3012 (230.12 %) is observed. This is, as mentioned in subsection 2.1, not a concern in light of frictionless markets, however, markets are generally not frictionless. This average turnover does stress the impor-tance of adding transaction costs to the model, since trading in general is not free. These results are presented in subsection 4.3.2.
The in sample performance measures are not surprising. The in sample average portfolio return of the PPP portfolio outperforms the benchmarks in terms of av-erage portfolio return, certainty equivalent return and the Sharpe ratio. Naturally, this extra return is only possible by taking extra on risk (see subsection 3.2), which is reflected in the standard deviation of the portfolio returns. The Sharpe ratio tells us how lucrative it is to take on the risky investment strategy in comparison to the risk free rate, which is positive in case of the parametric portfolio and highly nega-tive for the benchmarks. The out of sample performance is fairly promising. Notice that the certainty equivalent return is slightly lower out of sample, but the average portfolio return (and not surprisingly the standard deviation) and the Sharpe ratio increased. The increased Sharpe ratio can for a large part be attributed to the downward trend of the risk free rate throughout the dataset (see figure 3), which makes it less lucrative to invest in the risk free rate at later dates in the dataset.
For completeness, the results of the base case, where the benchmark weights in (2) are from the equally weighted portfolio, are presented in table 2. The in sample results are similar compared to table 1, parameter estimates differ only slightly and the differences between the weight statistics and performance statistics are negli-gible. Note that the out of sample parameters differ slightly more, but the weight statistics and the performance measures are practically the same. From this can be concluded that the choice of benchmark in (2) does not matter. Further conclusions that can be drawn are firstly that parametric policy outperforms the benchmark models in sample and secondly that applying the investment strategy out of sample produces lucrative results. Despite the fact that certainty equivalent return is less out of sample than in sample, the average portfolio return is higher and the Sharpe ratio has increased substantially (though part of this increase can be attributed to the downward trend of the risk free rate).
4.2.1 Sensitivity to sample size
Statistic/estimate VW EW PPP in sample PPP out of sample ˆ θme 0 0 1.4403 2.569 std. error - - (0.9567) -ˆ θbtm 0 0 5.8840 4.7326 std. error - - (2.1952) -ˆ θmom 0 0 10.7045 6.8104 std. error - - (2.4300) -mean |wi| 0.0089 0.0089 0.0722 0.0490 mean max wi 0.0201 0.0096 0.2937 0.2049 mean min wi 0.0016 0.0096 -0.3531 -0.2627 P wi1{wi < 0} 0.0000 0 -3.5465 -2.5746 P1{w i < 0}/Nt 0.0002 0 0.4584 0.3865 P |wit− wi,t−1| 0.0245 0.01502 2.6636 2.7100 CE 0.0034 0.0023 0.0370 0.0103 ¯ rp 0.0069 0.0057 0.0755 0.1073 sd(rp) 0.0363 0.0362 0.1314 0.2284 SR -1.5133 -1.547 0.0385 0.3305
Table 2: Results for the base case, where the benchmark model for the parametric policy is the equally weighted portfolio, the sample size is ten years and the relative risk aversion of the investor is λ = 5.
models. The objective of this increase in sample size is to investigate the difference in the in sample performance compared to the default sample size of 10 years. Note that in sample performance is of meager importance to the objectives of the present study, but the relevance of the sample size is of practical importance in relation to the out of sample results.
Statistic/estimate VW EW PPP (VW) PPP (EW) ˆ θme 0 0 1.5555 1.9559 std. error - - (0.3890) (0.3890) ˆ θbtm 0 0 4.5461 4.5391 std. error - - (1.8879) (1.8879) ˆ θmom 0 0 6.3949 6.3978 std. error - - (0.9000) (0.9000) mean |wi| 0.0077 0.0077 0.0409 0.0409 mean max wi 0.0171 0.0082 0.1752 0.1752 mean min wi 0.0013 0.0082 -0.2183 -0.2182 P wi1{wi < 0} -0.0001 0 -2.1559 -2.1569 P1{w i < 0}/Nt 0.0006 0 0.4068 0.4085 P |wit− wi,t−1| 0.0256 0.0165 1.5511 1.5521 CE -0.0089 -0.0108 0.0345 0.0345 ¯ rp -0.0023 -0.0039 0.0725 0.0725 sd(rp) 0.0475 0.0483 0.1362 0.1363 SR -1.1910 -1.2048 0.1096 0.1100
Table 3: In sample results for the base case, where results of the parametric portfolio with both the equally weighted and the value weighted portfolio are presented, with a sample size of 20 years and the relative risk aversion of the investor is λ = 5.
using a sample size of ten years to the parameter estimates using a sample size of 20 years suggests that sample size is irrelevant to the estimation results.
The weight statistics do not support this statement. The weight statistics reveal that increasing the dataset has mildened the allocation strategy. Also, note that the choice of benchmark model in (2) is irrelevant when looking at the weight and performance statistics in table 3, the differences are negligible. The mildened allo-cation is firstly reflected in the mean absolute weight, this is 0.0409 when using a sample of 20 years and 0.0722 when using a sample of 10 years. Also notice that the mean maximum allocation has decreased from 0.2937 to 0.1752 and the mean minimum selection has increased from -0.3531 to -0.2183. Moreover, the average aggregate short position and the average turnover have decreased substantially, the average aggregate short position from approximately -3.55 to approximately -2.16 and the average turnover from approximately 2.66 to approximately 1.55.
es-timates and consequently different weights. The real statistics of interest are the performance statistics and these are barely different. The differences in certainty equivalent return, average portfolio return and volatility of the portfolio returns are negligible. The only difference in performance is the Sharpe ratio, which is around 0.11 in the sample of 20 years and around 0.04 in the sample of 10 years. Again, this can be attributed to the fact that the Sharpe ratio is computed using the risk free rate, which is downward trending. Consequently, by increasing the sample size the risk free rate will be lower on average. Therefore, it is safe to conclude that the in sample performance of the parametric portfolios is robust to sample size, which justifies the use of the default sample size of 10 years in the subsequent sub-sections. Also, it is once more observed that the choice of benchmark in (2) is irrelevant. Consequently, all specifications in the subsequent subsections use the value weighted portfolio as the benchmark. Finally, the benchmark portfolios per-form worse in the sample of 20 years, hence it is also safe to conclude here that the parametric portfolio policy outperforms the benchmark portfolios in sample.
4.2.2 Sensitivity to risk aversion
PPP in sample Statistic/ estimate λ = 2 λ = 5 λ = 10 λ = 20 λ = 100 ˆ θme 4.3691 1.0454 -0.0900 -0.7871 -2.4017 std. error (6.7824) (0.9077) (0.3301) (0.1781) (0.6524) ˆ θbtm 15.4042 5.8909 2.6394 1.0264 -1.3947 std. error (25.0913) (2.3920) (0.6367) (0.3990) (1.6191) ˆ θmom 26.2231 10.7047 5.2385 2.5932 0.7368 std. error (22.0546) (2.6359) (0.6074) (0.2654) (0.5179) mean |wi| 0.1785 0.0722 0.0354 0.0184 0.0163 mean max wi 0.7032 0.2937 0.1495 0.0808 0.0580 mean min wi -0.9022 -0.3533 -0.1635 -0.0722 -0.0519 P wi1{wi < 0} -9.5000 -3.5471 -1.4819 -0.5333 -0.4127 P1{w i < 0}/Nt 0.4835 0.4584 0.4142 0.3278 0.2687 P |wit− wi,t−1| 6.5245 2.6636 1.3027 0.644 0.2420 CE 0.0922 0.0371 0.0164 0.0008 -0.0516 ¯ rp 0.1783 0.0755 0.0393 0.0214 0.0081 sd(rp) 0.3225 0.1315 0.0681 0.0431 0.0397 SR 0.3332 0.01387 -0.4318 -1.0113 -1.3716
Table 4: In sample results for the PPP model, where the benchmark model used is the value weighted portfolio, with sample size set to 10 years for different levels of risk aversion λ.
ob-served for the investor with relative risk aversion λ = 2 are rather extreme. Again, they should be taken with a grain of salt, as the parameter estimates are extremely noisy. The mean maximum and mean minimum weights are substantially larger in magnitude than those of the other investors. The same can be said about the average aggregate short position and the average turnover.
The in sample performance results are not promising for the more risk averse in-vestors. The certainty equivalent returns observed are lower for investors with a high level of relative risk aversion and it is even negative for an investor with a relative risk aversion of λ = 100. The same is observed with the average portfolio returns, which goes from 0.1783 corresponding to λ = 2 to 0.0081 corresponding to λ = 100. Not surprisingly, the standard deviation is also negatively related to the level of risk aversion, which is consistent with the theory that return is positively associated with risk (i.e. one can generate more return by taking on extra risk). The observed Sharpe ratios are particularly worrisome, the only positive Sharpe ratios are those of the investor with relative risk aversion λ = 2 and relative risk aversion λ = 5. The Sharpe ratios of the other investors are all negative and get worse for investors with higher levels of relative risk aversion. This means that more risk averse investors do not profit on average from the risk they take by using the parametric portfolio policy instead of investing in the risk free rate, which entails they are better off by investing in the risk free rate. In fact, the only investors that appear to benefit from the parametric portfolio policy are the most risk tolerant investor, the investors with relative risk aversion λ = 2 and λ = 5.
The out of sample results for the PPP model, where the benchmark model in pa-rameterisation (2) is the value weighted portfolio, with different levels of relative risk aversion are presented in table 5. The out of sample parameter estimates tell the same story as the in sample parameter estimates, the value of the parameter estimates are negatively related to the level of relative risk aversion. The differences are just of a lower magnitude compared to the in sample results. E.g. the momen-tum factor has an out of sample average coefficient estimate of 14.6389 for a relative risk aversion of λ = 2, compared to an in sample coefficient estimate of 26.2231, and decreases to an average coefficient estimate of 1.1691 for λ = 100, compared to a value of 0.7368 in sample. Furthermore, the coefficient estimate of the market equity characteristic does not go as far below zero for the investor with relative risk aversion λ = 100 out of sample, as it does in sample. Also, the coefficient estimate for the book to market ratio is positive at all times.
PPP out of sample Statistic/ estimate λ = 2 λ = 5 λ = 10 λ = 20 λ = 100 ˆ θme 6.4751 2.1925 0.6382 -0.2281 -0.7062 std. error - - - - -ˆ θbtm 11.0511 4.7527 2.4967 1.4662 0.9391 std. error - - - - -ˆ θmom 14.6389 6.7656 3.8422 2.3982 1.1691 std. error - - - - -mean |wi| 0.1108 0.0489 0.0270 0.0170 0.0116 mean max wi 0.4418 0.2039 0.1181 0.0780 0.0493 mean min wi -0.6061 -0.2625 -0.1369 -.0770 -0.0367 P wi1{wi < 0} -6.4462 -2.6274 -1.1961 -0.5672 -0.2333 P1{w i < 0}/Nt 0.4210 0.3859 0.3389 0.2770 0.2043 P |wit− wi,t−1| 3.7465 1.6994 0.9494 0.5836 0.3061 CE 0.1204 0.011 -0.0034 -0.00381 -0.2033 ¯ rp 0.2618 0.1068 0.0505 0.0220 -0.0021 sd(rp) 0.5581 0.2276 0.1139 0.0669 0.05620 SR 0.4128 0.33 0.1608 -0.1484 -0.5801
Table 5: Out of sample results for the PPP model, where the benchmark model used is the value weighted portfolio, with sample size set to 10 years for different levels of risk aversion λ.
with a relative risk aversion of λ = 2, the mean absolute weight, mean minimum weight, mean maximum weight and average turnover are considerably lower in ab-solute value compared to the in sample results. Moreover, the average aggregate short position has decreased substantially. For the other investors the difference is less grave, but still the same trend is observed.
relative risk aversion of λ = 10 now has a positive Sharpe ratio. Furthermore, the out of sample results dominate the in sample results in terms of average portfolio return, except for the most risk averse investor.
To conclude, the level of risk aversion has a clear effect on the estimation and performance results, both in sample and out of sample. Clearly, the level of risk aversion is reflected in the prudent investment allocation for the more risk averse investor. This can be deduced from the parameter estimates, but even more so the weight statistics. Performance wise it can be concluded that performance is nega-tively related to the level of relative risk aversion. In fact, it can be concluded that relatively risk averse investors are not well off using the PPP investment strategy
14.
4.3
Extensions
In this subsection, the results of the extensions of the base case, as proposed in subsection 2.1, are presented. The first extension is the imposition of weight re-strictions, specifically the restriction that investors can not assume short positions. This entails that the weights are truncated at zero (see equation (8)) and that the estimator of θ in (2) is given by (8). The second extension is the imposition of pro-portional transaction costs, which functional form depends negatively on a firm’s size. The estimator of θ in (2), that arises from this augmentation, is given by (10). Both the in sample and out of sample results of the extensions are presented in table 6.
4.3.1 Weight restrictions
The results for the PPP model, where the benchmark model in (2) is the value weighted portfolio, with long only restrictions imposed are presented in the first two columns of table 6. Compared to the results in table 1, the parameter esti-mates are much lower in magnitude, which is observed both in sample and out of sample. This is not surprising, the long only restriction does not permit big de-viations from the benchmark weights, as this could result in a short position in case of a negative deviation. Moreover, the variability of the parameter estimates is relatively low, which suggests that the in sample parameters are highly significant.
An interesting thing to note is that the long only mean absolute weight is equal to those of the benchmark models in table 1. This is because, like the value weighted portfolio, the long only portfolio divides all wealth over the available assets and consequently, the average weight will be that of the equally weighted portfolio. Note that the average aggregate short position and the average fraction of negative weights are now zero, as required. Also, the trading activity is substantially lower, 0.0795 on average in sample and 0.0402 on average out of sample, compared to
Statistic/estimate long only transaction costs in sample out of sample in sample out of sample ˆ θme -0.4031 -0.1841 1.1621 1.1562 std. error (0.0274) - (0.0000) -ˆ θbtm 0.0091 0.0126 4.4573 4.4584 std. error (0.0092) - (0.0000) -ˆ θmom 0.2711 0.0699 8.3322 8.3356 std. error (0.0045) - (0.0000) -mean |wi| 0.0089 0.0080 0.0569 0.0500 mean max wi 0.0174 0.0135 0.2307 0.2283 mean min wi 0.0031 0.0027 -0.2633 -0.2885 P wi1{wi < 0} 0 0 -2.6904 -2.6384 P1{w i < 0}/Nt 0 0 0.4486 0.3985 P |wit− wi,t−1| 0.0795 0.0402 2.0844 1.9535 CE 0.0044 -0.0189 0.0252 -0.0870 ¯ rp 0.0077 -0.0099 0.0513 0.1016 sd(rp) 0.0356 0.0543 0.1069 0.2361 SR -1.5164 -0.7225 -0.1738 0.2942
Table 6: Results for the extensions of the PPP model, where the benchmark weight is the value weighted portfolio, with relative risk aversion λ = 5 and a sample size of 10 years.
2.6636 in sample and 1.6994 out of sample for the base case.
All in all, adding the long only restriction unambiguously mildens the investor’s allocation behaviour. The in sample performance is reasonable compared to the benchmark models presented in table 1, but has deteriorated substantially com-pared to the base case. The out of sample performance of the long only portfolio is unambiguously poor, the certainty equivalent return, the average portfolio return and the Sharpe ratio are all negative, contrary to the base case, where all these measures are positive.
4.3.2 Transaction costs
The results for the PPP model, where the benchmark model in (2) is the value weighted portfolio, with transaction costs are presented in the last two columns of table 6. Compared to the parameter estimates of the base case presented in table 1, in sample parameter estimates of the PPP model with transaction costs are slightly lower in magnitude, the converse is true for the out of sample parameter estimates. This entails that the investor is less inclined to deviate from the value weighted portfolio for high (low) values of the characteristics, in sample. The converse is true out of sample. The former should make intuitive sense, since the introduced transaction costs should ‘keep the investor in check’. The latter statement does not make sense intuitively, however no hard conclusions should be drawn from the parameter estimates. Another interesting observation is the nearly zero variability of the parameter estimates. The estimator given by (10) picked nearly (not exactly) the same parameter estimates for the different sampled datasets. In fact, even the differences between the in sample and out of sample parameters are negligible.
between taking a certain return and the parametric policy.
5
Conclusion
In the present study, the parametric portfolio policy (PPP) model for equity portfo-lios, proposed by Brandt et al. [2009] [3], is tested in the Dutch stock market. The model parameterises portfolio weights as a function of observed firm characteristics. In the present case, weights were parameterised as an affine function of the firm’s market equity, book to market ratio and momentum. The model was tested in sam-ple, out of sample and was compared to two benchmark models, the value weighted and the equally weighted portfolio. The emphasis in the present study was on the out of sample performance of the model, due to its practical relevance. A sensitivity analysis was conducted where the sensitivity to sample size and the sensitivity to the level of relative risk aversion were tested. Further extensions to the base case were added, in order to make the model more realistic. These extensions were a restriction on short selling and the addition of transaction costs. Ultimately, the results answer the following research question.
How well does the PPP model perform in the Dutch stock market,in general and compared to several benchmark models?
The findings of the base case were promising, the in sample performance measures dominated those of the benchmark models. The out of sample results were worse in terms of certainty equivalent return, compared to the in sample results, but were better in terms of average portfolio return and the Sharpe ratio. The exhibited investing behaviour stressed the importance of extending the base case model, high turnovers and high aggregate short positions were observed. Adding short selling restrictions certainly mildens the allocations and turnover. However, the in sam-ple performance results were substantially worse, compared to the base case and are similar to the performance of the benchmark models. Out of sample perfor-mance was poor in general in terms of all perforperfor-mance measures provided. Adding transaction action costs desirably lowered the turnover, in sample. The in sample performance was slightly worse compared to the base case, but still better than the benchmark models. A negative Sharpe ratio revealed that it was more lucrative to invest in the risk free rate, though. The out of sample turnover was higher than that of the base case, contrary to expectation. The performance was similar to the base case, based on average portfolio return and the Sharpe ratio, but was worse in terms of certainty equivalent return. In fact, the certainty equivalent return was negative, which entails that the investor is indifferent between a certain loss and applying the parametric portfolio policy.
for higher levels of risk aversion. Consequently, the PPP model is not suitable for relatively risk averse investors.
The answers to the research question are as follows. Firstly, the base case version of the PPP model performs well, in general and compared to the value weighted and the equally weighted portfolio. The general performance results are positive both in and out of sample. However, adding the realistic assumptions of short sale restric-tions and long only restricrestric-tions produced undesirable performance results where the investor was better off not using the parametric portfolio policy. Consequently, due to the practical relevance of the extensions, the parametric portfolio policy does not work well and based on the provided empirical evidence, it is not advisable to use the model in practice in the Dutch stock market.
The contribution of the present study is an mainly empirical one. Despite that Brandt et al. [2009] [3] proposed the model more than eight years ago, the empir-ical evidence on the model is scarce. In fact, the only markets in which the model has been tested empirically (to my knowledge) are the US market (Brandt et al. [2009] [3]) and the Brazilian market (Medeiros et al. [2014] [35]), both of which are different from the Dutch market (e.g. in size). Both Brandt et al. and Medeiros et al. conclude positively in their works by highlighting the efficacy and adequacy of the model, whereas I remain unconvinced.
6
References
[1] Barberis, N., 2000. Investing for the long run when returns are predictable. The Journal of Finance, 55(1), pp.225-264.
[2] Brandt, M.W. and Santa-Clara, P., 2006. Dynamic portfolio selection by aug-menting the asset space. The Journal of Finance, 61(5), pp.2187-2217.
[3] Brandt, M.W., Santa-Clara, P. and Valkanov, R., 2009. Parametric portfolio policies: Exploiting characteristics in the cross-section of equity returns. The Review of Financial Studies, 22(9), pp.3411-3447.
[4] Brandt, M.W., 2009. Portfolio choice problems. Handbook of financial econo-metrics, 1, pp.269-336.
[5] Carhart, M.M., 1997. On persistence in mutual fund performance. The Journal of finance, 52(1), pp.57-82.
[6] C¸ elik, 2012. Theoretical and Empirical Review of Asset Pricing Models: A
Struc-tural Synthesis. International Journal Of Economics & Financial Issues (IJEFI), 2(2), pp. 141-178.
[7] Chan, L.K., Karceski, J. and Lakonishok, J., 1999. On portfolio optimization: Forecasting covariances and choosing the risk model. The Review of Financial Studies, 12(5), pp.937-974.
[8] Chen, L., Novy-Marx, R. and Zhang, L., 2011. An alternative three-factor model [9] Copas, J.B., 1983. Regression, prediction and shrinkage. Journal of the Royal
Statistical Society. Series B (Methodological), pp.311-354.
[10] Davis, M.H. and Norman, A.R., 1990. Portfolio selection with transaction costs. Mathematics of operations research, 15(4), pp.676-713.
[11] Dittmar, R.F., 2002. Nonlinear pricing kernels, kurtosis preference, and evi-dence from the cross section of equity returns. The Journal of Finance, 57(1), pp.369-403.
[12] Elton, E.J. and Gruber, M.J., 1974. Portfolio theory when investment relatives are lognormally distributed. The Journal of Finance, 29(4), pp.1265-1273. [13] Ergemen, Y.E. and Taamouti, A., 2015. Parametric Portfolio Policies with
Common Volatility Dynamics (No. 2015-41). Department of Economics and Business Economics, Aarhus University.
[14] Fama, E.F., 1965. Portfolio analysis in a stable Paretian market. Management science, 11(3), pp.404-419.
[15] Fama, E.F., 1970. Multiperiod consumption-investment decisions. The Ameri-can Economic Review, pp.163-174.
[16] Fama, E.F. and French, K.R., 1993. Common risk factors in the returns on stocks and bonds. Journal of financial economics, 33(1), pp.3-56.
[18] Fama, E.F. and French, K.R., 2004. The capital asset pricing model: Theory and evidence. The Journal of Economic Perspectives, 18(3), pp.25-46.
[19] Frost, P.A. and Savarino, J.E., 1986. An empirical Bayes approach to effi-cient portfolio selection. Journal of Financial and Quantitative Analysis, 21(3), pp.293-305.
[20] French, C.W., 2003. The Treynor capital asset pricing model. Journal of In-vestment Management, 1(2), pp.60-72.
[21] Frost, P.A. and Savarino, J.E., 1988. For better performance: Constrain port-folio weights. The Journal of Portport-folio Management, 15(1), pp.29-34.
[22] Hansen, L.P., 1982. Large sample properties of generalized method of moments estimators. Econometrica: Journal of the Econometric Society, pp.1029-1054. [23] He, H. and Pearson, N.D., 1991. Consumption and portfolio policies with
incomplete markets and short-sale constraints: The infinite dimensional case. Journal of Economic Theory, 54(2), pp.259-304.
[24] James, W. and Stein, C., 1961, June. Estimation with quadratic loss. In Pro-ceedings of the fourth Berkeley symposium on mathematical statistics and prob-ability (Vol. 1, No. 1961, pp. 361-379).
[25] Jobson, J.D. and Korkie, B., 1980. Estimation for Markowitz efficient portfo-lios. Journal of the American Statistical Association, 75(371), pp.544-554. [26] Jorion, P., 1986. Bayes-Stein estimation for portfolio analysis. Journal of
Fi-nancial and Quantitative Analysis, 21(3), pp.279-292.
[27] Hand, J.R. and Green, J., 2011. The importance of accounting information in portfolio optimization. Journal of Accounting, Auditing & Finance, 26(1), pp.1-34.
[28] Hjalmarsson, E. and Manchev, P., 2012. Characteristic-based mean-variance portfolio choice. Journal of Banking & Finance, 36(5), pp.1392-1401.
[29] Kane, A., 1982. Skewness preference and portfolio choice. Journal of Financial and Quantitative Analysis, 17(1), pp.15-25.
[30] Kraus, A. and Litzenberger, R.H., 1976. Skewness preference and the valuation of risk assets. The Journal of Finance, 31(4), pp.1085-1100.
[31] Lintner, J., 1965. The valuation of risk assets and the selection of risky in-vestments in stock portfolios and capital budgets. The review of economics and statistics, pp.13-37.
[32] Lintner, J., 1965. Security prices, risk, and maximal gains from diversification. The journal of finance, 20(4), pp.587-615.
[33] Markowitz, H., 1952. Portfolio selection. The journal of finance, 7(1), pp.77-91. [34] Markowitz, H., 1959. Portfolio selection: Efficient diversification of
[35] C Medeiros, M., M Passos, A. and FR Vasconcelos, G., 2014. Parametric portfolio selection: Evaluating and comparing to Markowitz portfolios. Revista
Brasileira de Finan¸cas, 12(2).
[36] Merton, R.C., 1969. Lifetime portfolio selection under uncertainty: The
continuous-time case. The review of Economics and Statistics, pp.247-257. [37] Merton, R.C., 1971. Optimum consumption and portfolio rules in a
continuous-time model. Journal of economic theory, 3(4), pp.373-413.
[38] Michaud, R.O., 1989. The Markowitz optimization enigma: Is ?optimized? optimal?. Financial Analysts Journal, 45(1), pp.31-42.
[39] Mossin, J., 1966. Equilibrium in a capital asset market. Econometrica: Journal of the econometric society, pp.768-783.
[40] Roll, R., 1977. A critique of the asset pricing theory’s tests Part I: On past and potential testability of the theory. Journal of financial economics, 4(2), pp.129-176.
[41] Samuelson, P.A., 1969. Lifetime portfolio selection by dynamic stochastic pro-gramming. The review of economics and statistics, pp.239-246.
[42] Sharpe, W.F., 1964. Capital asset prices: A theory of market equilibrium under conditions of risk. The journal of finance, 19(3), pp.425-442
[43] Sortino, F.A. and Price, L.N., 1994. Performance measurement in a downside risk framework. the Journal of Investing, 3(3), pp.59-64.
[44] Stein, C., 1956, January. Inadmissibility of the usual estimator for the mean of a multivariate normal distribution. In Proceedings of the Third Berkeley sympo-sium on mathematical statistics and probability (Vol. 1, No. 399, pp. 197-206). [45] Tobin, J., 1965. ”The Theory of Portfolio Selection,” The Theory of Interest
Rates, F. H. Hahn and F. P. R. Brechling (eds.) (London: Macmillan). [46] Treynor, J.L., 1961. Market value, time, and risk.
[47] Treynor, J.L., 1961. Toward a theory of market value of risky assets. Unpub-lished manuscript, 6.
A
Appendix: programming code (R)
# Packages required: # e1071 # lubridate # magic # ks # scales # tseries # foreach # doParallel # iterators## CAUTION: BOOTSTRAP ROUTINES ARE TIME CONSUMING!
################ ##### FUNCTIONS# ################
sumVar = function(data){
characteristics = unique(data$characteristic)
colnames=c("characteristic","mean", "median", "standard deviation",
"variance" ,"skewness", "kurtosis" , "min",
"max", "0.5% percentile", "99.5% percentile", "observations")
table = matrix(NA, nrow = length(characteristics), ncol = length(colnames))
library(e1071)
for (i in 1:length(characteristics)) {
index = which(data$characteristic == characteristics[i]) assign(paste0("frame ", characteristics[i]), data[index,]) useData = data[index, ]
delete = NULL
for (j in 1:ncol(data)){
if (class(data[,j]) != "numeric"){ delete = cbind(delete, j)} }
useData = useData[, -delete]
useData = as.numeric(unlist(useData))
rowValues = c(mean(useData, na.rm = T), median(useData, na.rm = T),
sd(useData, na.rm = T), var(useData, na.rm = T),
skewness(useData, na.rm = T), kurtosis(useData, na.rm = T),
min(useData, na.rm = T), max(useData, na.rm = T),
quantile(useData, c(0.005, 0.995), na.rm = T),
sum(!is.na(useData)))
newRow = c(characteristics[i], rowValues)
table[i,] = newRow }
table = as.data.frame(table)
names(table) = colnames return(table)
}
addColumnRight = function(place, data){ newCol = rep(NA, nrow(data))
addCol = cbind(data[,1:place], newCol, data[,(place+1):ncol(data)])
return(addCol) }
addRowUnder = function(place, data){ newRow = rep(NA, ncol(data))
addRow = rbind(data[1:place, ], newRow, data[(place+1):nrow(data),])
}
addNewCharacteristic = function(data, type){ vars = unique(data$characteristic)
nrow = nrow(data)
ncol = ncol(data) dates = unique(data$date)
firms = unique(names(data[,-c(1,2)])) output = data
for (k in 1:length(dates)){
rowNum = which(output$date == dates[k] & output$characteristic == vars[length(vars)])
if (rowNum < nrow(output)){
output = addRowUnder(rowNum, output) output$date[rowNum+1] = dates[k] output$characteristic[rowNum+1] = type
print(rowNum) } else {
output = rbind(output, rep(NA, ncol(output))) output$date[rowNum+1] = dates[k]
output$characteristic[rowNum+1] = type
print("Final row") }
}
vars = unique(output$characteristic)
rowNumbers = which(output$characteristic == type)
if (type == "return"){
prices = output[which(output$characteristic == "price"), -c(1,2)] output[rowNumbers[1], -c(1,2)] = rep(NA, ncol(prices))
for (i in 2:nrow(prices)){
return = (prices[i,]-prices[i-1, ])/prices[i,] output[rowNumbers[i], -c(1,2)] = return
}
} else if (type == "market equity"){
marketValues = output[which(output$characteristic == "market value"), -c(1,2)] marketValues = replace(marketValues, marketValues == 0, NA)
for (i in 1:nrow(marketValues)){ marketEquity = log(marketValues[i,])
output[rowNumbers[i], -c(1,2)] = marketEquity }
} else if (type == "book to market"){
library(lubridate)
totAssets = output[which(output$characteristic == "total assets"), -c(1,2)]
totLiabilities = output[which(output$characteristic == "total liabilities"), -c(1,2)] defTax = output[which(output$characteristic == "deferred income taxes & invest"), -c
(1,2)]
prefStock = output[which(output$characteristic == "preferred stock"), -c(1,2)] marketValues = output[which(output$characteristic == "market value"), -c(1,2)] marketValues = replace(marketValues, marketValues == 0, NA)
defTax[is.na(defTax)] = 0 prefStock[is.na(prefStock)] = 0
bookEqRaw = totAssets-totLiabilities + defTax - prefStock bookEqRaw = replace(bookEqRaw, bookEqRaw < 0 , NA)
lagamounts = c(13, 14, 15, 16, 17, 18, 7, 8, 9, 10, 11, 12)
for (i in 1:nrow(bookEqRaw)){
output[rowNumbers[i], -c(1,2)] = rep(NA, ncol(bookEqRaw)) } else {
currentMonth = month(as.Date(output$date[rowNumbers[i]])) lag = lagamounts[match(currentMonth, 1:12)]
bookEquity = bookEqRaw[(i-lag),] marketEquity = marketValues[i, ]
output[rowNumbers[i], -c(1,2)] = log(1 + bookEquity/marketEquity) }
}
} else if (type == "momentum"){
returns = output[which(output$characteristic == "return"), -c(1,2) ] returns = 1 + returns
for (i in 1:nrow(returns)){
if (i < 13){
output[rowNumbers[i], -c(1,2)] = rep(NA, ncol(returns)) } else {
returnSnap = returns[(i-12):i,] momentum = apply(returnSnap, 2, prod)-1 output[rowNumbers[i], -c(1,2)] = momentum } } } return(output) }
windsorise= function(data, type, quantiles, replace){ rowNumbers = which(data$characteristic == type) observations = data[rowNumbers, -c(1,2)]
cutoff = quantile(unlist(observations), quantiles, na.rm = T)
for (j in 3:ncol(data)){
for (i in rowNumbers){
if (!is.na(data[i,j])){
if (data[i,j] < cutoff[1] || data[i,j] > cutoff [2]){
data[i,j] = replace
} } } } return(data) }
eliminateFirms = function(data){
characteristics = unique(data$characteristic) colNumbers = NULL
for (i in 1:length(characteristics)){
subset = data[which(data$characteristic == characteristics[i]),]
for (j in 3:ncol(subset)){
if (sum(is.na(subset[,j])) ==nrow(subset)){ colNumbers = c(colNumbers, j)
} } }
colNumbers = unique(colNumbers) colNumbers = sort(colNumbers)
return(data[,-colNumbers])
}
rowNums = NULL
for (i in 1:nrow(data)){
if (sum(is.na(data[i, ])) == ncol(data)-2){ rowNums = c(rowNums, i)
} }
rowNums = unique(rowNums)
dates = unique(data$date[rowNums])
delete = which(data$date %in% dates)
return(data[-delete, ]) }
countFirms = function(data){ dates = unique(data$date) totalFirms = NULL
for (i in seq_along(dates)){ currentDate = dates[i]
subset = data[which(data$date == currentDate), -c(1,2)] firmsAvaiable = 0
for (j in 1:ncol(subset)){
if (sum(is.na(subset[,j])) == 0){firmsAvaiable = firmsAvaiable + 1} }
totalFirms = c(totalFirms, firmsAvaiable) }
return(totalFirms) }
removeUseless = function(data){ dates = unique(data$date)
for (i in seq_along(dates)){ currenDate = dates[i]
rowNumbers = which(data$date == currenDate)
for (j in 3:ncol(data)){
subset = data[rowNumbers, j]
if (sum(is.na(subset)) > 0){
data[rowNumbers, j] = rep(NA,length(rowNumbers)) }
} }
return(data) }
normaliseData = function(data, type){
rowNumbers = which(data$characteristic == type)
for (i in rowNumbers){
mean = mean(as.numeric(data[i,-c(1,2)]), na.rm = T)
sd = sd(as.numeric(data[i, -c(1,2)]), na.rm = T)
data[i, -c(1,2)] = (data[i, -c(1,2)] - mean)/sd
}
return(data) }
powerUtility =function(lambda, wealth){
return(((1+wealth)^(1-lambda))/(1-lambda)) }
constructOptim = function(data, lambda, method){
function(theta){
