1. Introduction
Many new methods and concepts have emerged in financial portfolio construction, risk management and performance evaluation since ^{Markowitz’s (1952)} pioneering work. The use of the variance, or standard deviation, of returns as a proxy for investment risk has been questioned, and alternative risk measures have been proposed, according to ^{Rom and Ferguson (1994)}; ^{Roman and Mitra (2009)} and ^{Araújo and Montini (2015)}. The use of the original sample covariance matrix to estimate the risk of an asset portfolio has also been questioned, according to ^{Chan et al. (1999)} and ^{Santos and Tessari (2012)}. Not even the use of quantitative methods to optimize portfolios or to allocate resources among different asset classes has been spared from criticism (^{DeMiguel et al. (2009)}.
Given the vast academic literature that proposes, explains and details numerous methods for portfolio construction and risk management and performance evaluation, the following central research question is addressed in this paper: among the major quantitative techniques for portfolio construction, risk management and performance evaluation suggested by the international and Brazilian academic literature, which ones are actually adopted by financial market practitioners? The main objective is to identify possible differences between what is taught in classrooms or discussed in academic conferences and what is involved in the daytoday practice of asset managers and to determine whether a mismatch exists.
In addition to the central research question, the present work aims to compare Brazilian data with European data to inform us about the degree of globalization of the Brazilian asset management industry. Another specific goal is to check the existence of different practices among subgroups in the industry, such as companies of foreign and Brazilian origin, and to investigate the determinants of optimization and risk budgeting.
To achieve these objectives, we conduct a literature review and obtain 78 responses from a field survey with 274 asset management companies. The number of respondents is considerable, especially given the limited universe of professional asset management companies and the difficulty of extracting information from participants in a highly competitive and resultsoriented segment. The field survey used an online questionnaire similar to that applied by ^{Amenc et al. (2011)} in Europe. To analyze the data, the present paper uses Pearson’s chisquare independence test, multiple regressions using ordinary least squares (OLS) and a probit model.
According to ^{Amenc et al. (2011)}, little is known about the dissemination and use of academic research findings among investment professionals. Surveybased papers can assist in guiding and developing empirical studies and new theories by academics and can alert practitioners of researchinformed recommendations that have not yet been adopted, according to ^{Graham and Harvey (2001)}. However, similar surveys for Brazil have not been found, thus confirming ^{Amenc et al.’s study (2011)} and justifying this paper.
This paper is divided into five sections, including the introduction. In Section 2, based on the existing theory, we formulate hypotheses regarding the practices of investment managers. Section 3 describes the empirical methodology of both the field survey and data analysis. Section 4 presents the results, and Section 5 provides final remarks and suggestions for future studies.
2. Literature review and Hypotheses
2.1 Measuring market risk
This paper uses the definition of market risk proposed by ^{Dowd (2002)}: it is the risk resulting from changes in prices, such as changes in public companies’ share quotes, and rates, such as the interest rate and exchange rate. Two ways to quantify this risk are using variance and using downside risk.
^{Markowitz (1952)}, who used the variance of returns as a measure of risk, was one of the pioneers in proposing a quantitative methodology for portfolio construction. His work, along with that of ^{Sharpe (1964)} and ^{Lintner (1956)}, generated discussions and publications that formed the modern theory of portfolios.
^{Rom and Ferguson (1994)} and ^{Roman and Mitra (2009)} argued in favor of more appropriate risk measures for asymmetric distributions of returns, socalled downside risk metrics. They proposed a new paradigm for portfolio construction and risk management, made possible by computational advances and made necessary by the increasing use of derivatives in portfolios. They termed this new paradigm the postmodern portfolio theory.
Among the downside risk metrics, semivariance, proposed by ^{Markowitz (1959)}, is equivalent to the variance using only belowaverage returns. ^{Bawa (1975)} and ^{Fishburn (1977)} subsequently developed lower partial moments (LPM). Series can be described using the first four moments: mean, standard deviation, kurtosis and asymmetry. LPM is for moments what semivariance is for variance, as LPM is based on only one side of the distribution.
While LPM and semivariance focus on returns below a target or the average, respectively, the other downside risk metrics, value at risk (VaR) and conditional value at risk (CVaR) or expected shortfall (ES) focus on extreme negative returns or tail risk. According to ^{Dowd (2002)}, the VaR is the maximum likely financial loss within a given period and confidence interval provided that there is no certainty of an extreme adverse event. The CVaR of ^{Rockafeller and Uryasev (2000)} is the expected loss conditional on the occurrence of an extreme adverse event.
Empirical studies comparing and ranking different risk metrics usually consider the outofsample return of portfolios optimized for various risks but with the same return in the sample period. Brazilian studies indicate that LPMoptimized portfolios generally yield the best results, independent of the normality of the distribution of returns; see ^{Andrade (2006)} and ^{Araújo and Montini (2015)}. In light of the literature, H1, to be tested is as follows:
H1.
Postmodern theory is used more than the modern theory in performance evaluation and in optimization and risk budgeting with absolute risk.
Risk budgeting is a risk management practice in which the portfolio is changed whenever necessary to contain the predicted risk within predetermined limits. Examples of absolute risk are variance, semivariance and VaR.
In addition to the absolute risk metrics described above, relative risk metrics, which measure the variability of the differences between portfolio returns and a benchmark, are essential for the evaluation of investment managers; see ^{Roll (1992)}. Relative risk metrics are of interest both to passive portfolio managers who aim to replicate the returns of an index or benchmark and to active portfolio managers. Metrics such as tracking error may be used to verify with statistical significance whether a fund with active management is adding value relative to a passive portfolio:
where TE_{p} is the tracking error, N is the number of observations, n is the index of the sum, T is the sample frequency, D_{n} = R_{portfolio,n} − R_{benchmark,n} and D is the average of the return differences. Thus, we have:
2.2 Covariance matrix estimation
The estimation of portfolio risk typically uses the covariance matrix, which contains the variance of each asset, as well as the covariance for all combinations of two assets, according to ^{Dowd (2002)}. The covariance matrix is a key part of both portfolio optimization, which aims to obtain the optimal weights of assets, and risk management, which aims to estimate the expected risk of a portfolio or the portfolio’s VaR, as in ^{Scherer (2002)}.
RiskMetrics, proposed by ^{Guldimann et al. (1996)}, uses an exponentially weighted movingaverage approach and essentially has only one difference relative to the sample covariance matrix; that is, most recent observations receive greater weight. An exponentially weighted movingaverage or EWMA conditional covariance matrix Σt is given by:
where R_{t} is the vector of asset returns at time t, λ is the decay factor, 0 < λ < 1 and the recommended value is λ = 0.94 for daily data.
^{Stein (1956)} showed that the individually estimated sample mean is not a good estimator of the population mean when it has a multivariate normal distribution. The implication is that in portfolio construction, the samplebased covariance matrix may have estimation errors, although it is not biased. In the same paper, the author suggests using a statistical method to reduce estimation errors, which results in the use of an estimator constructed from the unbiased estimator and a biased and structured estimator, which is subjective and based on previous knowledge and experience.
According to ^{Ledoit and Wolf (2003)}, the shrinkage method for the covariance matrix is based on the work of ^{Stein (1956)} and entails the adjustment of the sample covariance matrix, which typically presents more extreme coefficients, toward a fixed target known as the biased estimator, which ideally has more central values. ^{Frost and Savarino (1986)} and ^{Jorion (1986)} initially applied shrinkage in the construction of portfolios, but the method did not become practical and feasible until the work of ^{Ledoit and Wolf (2003)}. We can represent the covariance matrix as:
where F_{t} is the fixed target or biased estimator, α ∈ [0,1] and can be interpreted as the weight given to the biased estimator and Σ̂_{t} is the sample covariance matrix.
An alternative way to use the methods suggested above for the covariance matrix involves the use of factors and is based on the theoretical framework such as ^{Ross’s (1976)} arbitrage pricing theory (APT) and empirical studies such as those of ^{Fama and French (1992)}. In APT, the excess return of an asset is:
where there are k systematic risk factors, R_{i} is the rate of return of asset i, R_{f} is the riskfree rate, β_{n} is the sensitivity or load of asset i on risk factor n and f_{n} is the risk premium of factor n. Assuming that the factors are not correlated with the residual return and that the residual returns are not correlated, the covariance matrix for N assets according to ^{Chan et al. (1999)} is:
where B is the factor sensitivity matrix, Ω is the covariance matrix of the factors and D is a diagonal matrix containing the residual variances of the return.
Statistical factors are also known as implicit factors, as they are hypothetical variables constructed to explain the movement of a set of time series of returns; see ^{Dowd (2002)}. Principal component analysis or factor analysis (FA) can be used to identify factors, and both of these quantitative methods can identify independent sources of movement. One of the advantages of using implicit factors in calculating the covariance matrix is to reduce the computational burden since risk factors are independent and a small number of factors are sufficient to achieve high explanatory power; see ^{Alexander (2001)} and ^{Amenc and Martellini (2002)}.
In addition, ^{Engle (1982)} and ^{Bollerslev (1986)} introduced the univariate generalized autoregressive conditional heteroskedasticity (GARCH) model. Some versions of the multivariate model can dynamically estimate expected and conditional variance and covariance for portfolio assets, and these approaches are recommended when asset volatility is inconsistent over time.
Empirical studies such as those of ^{Santos and Tessari (2012)} and ^{Beltrame and Rubesam (2013)} point out that the covariance matrix shrinkage method, which combines sample covariance with structured estimators, yields the best results among all the methods. The third test of theory versus practice thus involves testing.
Studies in Brazil, such as ^{Santos and Tessari (2012)}, show that minimum variance portfolios obtained through quantitative optimization using the sample covariance matrix, RiskMetrics, explicit factors and GARCH generally have higher returns than an equalweighted portfolio and a market cap weighted portfolio. Although authors such as ^{DeMiguel et al. (2009)} obtained different results when studying listed companies in the USA, we consider H4 as follows:
H4.
Investment managers use optimization methods more than the simple rule of establishing a maximum concentration limit per asset.
We do not detail the specific methodology used to optimize portfolios for a risk metric given a covariance matrix. Basically, this entails the optimization of a function subject to constraints. The methodology for portfolio optimization can be found in the studies of ^{Roman and Mitra (2009)} and ^{Araújo and Montini (2015)}.
According to ^{Scherer (2002)}, managers who are constrained by a risk budget should perform portfolio optimization. The selection of assets and definition of their weights through optimization indicate to managers the optimal portfolio for the preestablished risk limit. Thus, we have:
2.3 Distribution of returns
According to ^{Roman and Mitra (2009)}, research on return distributions is as important as the study of risk measures and covariance matrices in maximizing returns or minimizing risks in a portfolio. The practical implications are better estimates of the parameters for the optimization of financial models, such as the mean and variance optimization of Markowitz or the capital asset pricing model (CAPM), and for option pricing models, according to ^{Leal and Ribeiro (2002)}.
One way to test the suitability of a return distribution is to use the VaR and the Kupiec failure ratio test, which essentially consists of counting the number of losses above the established VaR using outofsample testing.
Historical VaR assumes that the distribution of future returns is nonparametric and is based exclusively on historical data or, in other words, that the past provides all the information and the future probability distribution will be equal to the past distribution. The Monte Carlo VaR assumes that future returns will follow a known stochastic process and a nonparametric distribution. To obtain historical or Monte Carlo VaR at the fifth percentile (or a 95 per cent confidence interval), the historical or simulated returns are sorted in ascending order, and the return at the fifth percentile is the estimated VaR.
The simplest parametric VaR assumes that future returns will follow a normal, or Gaussian, parametric distribution that can be estimated with only two parameters: the mean and the expected standard deviation of the returns.
VaR with higher moments considers the existence of kurtosis, asymmetry or both in the distribution and is a more appropriate parametric method when returns do not follow a normal distribution. According to ^{Dowd (2002)}, one way to consider kurtosis in a distribution is through the Student’s t distribution, which has a kurtosis equivalent to 3 (υ − 2)/(υ − 4), where υ can be chosen based on the expected kurtosis of future returns such that 5 < υ < 9.
Extreme value theory (EVT) provides a basis for modeling events with significant economic consequences and very small probabilities of occurring. The generalized extreme value distribution is parametric and may be used to build the worstcase scenario in ten years in the equity market or the probability of the eurodollar exchange rate appreciating more than 20 per cent in a week.
Brazilian and international empirical studies such as ^{Cassettari (2001)}, ^{Leal and Ribeiro (2002)} and ^{Arraes and Rocha (2006)} show that financial asset returns demonstrate asymmetry and kurtosis, and therefore, VaR with a normal distribution is not the most appropriate method. H6 will test whether the methods suggested by empirical studies in academia are the most frequently used by practitioners but will focus on return distributions:
2.4 Management of estimation risk
^{Scherer (2002b)} pointed out that the portfolio optimization process suffers from error maximization since assets with higher returns, low risk or low covariance tend to be chosen. Extreme results have a higher probability of estimation errors; in other words, they tend to be unsustainable in the long term. In addition, there is a consensus among scholars about the inability to predict future asset returns and a belief that the optimization process is very sensitive to differences in the expectation of future returns (^{Michaud, 1989}). Some alternatives proposed to address estimation risk are presented below.
The Monte Carlo resampling of ^{Michaud (1989)} and ^{Michaud and Michaud (2008)} typically uses the drawwithreplacement method to simulate asset returns based on their historical distribution. Usually, the simulation is performed hundreds of times and for all assets to test investment strategies.
Bayesian models and the Black Litterman model – see ^{Jorion (1986)} and ^{Black and Litterman (1992)} – allow us to add subjective individual convictions to the quantitative financial models. The model will make recommendations after confidence levels for the quantitative model and the subjective individual conviction are determined.
A different strategy to reduce estimation risk involves maintaining a portfolio with the lowest expected risk. The minimum variance portfolio depends only on the covariance estimation and is subject to a more moderate estimation error than other meanvariance portfolios are; see the studies of ^{Chan et al. (1999)}; ^{DeMiguel et al. (2009)} and ^{Caldeira et al. (2013)}. Given the importance of adopting techniques to manage estimation risk, we thus test:
2.5 Performance evaluation
The use of riskadjusted returns to evaluate performance was proposed at nearly the same time as the CAPM model. ^{Treynor (1965)}; ^{Sharpe (1966)} and ^{Jensen (1968)} proposed riskadjusted performance measures based on the theoretical framework of Markowitz’s meanvariance model. The desire for a portfolio optimization method to account for market risk and investment analysts’ opinions led ^{Treynor and Black (1973)} to create a ratio that would later be known as the information ratio. Last on our list of performance measures that are based on the modern portfolio theory is the ^{Modigliani and Modigliani (1997)} ratio, which can be interpreted as the return that a fund would have if its risk were equivalent to market risk.
Among the performance metrics that are based on downside risk or postmodern portfolio theory, the ^{Sortino and Van Der Meer (1991)} ratio and the return relative to the VaR are worth noting. In the table below, it is important to note that for the Sortino ratio, an investor can replace the riskfree return with a minimum acceptable return; the LPM can replace the semivariance, and the investor can define the degrees of freedom. ^{Dowd (2000)} proposed the return relative to VaR and defined it as the return above the riskfree rate divided by the portfolio’s VaR (Table I).
Name  Risk metric  Portfolio theory  Formula 

Absolute return  No risk adjustment  None  R_{p} 
Excess return on benchmark  No direct risk adjustment  None  R_{p} − R_{b} 
Sharpe ratio  Standard deviation  Modern* 

Treynor ratio  Beta CAPM  Modern 

Jensen's alpha  Beta CAPM  Modern  R_{p} − (R_{f} + b_{p} (R_{m} − R_{f}) 
Information ratio  Standard deviation of residual or tracking error  Modern* 

Modigliani and Modigliani – M^{2}  Beta CAPM  Modern* 

Sortino ratio  Semivariance  Postmodern 

Return relative to VaR  VaR  Postmodern 

Note: *The empirical tests reported in Section 4 compare the adoption of these performance measures based on the modern theory of portfolios with the adoption of the Sortino index and the return relative to VaR
Estimating the alpha of a portfolio is not restricted to the method proposed by ^{Jensen (1968)}, where the alpha is the portfolio return adjusted for the market risk incurred. According to ^{Bailey (1992)}, the simplest method to estimate alpha is to compare the returns obtained with other similar funds. ^{Fama and French (1992)} suggested capturing alpha through multifactor models that consider, for example, size and value factors. Performance attribution and style analysis facilitate the decomposition of excess returns into various components. These components may include, for example, asset class allocation and stock selection or allocation to risk factors (^{BARRA, 1990}; ^{Sharpe, 1992}).
H8 refers to the popularity of riskadjusted returns:
3. Empirical methodology
To test the hypotheses, we collected data via online questionnaires hosted on the website of the Centro de Estudos em Finanças CEF of Fundação Getúlio Vargas’ Business Administration School. Invitations and links were sent to key executives, portfolio and risk managers representing 274 asset management companies between August and September 2015. To increase the number of responses, followup telephone calls were made to the managers. We obtained telephone numbers and email addresses from the CEF database.
The questionnaire had a total of 3 sections and 17 questions that required the analysis of 65 nonmutually exclusive responses. The questionnaire is based on the work of ^{Amenc et al. (2011)}, who surveyed investment managers in Europe in 2007 on portfolio management and performance evaluation. Although based on a previous survey, a pretest of the questionnaire was conducted with a group of investment managers with the objective of evaluating the terminology, the clarity of the questions and the average response time, with the main goal of improving the questionnaire.
All questionnaires, according to ^{Graham and Harvey (2001)}, are subject to potential problems; for example, the responses may not reflect respondents’ actions. One concern regarding this work is that some respondents could omit certain information for fear of being copied. Therefore, all the questions were obligatory, except for some identification data: name and email, name and origin of the asset management company and whether the asset management company is affiliated with a bank. Ten of the respondents chose not to identify themselves.
In the present study, all variables of interest are categorical, the majority being binary, with only two categories of responses, such as “uses” and “does not use”. The first type of test was the independence test of two variables, which was used to compare categorical responses between two groups of the same sample or population. We performed independence tests, for example, to compare the rates of utilization of different portfolios and risk management methods. The statistic of the independence test, more specifically of the chisquare test, of Pearson, is given by χ^{2} (Table II):
Contingency table  Column variable (groups)  

Line variable (answers)  1  2  Total 
1: Uses  X_{1}  X_{2}  X 
2: Does not use  n_{1} − X_{1}  n_{2} − X_{2}  n − X 
Total  n_{1}  n_{2}  N 
where e_{ij} is the expected frequency based on the independence hypothesis for the category in Row i and Column j of the contingency table and f_{ij} is the frequency observed for the category in Row i and Column j of the contingency table. When we analyze two groups and binary responses, the test statistic has approximately a chisquare distribution for large samples with one degree of freedom.
Notably, using a Ztest is preferable when one wishes to analyze differences in the proportions of two different populations. An example is the comparison of proportions verified in the present research carried out in Brazil, with the proportions verified in the research of ^{Amenc et al. (2011)} in Europe. The Ztest uses two normal distributions rather than independence tests. In cases where the alternative hypothesis is the inequality of proportions and where the chisquare distribution has one degree of freedom, both the Ztest and the chisquare test produce exactly the same pvalues and the same statistical inferences.
^{Agresti (1996)} suggested the use of Fisher’s exact test, which is based on an exact distribution, for statistical inference when the sample is small and the expected value is less than five in any of the categories of the contingency table. To ensure robustness, Fisher’s exact test was performed to confirm the chisquare test results, and because we did not find significant differences, we omitted the results of Fisher’s exact tests.
The second type of test used multiple regressions to reveal the determinants of the use of quantitative optimization and risk budgeting by investment managers. OLS regressions, known as linear probability models when dependent variables are binary, can be used to reveal causal relationships, according to ^{Angrist and Pischke (2009)}. We use the probit model for robustness tests. In both regressions, our interest lies mainly in the probability that an observation with particular characteristics will fall into one of two categories:
where y is the dependent variable and binary indicator and x_{1} is the set of explanatory or independent variables. In the regressions, it is assumed that the probability of response is linearly determined by a set of parameters, and in a probit model, it is also assumed that the probability of response adheres to a standard normal cumulative distribution function. All tests and regressions were conducted using the software Stata version 13.
4. Analysis and discussion of results
4.1 Descriptive statistics and comparison of results with European data
There were 78 respondents; 21 per cent of the asset management companies had a foreign origin, and 29 per cent were affiliated with a bank. Regarding the size of the assets under management, 18 per cent of the respondents play an active role in the management of up to R$250m, 27 per cent between R$250m and R$1bn, 20 per cent between R$1bn and R$5bn and 35 per cent above R$5bn. Regarding the type of managed fund, 62 per cent have active equity funds, 21 per cent indexed equity funds, 26 per cent shortterm government bond funds or fundos DI, 47 per cent fixed income funds, 73 per cent hedge funds or fundos multimercados and 15 per cent other funds.
In Table III, we present a comparison of the results obtained with those of ^{Amenc et al. (2011)}, as well as the pvalue of the equality of proportions test, which will, in this case, be identical to the pvalue of the chisquare test, as explained in Section 3 of the paper.
Observations  Brazil 78 (%)  Europe 229 (%)  Δ (%)  pvalue  Sig 

In portfolio optimization, are objectives set for absolute risk?  
Yes, variance, volatility or standard deviation  21.8  45.9  −24.1  0.000  *** 
Yes, VaR or CVaR  2.6  50.7  −48.1  0.000  *** 
Yes, semivariance or LPM  5.1  23.1  −18.0  0.000  *** 
In portfolio optimization, are objectives set for relative risk?  
No  51.3  33.6  17.7  0.007  *** 
Yes, tracking error  34.6  49.8  −15.2  0.025  ** 
Yes, tracking error VaR or BVaR  19.2  18.8  0.5  0.930  
Yes, semivariance or LPM  11.5  12.7  −1.1  0.794  
Yes, other  0.0  2.6  −2.6  0.149  
How is the covariance matrix estimated?  
Sample covariance or RiskMetrics  53.8  59.8  −6.0  0.355  
Explicit factors, such as CAPM  19.2  29.3  −10.0  0.084  
Factor analysis or principal component analysis  3.8  12.7  −8.8  0.028  ** 
Shrinkage methods  1.3  3.9  −2.6  0.255  
GARCH and others  17.9  7.9  10.1  0.012  ** 
How is VaR estimated?  
Normal distribution  62.8  41.0  21.8  0.001  *** 
Higher moments  7.7  16.6  −8.9  0.053  * 
Extreme value theory  5.1  8.3  −3.2  0.359  
CVaR or expected shortfall  29.5  22.3  7.2  0.198  
Other  15.4  12.7  2.7  0.542  
What methods are used to address estimation risk?  
Maximum concentration limit per asset  64.1  67.7  −3.6  0.562  
Global minimum variance portfolio  5.1  17.0  −11.9  0.009  *** 
Bayesian methods  7.7  15.3  −7.6  0.089  
Resampling  9.0  13.5  −4.6  0.291  
Other  9.0  6.1  2.9  0.387  
How is performance evaluated?  
Sharpe ratio  78.2  77.3  0.9  0.868  
Treynor ratio  7.7  10.9  −3.2  0.414  
Sortino ratio  14.1  27.5  −13.4  0.017  ** 
Absolute return  65.4  41.5  23.9  0.000  *** 
Modigliani and Modigliani  5.1  3.1  2.1  0.395  
Jensen's alpha  15.4  34.1  −18.7  0.002  *** 
Information ratio  19.2  65.1  −45.8  0.000  *** 
Excess return relative to benchmark  75.6  31.4  44.2  0.000  *** 
How is alpha measured?  
Multifactor models  9.0  21.4  −12.4  0.014  ** 
Singlefactor models (such as CAPM)  17.9  26.6  −8.7  0.123  
Performance attribution or style analysis  69.2  35.4  33.9  0.000  *** 
Peer group analysis  66.7  56.8  9.9  0.124  
Other  5.1  2.2  2.9  0.183 
Notes: The pvalues provided are from Pearson's chisquare test with one degree of freedom. The null hypothesis is the equality of proportions, and the alternative hypothesis is inequality. We reject the null hypothesis at p < 0.01 (***); p < 0.05 (**) and p < 0.1 (*)
4.2 Adoption of modern and postmodern theory
Three independence tests were performed to compare investment managers’ adoption of the modern theory with that of postmodern theory. No statistically significant differences in risk management were found, specifically in the use of risk budgeting; 51.3 per cent are based on the modern theory and 59.0 per cent on the postmodern theory. Statistically significant differences in favor of the modern portfolio theory compared with postmodern theory were found:
in the construction of portfolios (21.8 per cent versus 6.8 per cent respectively) and
in performance evaluation (80.8 per cent versus 35.9 per cent, respectively).
Despite the results of academic studies such as ^{Araújo and Montini (2015)}, the results reported in Table IV lead not only to the rejection of H1, but also to the conclusion that the modern theory remains the dominant paradigm among practitioners.
ID  Hypothesis  Test  Result  

H1  Postmodern theory is used more than the modern theory in performance evaluation and in optimization and risk budgeting with absolute risk  Chisquare statistic  Hypothesis rejected based on tests in full sample and subsamples  
Tests performed  Hypotdeses  pvalue  Sig  Result  
T1a  Comparison of the proportions of respondents using modern (θ1) and postmodern (θ2) theory for absolute risk budgeting  h0: θ_{1} = θ_{2} h1: θ_{1} > θ_{2} 
0.334  H0 Not Rejected  
T1b  Comparison of the proportions of respondents using modern (θ1) and postmodern (θ2) theory for portfolio optimization with absolute risk  h0: θ_{1} = θ_{2} h1: θ_{1} > θ_{2} 
0.006  ***  H0 Rejected 
T1c  Comparison of the proportions of respondents using modern (θ1) and postModern (θ2) Theory for performance evaluation  h0: θ_{1} = θ_{2} h1: θ_{1} > θ_{2} 
0.000  ***  H0 Rejected 
We tested H1 in several subsamples: large asset management companies, with over R$1bn in assets under management, and small companies; companies with hedge funds and without; companies affiliated with banks and those that are independent; and companies with (a) one or two and three or (b) more types of funds. The results reported in Table V reject H1 for all subgroups in the sample as they are similar to the ones obtained for the full sample.
Subsample  T1a  T1b  T1c  

Obs  pvalue  Sig  pvalue  Sig  pvalue  Sig  
Large  43  0.822  0.024  **  0.000  ***  
Small  35  1.000  0.101  *  0.001  ***  
With hedge funds  57  0.338  0.003  ***  0.000  ***  
Without  21  0.753  1.000  0.002  ***  
Banks  20  0.490  0.633  0.010  **  
Independent  46  0.835  0.036  **  0.000  ***  
1 or 2 funds  32  0.035  **  0.03  **  0.002  *** 
3 or more funds  46  0.676  0.079  *  0.000  ***  
Total  78  0.334  0.006  ***  0.000  *** 
Notes: The pvalues provided are from chisquare tests of independence. The null hypothesis is equal use of modern and postmodern theory. T1a, T1b and T1c, respectively, address absolute risk budgeting, absolute risk optimization and performance evaluation. We reject the null hypothesis at p < 0.01 (***); p < 0.05 (**); p < 0.1 (*)
4.3 Determinants of relative risk budgeting and portfolio optimization
The importance of relative risk and return for Brazilian investment managers is evident: 75.6 per cent target a return above the benchmark, 64.1 per cent and 48.7 per cent perform, respectively, budgeting and portfolio optimization with relative risk. For comparison purposes, 65.4 per cent aim for the absolute return, and 55.1 per cent and 25.6 per cent perform budgeting and portfolio optimization with absolute risk. We conduct OLS and probit regressions to better understand the determinants of the use of relative risk and to test H2. The same approach was used to analyze the determinants of portfolio optimization and to test H5; see Table VI.
ID  Hypotheses  Tests  Results  

H2  The usage of relative risk is independent of whether the manager has passive indexed funds  Coefficient of the variable of interest in the regression  Rejected  
H5  Managers who comply with a risk budget tend to optimize their portfolios  Coefficient of the variable of interest in the regression  Partially rejected  
Regression equations and tests performed  
Model  Regression equations  Hypotheses  pvalue  *Sig  Results 
(1)  π(x) = α + β_{1}x + γΓ : where π(x) is the probability of setting a budget based on relative risk and x is the presence of indexed funds. See legend below  h0: β_{1} = 0 h1: β_{1} ≠ 0 
0.096  *  H0 Rejected 
(2)  probit[π(x)] = α + β_{1}x + γΓ : see legend in (1)  h0: β_{1} = 0 h1: β_{1} ≠ 0 
0.099  *  H0 Rejected 
(3)  π(x) = α + β_{1}x + β_{2}w + γΓ : where π(x) is the probability of performing relative risk optimization, x is the presence of indexed equity funds and w is the use of relative risk budgeting. See legend below  h0: β_{1} = 0 h1: β_{1} ≠ 0 h0: β_{2} = 0 h1: β_{2} ≠ 0 
0.058 0.000 
* *** 
H0 Rejected H0 Rejected 
(4)  probit[π(x)] = α + β_{1}x + β_{2}w + γΓ: see legend in (3)  h0: β_{1} = 0 h1: β_{1} ≠ 0 h0: β_{2} = 0 h1: β_{2} ≠ 0 
0.046 0.000 
** *** 
H0 Rejected H0 Rejected 
(5)  π(x) = α + β_{1}v + γΓ: where π(x) is the probability of performing absolute risk optimization and v is the presence of an absolute risk budget. See legend below  h0: β_{1} = 0 h1: β_{1} ≠ 0 
0.800  H0 Not rejected  
(6)  probit[π(x)] = α + β_{1}v + γΓ: see legend in (5)  h0: β_{1} = 0 h1: β_{1} > 0 
0.571  H0 Not rejected 
Notes: We reject the null hypothesis of β = 0 at p < 0.01 (***); p < 0.05 (**); p < 0.1 (*). The probit [π(x)] is a function or link that transforms π(x) into a Z statistic, α is the constant, β_{1} and β_{2} are the coefficients of interest, Γ is the vector of control variables and γ is the vector of coefficients of the control variables
In Models (1) and (2) in Table VII, the dependent variable is binary with a value of 1 (one) when the manager has a budget for relative risk and 0 (zero) when there is no such budget. In the other models, the dependent variables are also binary and take a value of one when the method is adopted and zero otherwise. The dependent variable is portfolio optimization with relative risk in Models (3) and (4) and is optimization with absolute risk in Models (5) and (6). Models (1)(4) will be used to test H2 and Models (3)(6) to test H5.
Dependent variables  

Independent variables  Budgeting with relative risk  Optimization with relative risk  Optimization with absolute risk  
(1)  (2)  (3)  (4)  (5)  (6)  
OLS  PROBIT  OLS  PROBIT  OLS  PROBIT  
Indexed equity funds  0.293* (0.173)  1.064* (0.644)  0.324* (0.168)  1.772** (0.889)  
Active equity funds  −0.0936 (0.138)  −0.242 (0.369)  −0.0300 (0.0902)  −0.328 (0.455)  0.0231 (0.126)  0.0416 (0.392) 
Large  −0.0646 (0.154)  −0.157 (0.399)  0.0504 (0.104)  0.242 (0.448)  −0.218* (0.126)  −0.768* (0.462) 
Foreign  0.131 (0.132)  0.451 (0.477)  −0.111 (0.141)  −0.321 (0.622)  0.125 (0.153)  0.431 (0.485) 
Bank  0.130 (0.138)  0.367 (0.440)  −0.0398 (0.0940)  −0.531 (0.622)  0.0198 (0.128)  0.120 (0.465) 
Hedge funds  0.104 (0.131)  0.267 (0.361)  −0.0915 (0.0825)  −0.667 (0.499)  0.213** (0.104)  0.814* (0.455) 
Budgeting with relative risk  0.692*** (0.0914)  2.798*** (0.511)  
Budgeting with absolute risk  −0.0269 (0.106)  −0.197 (0.348)  
Constant  0.530*** (0.142)  0.0618 (0.384)  0.0979 (0.0779)  −1.516*** (0.492)  0.164 (0.141)  −1.019* (0.526) 
Observations  68  68  68  68  68  68 
R^{2} or probit's pseudo R^{2}  0.128  0.112  0.547  0.499  0.086  0.083 
Notes: Robust standard errors are in parentheses. We reject the regression coefficient β = 0 at p < 0.01 (***); p < 0.05 (**); p < 0.1 (*)
The independent variables are as follows: indexed equity funds, which is an indicator variable for whether or not the manager has passive indexed equity funds; active equity funds; large, if the assets under management are above R$1bn; foreign, which describes the origin of the asset management company; bank, if the asset management company is affiliated with a private or public bank; hedge funds, which includes any type of hedge fund; and budgeting with relative and absolute risk indicates whether the manager sets a risk budget or not.
With regard to the regression results, Models (1) and (2) indicate that the only variable that explains relative risk budgeting with statistical significance is whether investment managers have passive or indexed equity funds. The presence of an index equity fund increases the probability of using a relative risk budget by 29.6 per cent according to Model (1) OLS and by 31.2 per cent according to Model (2) probit. All marginal probabilities with probit are estimated considering the other variables at their mean.
Models (3) and (4) of Table VII have significant explanatory power. The R^{2} and pseudo R^{2} values for the probit model are 0.50 and 0.55, respectively. The models show that both the presence of indexed equity funds and the use of relative risk budgets explain optimization with relative risk. The presence of indexed funds increases the probability of relative risk optimization from 32.4 per cent in the OLS model (3) to 59.3 per cent in the probit model (4). The use of relative risk budgeting increases the probability of relative risk optimization from 69.2 per cent in the OLS Model (3) to 77.0 per cent in the probit model (4).
Two variables explain portfolio optimization with absolute risk with statistical significance, according to Models (5) and (6) in Table VII. Neither of them is absolute risk budgeting. Respondents with hedge funds are 21.320.2 per cent more likely than respondents without hedge funds to perform this procedure, according to the (5) OLS and (6) probit models, respectively. Respondents from large companies are between 21.8 per cent and 22.4 per cent less likely than those from small companies to perform absolute risk optimization, according to Models (5) and (6), respectively.
Before we conclude, it is important to make three brief technical notes and address some endogeneity concerns. First, the probit model assumes that the errors in the probability of response follow a normal distribution. The LPM model is more appropriate in our case because it is more robust to specification errors, and because the independent variables are all binary, all estimated probabilities will be within the range of [0,1]; see ^{Angrist and Prischke (2009)}. Second, the addition of relative risk budgeting and indexed equity funds as independent variables in the same regression generates multicollinearity because both are significantly correlated. This may increase the standard errors, making it harder to detect statistical significance in the regressors, but it will not bias the regression coefficients. Finally, the constants of regressions (1) to (6) theoretically indicate the probability of an asset management company performing a risk budget or optimization if the company is small and local; has no index, active equity or hedge funds; and is not affiliated with a bank. In practice, however, there is arguably considerable noise in the determination of the constants, and therefore, we will refrain from interpreting the constants.
It could be argued that the use of budgeting and optimization is what causes an asset management company to be large, which would create endogeneity due to reverse causality. The argument that investors seek riskefficient portfolios is plausible and supported by studies in the USA. ^{Ippolito (1992)} and ^{Sirri and Tufano (1998)} found a negative impact of risk on fundraising. However, only a marginal impact was found using US data, and ^{Muniz (2015)} found no impact at all in Brazil.
In Models (3)(6) of Table VII, it is unlikely that optimization causes the risk budget and that models suffer from reverse causality. It seems more plausible that investment managers who need to adhere to risk limits imposed by customers rely on optimization to help them determine the portfolio with the highest expected return for a certain amount of risk.
Finally, since the presence of an indexed or passive equity fund increases the probability of budgeting or optimizing with relative risk, the 16 respondents who had equity index funds are more likely to use relative risk than the remaining respondents, and thus, H2 can be rejected.
As relative risk budgeting explains optimization with relative risk but absolute risk budgeting does not explain optimization with absolute risk, we can partially reject H5. Risk budgeting was found to affect only optimization with relative risk, not optimization with absolute risk.
Although we do not have all the necessary elements, we will speculate on a possible explanation for the partial rejection of H5. It may be the case that there are simple methods, such as limiting the maximum concentration per individual asset, that work sufficiently well for the management of absolute risk, but the same does not occur for the management of relative risk. An alternative, albeit longer, explanation is that it is well known that performance fees to asset management companies are generally tied to the excess return relative to the benchmark. Investment managers who do not perform optimization with relative risk may achieve lower excess returns, bring less revenue to the asset management company or fall outside of the preestablished risk limits. This would culminate in their dismissal, and investment managers who remain are those who optimize with relative risk. However, this may not be the case for portfolio managers who do not perform optimization with absolute returns, as performance fees are not usually linked to absolute returns.
4.4 Quantitative optimization and return distribution
Starting with H3, while only 1.3 per cent uses the shrinkage matrix, 34.6 per cent use RiskMetrics, the most popular method to produce a covariance matrix. Independence tests reject H3. Despite the excellent results in empirical tests reported by ^{Ledoit and Wolf (2003)} and ^{Santos and Tessari (2012)}, even in Europe, the method is used by only 3.9 per cent of managers (Table III). Attributing this low adoption to a costly learning curve does not seem reasonable since models such as GARCH are possibly more complex and are used by 17.9 per cent of managers.
Imposing asset concentration limits is a simple and intuitive method but is relatively arbitrary and without scientific backing, according to ^{Amenc et al. (2011)}. H4 tests whether this method is more popular than quantitative optimization. Of the respondents, 55.1 per cent performed optimization by absolute, relative risk or both, whereas 64.1 per cent set restrictions on asset concentration. The difference between the two proportions is not statistically significant, and we thus reject H4.
Only 5.1 per cent of managers use EVT, 7.7 per cent use distributions with higher moments and 9.0 per cent use at least one of the two methods to estimate VaR. Because 62.8 per cent of the sample, a much higher percentage, uses the normal distribution, we can reject H6. See Table VIII. According to ^{Cassettari (2001)} and ^{Leal and Ribeiro (2002)}, the preference for a normal parametric distribution is probably associated with simplicity and ease of use, but the asymmetry of the return distribution in the Brazilian market implies that simplification may lead to underestimation of the tail risk and may cause unpleasant surprises. Surprisingly, EVT and higher moments are more common in developed countries, where the return distribution resembles more a normal curve, than in Brazil (8.3 per cent and 16.6 per cent of respondents, respectively), according to Table III.
ID  Hypotheses  Tests  Results  

H3  The method most commonly used to determine the covariance matrix is shrinkage  Chisquare statistic  Hypothesis rejected  
H4  Investment managers use optimization methods more than the simple rule of establishing a maximum concentration limit per asset  Chisquare  Hypothesis not rejected  
H6  Investment managers use parametric distributions, such as extreme value theory or distributions with higher moments, more than the normal distribution to estimate VaR  Chisquare  Hypothesis rejected  
H7  Estimation risk management methods are more commonly used than the simple rule of imposing a maximum concentration per asset  Chisquare  Hypothesis rejected  
Tests performed  Hypotheses  pvalue  Sig*  Results  
T3  Comparison of the proportions of respondents using shrinkage (θ1) and those using RiskMetrics (θ2) in determining the covariance matrix  H0: θ_{1} = θ_{2} h1: θ_{1} < θ_{2} 
0.000  ***  H0 Rejected 
T4  Comparison of the proportions of respondents using quantitative optimization (θ1) and those using maximum concentration per asset (θ2) in the construction of portfolios  H0: θ_{1} = θ_{2} h1: θ_{1} > θ_{2} 
0.253  H0 Not rejected  
T6  Comparison of the proportions of respondents using extreme value theory or a distribution with upper moments (θ1) and those using the normal distribution (θ2)  H0: θ_{1} = θ_{2} h1: θ_{1} < θ_{2} 
0.000  ***  H0 Rejected 
T7  Comparison of proportions of respondents using advanced estimation risk management methods (θ1) and those using maximum concentration per asset (θ2)  h0: θ_{1} = θ_{2} h1: θ_{1} < θ_{2} 
0.000  ***  H0 Rejected 
T7b  Same variables as T7. Test performed in the subsample of 43 respondents who perform portfolio optimization  h0: θ_{1} = θ_{2} h1: θ_{1} < θ_{2} 
0.000  ***  H0 Rejected 
Note: We reject the null hypothesis at p < 0.01 (***)
Investment managers in Brazil are much less likely to adopt the selected methods to manage estimation risk than are their counterparts in Europe. The respective values are 5.1 per cent versus 17.0 per cent that adopt minimum variance portfolios, 7.7 per cent versus 15.3 per cent that adopt Bayesian methods and 9.0 per cent versus 13.5 per cent that adopt resampling. See Table III. Only 24.4 per cent use some method to manage estimation errors, thus leading us to reject H7. The results are robust for the subsample of 43 respondents who perform optimization with relative or absolute risk; see T7b in Table VIII.
4.5 Riskadjusted versus unadjusted return
Previous tests suggest that there is a significant gap between best practices and current practices in the Brazilian asset management industry. One could argue that the comparison is of little use, as the ideas conveyed in academic publications will always be years ahead of the market. If best practices take years to be fully understood, disseminated and implemented and new ideas are constantly being created, then a gap should naturally be expected (Table IX).
ID  Hypotheses  Tests  Results  

H8  Riskadjusted returns are used more than unadjusted returns  Chisquare statistic  Hypothesis rejected  
Tests performed  Hypotheses  pvalue  Sig  Results  
T8  Comparison of proportions of respondents using riskadjusted (θ1) and those using unadjusted (θ2) returns  h0: θ_{1} = θ_{2} h1: θ_{1} < θ_{2} 
0.151  H0 not rejected 
The analysis of riskadjusted returns was introduced in ^{Treynor (1965)}; ^{Sharpe (1966)} and ^{Jensen (1968)} and cannot be considered new. Furthermore, it cannot be considered very complex, since in Europe, according to ^{Amenc et al. (2011)}, 77 per cent of managers use the Sharpe index and 65 per cent use the information ratio. They are the most commonly used metrics to evaluate performance in Europe, according to Table III.
Although it is relatively old and not very complicated concept, the riskadjusted return is not as popular as the unadjusted return in Brazil, according to the survey, and we can thus reject H8. Of the Brazilian managers, 91.0 per cent use absolute return or excess return relative to the benchmark; these are measures of return not directly adjusted for risk. In comparison, 83.3 per cent use at least one riskadjusted return measure such as the Sharpe or information ratio.
5. Final remarks
This paper aims to identify a possible mismatch between the theory found in academic research and the practices of investment managers in Brazil. For this purpose, a bibliographical and field survey was carried out with 78 respondents to a questionnaire posted online, out of a total of 274 asset management companies. This study may be considered a pioneering work in portfolio construction, risk management and performance evaluation in Brazil.
The results of the tests performed indicate that practice departs from theory in the country: of the eight hypotheses tested, we rejected seven hypotheses and partially rejected one hypothesis. One possible explanation is that few Brazilian academic studies consider transaction costs such as brokerage fees, bidask spreads and liquidity when studying the benefits of quantitative portfolio optimization. ^{Santos and Tessari (2012)} and ^{Caldeira et al. (2013)}, for example, considered three types of rebalancing, daily, weekly and monthly, which would generate very high turnover and cost.
Compared with investment managers in Brazil, those in Europe seem to be closer to the best practices propagated by academia. During the data collection process, some of the respondents mentioned that the low liquidity and quantity of financial assets in Brazil did not justify the use of quantitative methods for portfolio construction. Notably, the European managers researched by ^{Amenc et al. (2011)} represented companies with greater assets under management overall.
This work aims not only to suggest improvements to practitioners but also to help researchers better understand the reality in Brazil. As an example, although ^{Santos and Tessari (2012)} noted that the sample covariance matrix is seldom used by practitioners, only RiskMetrics is more popular than the sample covariance matrix.
Perhaps the most important contribution in this sense is that managers attach great importance to relative risk and return, but few empirical studies focusing on relative risk were found. In future work, we suggest that researchers compare optimized portfolios for tracking error, benchmark VaR and other optimization methods with relative risk. We also suggest studies that help determine how international diversification or the financial education of clients of asset management companies relates to the adoption of sophisticated methods for portfolio construction, risk management, and performance evaluation. Because the present study is limited in that it reflects the state of the industry in only a short time interval, longitudinal studies are suggested because they do not suffer from this restriction.
This paper also aimed to investigate the determinants of the risk budget and the optimization of portfolios. The only variable that significantly explained risk budgeting was the presence of passive or indexed equity funds. For portfolio optimization, the key explanatory variables differed depending on the type of risk: absolute or relative risk. While the presence of passive equity funds and risk budgeting increases the likelihood of relative risk optimization, the presence of hedge funds and assets under management of less than R$1bn increases the probability of absolute risk optimization.