Efficiency Implementation in the Construction of Active Portfolios: An Assessment through the Fundamental Law of Active Management
FREDY ALEXANDER PULGA VIVAS*
ABSTRACT
Active portfolio management aims to deliver superior returns through using an extensive analysis of securities in order to identify mispriced stocks and estimate alphas. Moreover, an active strategy relies on the capability of the active manager to transform his/her forecasting skills into an active portfolio. This thesis assesses the effects of investment constraints in the ex ante capabilities of an active manager to construct portfolios through the fundamental law of active management, and provides evidence on the efficiency loss of the manager in presence of such constraints through a Monte Carlo simulation.
Portfolio management theory debates between two main approaches to deliver returns to investors. Investment strategies can be identified as either passive or active, in line with the investor’s beliefs on the degree of efficiency in the financial markets and on how securities are priced.
The efficient market hypothesis introduced by Fama (1970) serves as a baseline to distinguish a passive from an active portfolio strategy. The broader definition of market efficiency asserts that security prices incorporate all available information. Under the main assumption of perfect competition, a market is efficient in the strong sense, when a large number of rational investors, with total access to information, compete with each other by forecasting future security returns and when the frictionless interaction between supply and demand allows for asset prices to fully reflect such information instantaneously.
In a completely efficient market there is no room for mispriced securities, thus there is no point for an investor to try achieving superior performance. In this line of
thinking arise the passive investment strategies, where the manager invests in the market, which is normally defined by the collection of a representative number of securities into an index, i.e. the benchmark.
In contrast, active investing aims to deliver returns above the benchmark by exploiting the inefficiencies in the market. Such malfunctioning mainly arises from the incompleteness of information, among other factors; therefore the theory underlying active portfolio management assumes that security prices do not necessarily reflect all available information.
According to the theoretical and empirical work on efficient capital markets by Fama (1970), security prices reflect all past and public available information under the weak and semi strong form of market efficiency, thus investors do not have access to the inside information. Once considered a lower degree of efficiency in the market (weak and semi strong), active management becomes operational through price estimates for the set of securities within the market. The underlying assumption of any active strategy is that market prices might converge to these estimated values. As a consequence, the purpose of active investing is to identify under and overvalued securities and exploit such mispricing to deliver exceptional returns. Consequently, an active strategy completely relies on the quality of the information gathered for the purpose of forecasting future prices and returns to outperform the market.
A lot of attention has been given to the identification and estimation of mispriced securities. Nevertheless, the success of an active strategy depends on how these estimates are efficiently transformed into active bets on the over and under valued securities in order to construct portfolios with superior performance.
portfolios depends on these restrictions. As a result, the performance of active portfolio management is sensitive to the information that is used to forecast returns and to the capability of the manager to transform these forecasts into a portfolio.
Through “The Fundamental Law of Active Management” (Grinold (1989)) and “The Generalized Law” (Clarke et al. (2002)), one can assess the quality of the information about the forecasts as well as the competitive advantage of a manager to transform such information into active portfolios. The generalized law (Clarke et al. (2002)) posits an ex ante relationship between the ratio of the forecasted abnormal returns to the risk for bearing them (the information ratio), and (i) the skills of active managers to forecast abnormal returns (information coefficient), (ii) their ability to transform these skills into active portfolios (transfer coefficient) and (iii) the number of independent bets on the forecasts (breadth). Under this framework, it is possible to evaluate efficiency implementation (Clarke et al. (2005)), which is the ability of the manager to transform forecasts on abnormal returns into active bets in a portfolio.
In this thesis we intend to investigate whether investment constraints, such as transaction costs, the prohibition of short sales and investment style, diminish the ex ante efficiency of the active portfolio manager. In order to assess the impact of these restrictions, we apply a Monte Carlo simulation to generate random forecasts on abnormal returns and to construct unconstrained and constrained active portfolios. Moreover, we perform these simulations for different levels of active risk in order to evaluate its impact on efficiency under each constraint. As a result, we calculate transfer coefficients, expected returns and information ratios for every case study based on the generalized law.
on unconstrained and constrained active portfolios in terms of efficiency implementation, and the final section concludes.
I. Literature Review
The accuracy of forecasts on abnormal returns plays a major role to enhance performance and to add value in active portfolios. For this reason, a lot of attention has been given to the identification of mispriced securities and the estimation of returns. Nevertheless, it is also important how these forecasts are efficiently transformed into active portfolios.
“The fundamental law of active management” (Grinold (1989)) focuses on the quality of the information and the skill of the manager to forecast abnormal returns to construct active portfolios and add value. Similarly, “the generalized fundamental law” (Clarke et al. (2002)) adds to the aforementioned analysis, the capabilities of the active manager to transform the forecasts on abnormal returns into active portfolios. Moreover, the generalized law provides a framework to assess the effects of imposing constraints to the manager on such capabilities.
A. The fundamental law of active management
One of the main contributions to the finance literature on portfolio management is “The fundamental law of active management”. Formulated by Grinold (1989), the fundamental law postulates an ex ante relationship to assess the forecasting skills of an active manager to deliver exceptional returns and add value relative to a benchmark1.
Specifically, the law relates the expected performance of the manager measured by the ex ante information ratio, 𝐼𝑅, to his/her ability to forecast abnormal returns, as gauged through the ex ante information coefficient, 𝐼𝐶, and the number of independent bets that he/she makes on these forecasts or the breadth, 𝐵𝑅, of the investment strategy:
With respect to the ex ante information ratio of an active portfolio, 𝐼𝑅, Grinold (1989), Sharpe (1994) and Goodwin (1998) define it as the expected excess return of the active portfolio over the benchmark portfolio, per unit of volatility on excess return. Specifically, it is the ratio between the expected active return, 𝐸[𝑅!], to active risk2, 𝜎
!:
𝐼𝑅 =![!!]
!! (2)
The ex ante information ratio measures the quality of the proprietary information of the active manager, given the risk he/she faces to build active portfolios: the better his/her information, the higher the expected excess return for a given level of active risk. Therefore, the main objective of the active manager is to achieve the highest ex ante information ratio within an active portfolio.
Moreover, the fundamental law approximates and decomposes the expected information ratio into two separate components linked to the performance of the active manager. First, the ex ante information coefficient measures the accuracy of the manager to forecast abnormal returns3. Basically, it relates these forecasts to the
subsequent realized abnormal returns through a correlation coefficient. Grinold (1989), Grinold et al. (1992) and Grinold et al. (2000) consider the information coefficient as a measure of the skillfulness and accuracy of the manager to identify under or overpriced securities, to the extent that he/she is able to exploit such mispricing.
Second, the ex ante information ratio depends on the scope of the active investment strategy. Grinold (1989) asserts that breadth is the number of independent decisions made by the manager based on his/her forecasts4 within the active
(2007) use an econometric framework to interpret the fundamental law and conclude that breadth is equivalent to the number of explanatory variables used in a regression analysis to forecast abnormal returns, and Grinold et al. (2011) develop the fundamental law under a dynamic optimization model and find that breadth is the number of assets in the portfolio times the rate of information turnover, which is the equilibrium rate of arrival and declining of new and old information on the forecasts.
Besides the independence of the forecasts, the underlying assumptions of the fundamental law also relate to the skill of the manager and how he/she exploits his/her information for the investment process. With regard to the skillfulness of the manager, the law assumes that the information coefficient is an average; therefore it is constant between time and securities5. Additionally, the stronger assumption of the law is that
the active manager adequately assess his/her proprietary information and constructs active portfolios efficiently, thus he/she is able to completely transform his/her skills into active weights.
As an insightful tool of the investment process, the law affirms that an active manager has to be accurate in his/her forecasts and has to augment his/her investment opportunities as well, by analysing more assets or increasing the frequency of the forecasts, in order to improve his/her information ratio. Moreover, a modest skill is to be implemented more frequently among a large number of securities.
B. The active manager’s opportunity set and preferences: a perspective from the fundamental law
The active manager opportunity set is determined by the relationship between expected active return and active risk determined by the information ratio:
𝐸[𝑅!] = 𝐼𝑅 ∙ 𝜎! (3)
the manager has to bear more active risk6. Moreover, from equation 3, the information
ratio is the slope of the residual frontier, therefore, when the active manager increases it, he/she augments the scope of the investment opportunities in the return risk continuum.
Additionally, the fundamental law sets an important relationship between the skills of an active manager and the value that he/she adds throughout the investment process. The value added function, 𝑉𝐴, represents the trade off between the forecasts on abnormal returns and active risk:
𝑉𝐴 = 𝐸 𝑅! − 𝜆!𝜎!! (4)
Where the parameter 𝜆! is a measure of active risk aversion7. For higher levels of
active risk aversion, the valued added from active management is lower for a fixed amount of active risk and vice versa, as can be seen in figure 1. Thus, the goal of the active manager is to maximize the value added from expected active return and its inherent risk given his/her aversion.
The set of preferences of the active manager is derived from the value added function and plots all possible active portfolios with different combinations of expected active return and active risk for a constant magnitude of value added, as described in figure 2.
With the residual frontier and the set of preferences, the optimization problem of the active manager can be solved, as shown in figure 3: the maximum value added, 𝑉𝐴∗,
by the manager is proportional to the square of the information ratio of the active portfolio8, under a mean variance approach for constructing active portfolios:
𝑉𝐴∗ = !
!!!∙ 𝐼𝑅!
! (5)
Grinold (1989) acknowledges that the fundamental law is not intended to be an operational tool for practitioners, but to reveal the insights of value added by active portfolio managers through active investing. Actually, the original version of the law is formulated as an approximation thus it is not an equality. As a consequence of the aforementioned assumptions of the fundamental law, it sets the upper boundary for information ratios and value added (Grinold (1989), Goodwin (1998), Clarke et al. (2002)).
C. The fundamental law and the transfer coefficient
In an attempt to explain the differences between the information ratios estimated through the fundamental law and the observed information ratios of active portfolios, Clarke et al. (2002) introduce the ex ante concepts of transfer and performance9 coefficients and present “The generalized fundamental law”.
assess the transformation of the manager skills into the portfolio construction process, Clarke et al. (2002) define the ex ante transfer coefficient as the correlation between the forecasts on abnormal returns and the active weights10 of the securities that constitute
the active portfolio.
Under this framework, the transfer coefficient, 𝑇𝐶, becomes a scale factor for the information coefficient in the generalized fundamental law:
𝐼𝑅 ≈ 𝑇𝐶 ∙ 𝐼𝐶 ∙ 𝐵𝑅 (6)
In the original version of the fundamental law, it is assumed that there is a perfect correlation between the active weights and the forecasts on abnormal returns; therefore the active manager is capable of fully exploiting his/her abilities to forecast returns and transforming them into active weights. In the generalized version, there is room for inefficiencies, which arise from the constraints imposed to the active manager through the investment constraints11 of the managed portfolio. In presence of such
constraints, the transfer coefficient becomes lower than one thus the manager is not able to completely transform his/her skills into the active portfolio. Therefore, the information ratios and the value added of the portfolio manager become lower than those estimated from the original version of the fundamental law.
The generalized law adds a new component to active management in order to achieve outstanding portfolio performance. The information ratio improves by means of a better ability to construct an active portfolio, which nearly reflects the proper weights of the forecasts; an increase in the forecasting skill, and the greater the investment set.
efficiency changes in presence of different constraints when constructing active portfolios (Clarke et al. (2005)).
II. Methodology
We construct active portfolios using a mean variance optimization framework. The objective is to find efficient portfolios that maximize expected returns controlling for risk, which is measured by the standard deviation of expected returns. This quantitative method requires the estimation of a variance covariance matrix of stock returns, a vector of betas, a vector of expected returns and a vector of alphas or expected residual returns12 for the securities that constitute the index. Moreover, the
optimization model needs further assumptions on the risk free rate and the expected benchmark return13, which are also used as inputs.
A. Estimating the variance covariance matrix: the average constant correlation model
As a starting point, we calculate the elements in the historical variance covariance matrix from the daily logarithm returns of the stocks quoted in the benchmark14 as follows:
𝜎!" =!! ! 𝑅!"− 𝑅!
!!! 𝑅!"− 𝑅! (7)
Where the ijth element of the variance covariance matrix is the term 𝜎!" which denotes the covariance between securities i and j; 𝑅!" and 𝑅!" are their corresponding logarithm returns in time t; 𝑅! and 𝑅! are their time mean logarithm returns, and T is the
number of total observations. The elements in the diagonal of the historical variance covariance matrix represent the variance of each stock and the off diagonal elements are the covariance among securities.
Similarly, we compute the elements of the historical correlation matrix with the information of the historical variance covariance matrix, as defined by:
𝜌!" = !!"
In this case, the ijth element of the historical correlation matrix is defined by 𝜌!", which is the correlation coefficient between securities i and j. The off diagonal elements of the historical correlation matrix are the correlation coefficients, whereas a set of ones (the correlation between the logarithm returns of a given security and itself) constitutes the diagonal.
Afterwards, we estimate a variance covariance matrix by using the average constant correlation model (Elton et al. (2010)). First, we calculate the mean value of the correlation coefficients, 𝜌, of the off diagonal elements from the historical correlation matrix. Next, we construct the average correlation matrix in such way that the off diagonal elements are 𝜌, and the diagonal elements are ones. Using these estimated correlations, we calculate the ijth element of the estimated variance covariance matrix, which is the estimated covariance between securities i and j, as follows:
𝜎!" = 𝜌𝜎!𝜎! (9)
As before, the off diagonal elements are the estimated covariance between the securities, and the diagonal elements are the historical variances of each security.
B. Expected returns: a market model
Assume that the total excess return on a given security, 𝑟!, defined as the return on excess of security i over the risk free rate, is determined by two components: (i) a part, which is related to the benchmark, and (ii) a portion that does not depend on it:
𝑟! = 𝛽!𝑟!+ 𝜃! (10)
The first expression in equation (10) is the portion of the excess return of security i related to the benchmark, i.e. systematic. 𝛽! is a measure of the sensitivity of
the excess return of stock i due to changes in the benchmark excess return15, 𝑟
!. The
return is of the most importance in this model and in the forthcoming analysis since it defines the aim of active management.
From equation (10), the expected return on any stock i, 𝐸 𝑅! , is defined by:
𝐸 𝑅! = 𝑅!+ 𝛽! 𝐸 𝑅! − 𝑅! + 𝛼! (11)
Where 𝑅! is the risk free rate, 𝛽! is the beta of stock i, as previously defined, 𝐸 𝑅! is the expected return on the benchmark and 𝛼! is the expected residual return for security i.
Note that the first two expressions on the right hand of equation (11) refer to the standard version of the CAPM model (Sharpe (1966)), specifically to the Security Market Line, where the expected return on a given security is determined by the risk free rate added to its beta times the benchmark excess return. Additionally, the third term on the right hand of the market model represents a residual return for estimating stock returns: the so called alpha.
B.1. Constructing betas on a market model
As early noticed, we calculate the betas for each security in order to estimate returns. We retrieve these betas from historical data using regression analysis on the realized returns of the benchmark and its constituent securities. The regressed equation, consistent with the market model, is:
𝑟!" = 𝜃!"+ 𝛽!𝑟!" + 𝜖!" (12)
The expression 𝜃!" is the estimated historical residual return, 𝛽! is the estimated
historical beta, and 𝜖!" is the error term on the regression on security i.
𝛽! = !!"
!!! (13)
As the reader can observe, the regression analysis yields historical estimations on betas and residual returns for each security listed in the benchmark. Nevertheless, we make two assumptions in order to forecast expected returns using the market model: first, historical betas are representative of the future sensitivity of excess returns due to changes in the benchmark return and, second, forecasts of residual returns are not the result of regression analysis, but of the estimation by skillful active managers. For the latter, the next subsection explains how alphas are estimated.
B.2. Expected residual returns
When looking into the future, an active manager forecasts residual returns on stocks in order to construct active portfolios. These estimations make the difference between the consensus on expected returns, as those provided by the CAPM, and the beliefs of the active manager to produce a better performing portfolio by identifying over and undervalued securities.
Grinold (1994), in his article “Alpha is Volatility times IC times Score”, presents a technique to forecast residual returns or alphas, 𝛼!. With this methodology, we estimate expected residual returns by means of the residual risk of each security, 𝜎!", the
information coefficient, 𝐼𝐶, and a score, 𝑆!. Specifically, the expected residual return for the ith stock is17:
𝛼! = 𝜎!" ∙ 𝐼𝐶 ∙ 𝑆! (14)
With regard to the residual risk for each security, the market model provides the background to decompose the total risk on stock i, as measured by the variance of its returns, into two components: first, the systematic risk, 𝛽!!𝜎
!!, which is the part of the
residual risk, in terms of the variance for each security, as the difference between its total and systematic risk:
𝜎!"! = 𝜎
!!− 𝛽!!∙ 𝜎!! (15)
Similarly, the second component of alpha relates to the information coefficient defined in the fundamental and the generalized law, which is a measure of the forecasting skills of the active manager on residual returns18.
Finally, the score is a measure of the confidence of the active manager about a particular security at a particular moment in time, thus it can be addressed as “the personal bet” of the manager on each stock.
Unlike the information coefficient, the scores change over time and across stocks. Additionally, the scores are normally distributed and standardized, therefore, the average and the standard deviation of the scores is approximately zero and one respectively, for a given set of securities. Moreover, it also holds for a set of scores on a specific stock over time.
Since the score reflects the confidence of the manager on a particular stock, it may incorporate his/her beliefs into the form of a “tip”, a buy and sell recommendation or a numerical forecast based on technical or fundamental analysis, among other methodologies to value stocks (Grinold (1994))19.
In order to simulate the scores, we use a Monte Carlo simulation to generate a sample of normal distributed random numbers. Specifically, we create a thousand scores for every security in the benchmark and we compute the same number of alphas and portfolios for each constraint and level of active risk we analyse20.
C. The optimal standard active portfolio
optimal active portfolio under the mean variance framework. This is the standard active portfolio.
We carry out further optimization experiments to construct active portfolios that consider different types of investment constraints on the active manager using the mean variance approach. The following subsections outline the main characteristics of the basic optimization problem, which is narrowly altered when investment constraints are introduced.
C.1. Expected excess returns and tracking error
Denote the weight of stock i in the benchmark portfolio as 𝑤!", and its portfolio holding as 𝑤!". Now, define the active weight of security i, ∆𝑤!, as the difference
between the portfolio and the benchmark holdings:
∆𝑤! ≡ 𝑤!" − 𝑤!" (16)
Active weights represent the increasing (decreasing) position on a given security by changing its investment loading from the benchmark to the managed portfolio, thus they are the result of active management. Note that the active holdings can be either positive or negative, which means that more or less investing on a specific stock is required to attain the active portfolio relative to its initial position on the benchmark.
As early noticed, the sum of the security holdings on the benchmark portfolio equals one. Henceforth, we assume that the sum of the stock weights in the active portfolio equals one, implying that this portfolio is fully invested21. As a consequence,
the sum of the active weights equals zero.
On the other hand, the expected return on the benchmark portfolio 𝐸 𝑅! is the weighted average sum of the expected returns by the benchmark weights:
𝐸 𝑅! = ! 𝑤!"∙ 𝐸 𝑅!
!!! (17)
𝐸 𝑅! = ! 𝑤!" ∙ 𝐸 𝑅!
!!! (18)
Then, we define the expected excess return as the difference between the expected return on the managed portfolio and the benchmark. The resulting difference is the expected active return, 𝐸 𝑅! , on the managed portfolio, which is the weighted
average sum of the expected stock returns by the active holdings:
𝐸 𝑅! = 𝐸 𝑅! − 𝐸 𝑅! = ! ∆𝑤!∙ 𝐸 𝑅!
!!! (19)
Correspondingly, the expected active return has an inherent risk measured by the standard deviation of the excess returns of the managed portfolio over the benchmark, i.e. the tracking error22. Basically, the active holdings and the estimated
variance covariance matrix determine the tracking error or active risk, 𝜎!, as follows (using matrix notation):
𝜎! = ∆𝑊!∙ 𝑉 ∙ ∆𝑊 (20)
Where ∆𝑊 is the vector of active holdings and 𝑉 is the variance covariance matrix as discussed previously in the methodology section, subsection A23.
C.2. The ex ante information ratio and the optimization problem
The conventional approach to construct efficient portfolios based on the mean variance framework focuses on the maximization of the Sharpe ratio24. The investment
strategy consists in selecting the portfolio that maximizes it; hence, the result is an optimal combination of portfolio holdings. In general, maximizing the Sharpe ratio implies an asset mix of the risky portfolio and the risk free asset, where the optimal portfolio is fully invested and financed at the risk free rate (Sharpe (1994) and Goodwin (1998)).
his/her objective, the active manager chooses a portfolio that maximizes its ex ante information ratio (equation 2).
As explained section I, the ex ante information ratio is a measure of opportunity (Grinold et al. (2000)) from a forward looking perspective. As an ex ante measure, it represents the maximum expected active return per unit of active risk that can be achieved for the manager by using efficiently his/her private information, thus it sets the upper boundary on the ex post information ratios for the portfolios under management.
We construct the standard active portfolio in order to maximize its information ratio subject to the constraint that the sum of the active weights equal zero, hence the managed portfolio is fully invested. The optimization problem yields a vector of active holdings, which reflects the forecasts on residual returns made by the manager and determines the under or over weighting of the securities in the active portfolio relative to the benchmark.
Regarding the forthcoming experiments, we modify the constraints of the basic optimization problem in order to incorporate the investment limitations imposed to the active manager for each case study. Moreover, we perform a sensitivity analysis by adding more restrictions on the amount of active risk that the active manager is allowed to face. We perform the forecasts of residual results through the Monte Carlo simulation, and the subsequent optimizations using Matlab®25.
III. Data
Data consists of a set of closing prices for the benchmark, which in this case is the Amsterdam Exchange Index (AEX Index®) produced by NYSE Euronext, and for its 25 constituent securities26.
We collect daily observations from datastream® for the period beginning on the 20th of June 2011 as to the 20th of July 201227, for a total of 282 observations for each,
Table I presents descriptive statistics for the time series of closing prices. As can be seen, more than half of the securities display coefficients of variation above 10%. Actually, the stocks that exhibit the highest dispersion relative to their means are Air France Klm, Aperam and PostNl (37%, 27% and 26% respectively). On the other hand, Reed Elsevier, Unilever and Unibail Rodamco display the lowest coefficients of variation (5% in average). Similarly, the AEX index® exhibits a 6% coefficient of variation.
We transform the original time series into daily logarithm returns for the index and the stocks, as defined by:
𝑅! = 𝐿𝑛 !!"
Where 𝑅! denotes the logarithm return28 for security i, or the index, and 𝑃
!
represents the corresponding daily closing prices at time periods 𝑡 and 𝑡 − 1 respectively. In terms of daily logarithm returns, the transformed data set consists of 281 daily observations for the benchmark its 25 constituent securities.
returns (5,696; 5,167 and 4,821 basis points respectively)29. Accordingly, the returns on
these securities also display the greatest daily standard deviations (4% in average). On the other hand, the returns on the AEX index® present a range of 899 basis points and a standard deviation of 1% approximately. These statistics circumscribe the index amid the less volatile securities.
In order to estimate the risk free rate of the optimization model, we gather monthly data on the Euribor three month rate from January 2007 to June 2012. We compute its arithmetic mean and calculate the equivalent continuously compound rate on a daily basis. As a result, the risk free rate is assumed as 0.009% on a daily basis. We collect this information from datastream®.
Similarly, we estimate the expected benchmark return from the daily logarithm returns of the AEX Index®. We calculate the arithmetic mean of the time series from the 1st of September 2011 to the 1st of March 2012. As a result, the expected benchmark
return is assumed as 0.083% on a daily basis.
IV. From unconstrained to constrained active portfolios: an efficiency assessment We conduct a first optimization to build the standard active portfolio without restrictions, except for the full investment constraint. The standard active portfolio is the baseline to analyse the impact of imposing investment constraints to the active manager.
For the case studies we conduct further optimizations subject to different restrictions and we compare these results with the unconstrained portfolio in terms of expected active returns, expected adjusted returns, information ratios and transfer coefficients30. These optimizations are performed at a 5% level of active risk on an
A. Expected adjusted returns
As a starting point, we estimate alphas and betas and use them as inputs in the market model to calculate the expected adjusted returns for each security quoted in the AEX Index®.
we perform a Monte Carlo simulation in order to simulate scores for every security in the index32. Since each score represents a personal bet on a particular stock at any
moment in time, the set of scores define a large sample of the beliefs of the active manager on the expected performance of these securities. Table III exhibits the residual risk for each security, the results of the Monte Carlo simulation on the scores and the construction of alphas.
As a result of the simulation process to determine the scores, alphas have a random component, which is scaled by the information coefficient of the manager. Furthermore, the residual risk of each security incorporates idiosyncratic elements. In this sense, one can interpret these forecasts as random private information to construct active portfolios to outperform the benchmark.
Additionally, we estimate betas for each security from historical data on logarithm returns through regression analysis as explained in the methodology section. With the information on alphas and betas, we construct a matrix of expected adjusted returns that has a thousand estimations for each security. Table IV exhibits estimated historical betas; expected returns derived from these betas, and mean values and standard deviations of expected adjusted returns for each security.
B. The benchmark and the standard active portfolio: a first assessment on private information
We perform an initial optimization without investment constraints based on the matrix of alphas33. The result is a matrix (25 x 1000) of active portfolio weights where
the rows represent the stocks quoted in the index and the columns the corresponding active weights for every standard active portfolio.
This matrix represents a set of different scenarios where the active manager freely assesses which securities are over or undervalued relative to the benchmark portfolio based on the forecasts on residual returns. Each optimized unconstrained portfolio is in line with the original version of the fundamental law of active management (Grinold (1989)), where there is a perfect correlation between the forecasts of residual returns and the active weights, thus there is no loss in efficiency when the active manager transform his/her skills into active portfolios34.
The holdings of the benchmark portfolio correspond to the relative value of the market capitalization of each security quoted in the AEX Index®35. In addition, the
benchmark holdings are positive, which implies that there are no short positions in this portfolio.
by the incremental participation in the majority of the stocks, despite a lessening in some large, mid and low cap securities. The proceedings of short sales are used to fund the purchase of undervalued stocks within the standard active portfolio.
The composition of the standard active portfolio is the result of the forecasts on residual returns and the estimated variance covariance matrix. As a consequence, the active weights reflect the optimal combination of the excess return of the standard active portfolio over the benchmark and the tracking error.
As can be seen in Table VI, the expected return of the standard active portfolio is greater than the expected return of the benchmark portfolio. Actually, the expected active return of the standard active portfolio is 2.2 basis points on a daily basis. On the other hand, the risk of the standard active portfolio is higher relative to the benchmark. Nevertheless, the Sharpe ratio of the active portfolio suggests that it is more efficient than the benchmark in terms of a risk to reward ratio.
In resume, the interrelation of systemic and idiosyncratic factors allows for a standard active portfolio, which displays greater expected returns and Sharpe ratios with a marginal increase in overall risk, derived from the fact that the optimization was conducted subject to the constraint of a 0.315% level of active risk on a daily basis. Under the random believes of the manager and the active risk allowed, the result is an ex ante active portfolio that dominates the benchmark in the return risk continuum.
C. Case study 1: including transaction costs
We include transactions costs in the standard active portfolio by calculating its total turnover. First, we define the active portfolio turnover, 𝑇, as the sum of the absolute values of the active weights in the managed portfolio. The portfolio turnover incorporates the total change in the portfolio loadings as a consequence of active management, setting as a starting point the benchmark portfolio for calculations.
𝑇 = !!!! ∆𝑤! (22)
Once we compute portfolio turnover, we calculate transaction costs by applying a constant rate among securities. In this scenario, the expected active return of the managed portfolio is lowered by the total amount of the transaction costs, which is equivalent to the portfolio turnover times the cost rate. Therefore, the optimization problem entails portfolio turnover and transaction costs36.
Figure 5 exhibits the active weights of the managed portfolio in presence of transaction costs. Compared with the standard active portfolio, the new active weights are adjusted in such way that the majority of purchases and sells decrease in absolute terms37, thus the first impact of transaction costs is a lower exposition of the managed
transaction costs in order to attain the highest expected active return given a level of active risk.
As Table VII illustrates, transaction costs restrict the manager in achieving the highest active return in the managed portfolio relative to the standard active portfolio, as measured through the information ratio. Basically, the limitation of the manager arises because there is a trade off between maximizing forecasted returns and minimizing transaction costs in the optimization process.
the active weights do not only reflect which securities are over or undervalued, but the relative cost of assuming active positions.
We assess the loss in efficiency originated from the presence of transaction costs through the transfer coefficient, which falls from 0.993 in the standard active portfolio to 0.876 in the managed portfolio. The limitations introduced by transaction costs imply a relative reduction in the information ratio of 11.76% from the standard active portfolio. Moreover, transaction costs induce a decline in total expected return equivalent to four basis points on an annual basis, and an overall increase in risk relative to the standard active portfolio. Consequently, the resulting active portfolio displays a lowered Sharpe ratio.
D. Case study 2: the long only constraint
Though it seems innocuous, the long only investment policy sets up a constraint for constructing active portfolios. In practical terms, the long only constraint may not be seen as such since it is assumed that the available monetary resources are completely allocated among the asset classes, suggesting that the weights of the managed portfolio are non negative. Nevertheless, the absence of short positions limits the ability of the manager to fully exploit his/her private information on the securities that constitute the active portfolio.
When short sales are not allowed, the manager is just permitted diminishing the weight of an overvalued security to zero in the managed portfolio, meaning that it is completely sold. Therefore, the minimum value of an active weight for a given stock is the negative of its corresponding benchmark weight.
When we perform the optimization subject to the long only constraint in presence of transaction costs, the short positions in the standard active portfolio are now positive weights in the managed portfolio. Moreover, since the manager is only allowed to completely sell an overvalued security in the managed portfolio, the weights on the previous short positions are now negligible, as can be seen in Figure 6.
The long only constraint produces a less diversified portfolio, where those securities that exhibit higher forecasted residual returns display higher weights38, as
depicted by figures 6 and 7. Furthermore, figure 7 shows that the active weights for those securities with relatively low forecasted residual returns reduce monotonically towards zero in the managed portfolio.
As a result, there is a reallocation of funds towards those securities that exhibit the highest forecasts, thus the active manager is forced to augment the investment loads in these securities in order to compensate the relative loss imposed by the absence of short sales in the managed portfolio.
The combined impact of the long only constraint and the presence of transaction costs in the active weights is a deepening loss in efficiency, as can be seen in Table VIII. The fact that the securities with negative adjusted excess returns are not allowed to be short sold, and that even these stocks are practically not held in the managed portfolio, results in a decrease in the portfolio turnover, which accounts for 48.28%.
Moreover, the long only constraint coins a further decrease in efficiency, as measured by the transfer coefficient, compared with the standard active portfolio and the scenario in case study 1. The long only constraint produces a managed portfolio with a transfer coefficient of 0.840, which is lower relative to the standard active portfolio. This transfer coefficient indicates that 15.48% of the skills of the manager are lost during the portfolio construction process.
Such circumstance negatively affects the information ratio of the managed portfolio, which falls from 0.069 to 0.058 as a consequence of the deterioration in expected active returns. In general, the total return on the managed portfolio decreases 33 basis points on an annual basis and its total risk increases relative to the standard active portfolio, leading to an overall decline in its Sharpe ratio, which falls from 0.056 to 0.054.
E. Case study 3: industry oriented investing
In some cases, the investment problem is circumscribed to a specific subset of securities within the benchmark. More specifically, the active portfolio must be invested in an industry, a sector or a sub sector of the index39 due to the investment policies and
objectives. For example, the portfolio must focus just in technology or consumption or energy companies in the index, setting aside the rest of the securities that constitute the benchmark. This type of constraint constitutes a restriction to the manager in two aspects: first, he/she faces a reduced set of investment alternatives, thus breadth is lower; second, the active manager is not able to fully transform his/her skills into the active portfolio due to his/her forecasts are now restricted to some specified categories within the index.
transaction costs under the long only constraint, thus the manager is restricted to have only long positions in the managed portfolio.
As discussed earlier, the forecast of residual returns of each security determines the magnitude of the active weights. The main difference with the previous case studies is that there are active holdings, which are equal to the negative weight on the benchmark as a result of the limitation of the manager to invest in the excluded securities, independently on the magnitude of the forecasted residual returns. Therefore, these stocks must be completely sold to construct the active portfolio, no matter the beliefs of the manager on such securities.
The manager focuses on the forecasted residual returns of those securities available among the selected industries, given the industry oriented constraint and the long only constraint. Under this consideration, the stocks that exhibit higher alphas increase their share in the managed portfolio relative to the standard active portfolio, as shown in the left side of Figure 841. Similarly, the active weights for those securities with
