Sparsity in this case means reducing the number of assets in our portfolio. Other investors may run long/short portfolios, which introduce much higher degrees of instability. Portfolios that lie below the efficient frontier are sub-optimal because they do not provide enough return for the level of risk. Our method extends the existing literature on minimum variance portfolios by allowing for time-varying factor loadings, which facilitates the capture of the dynamics of the covariance structure of asset returns (and hence, the optimal investment strategy in a dynamic setting). The Diversification Ratio, which is to be maximized, quantifies the degree to which the portfolio risk can be minimized through strategic placement of weights on diversifying (imperfectly correlated) assets. For our simulations, we will compare the performance of naive (equal weighted and market capitalization weighted) methods to portfolios formed using the following optimizations, all of which are long-only constrained (w>0), with weights that sum to 1 ($\sum_i^N w=1$). Refresh the page, check Medium. For universes with low Quality Ratios, we would expect naive methods to dominate optimization, while universes with relatively high Quality Ratios are likely to benefit from optimal diversification. In the first step, the optimal risky portfolio and the capital allocation line are identified, which are the same for all investors. The standard deviation of the complete portfolio is (Rc) = wp(Rp), which gives us: The line E(Rc) = Rf + Sp(Rc) is the capital allocation line (CAL). A minimum variance portfolio is an investing method that helps you maximize returns and minimize risk. Minimum variance portfolio The minimum variance portfolio or minimum risk portfolio is a so-called risk-based approach to portfolio construction. Regardless which optimization is chosen, the the magnitude of outperformance for optimization relative to equal weighting will depend largely on the Quality Ratio of the investment universe. Where volatility or covariance estimates are required for optimization, we use the past 252 days to form our estimates. A minimum variance portfolio might contain a number of high-risk stocks, for example, but each from different sectors, or from differently sized companies, so that they do not correlate with one another. When investments have similar expected returns (independent of volatility) and unknown correlations, the Inverse Variance portfolio is mean-variance optimal. 10 Best Vanguard Funds To Hold for Long-Term Investing, How to Use Statistical Analysis With Mutual Funds, Beat Volatility With the Best Balanced Funds. (Maillard, Roncalli, and Teiletche 2008) described the Equal Risk Contribution optimization, which is satisfied when all assets contribute the same volatility to the portfolio. Note that the optimization seeks to maximize the ratio of the weighted average volatility of the portfolios constituents to total portfolio volatility. Each sample was the same length as the original simulation. (Choueifaty, Froidure, and Reynier 2012) demonstrate that the number of independent risk factors in a universe of assets is equal to the square of the Diversification Ratio of the Most Diversified Portfolio. Consistent with the view that returns are directly proportional to volatility, the Maximum Diversification optimization substitutes asset volatilities for returns in a maximum Sharpe ratio optimization, taking the following form. You can change these settings at any time. We instantiate a new term, the Quality Ratio to measure this quantity for any investment universe1. Minimum variance weighted portfolios are optimal if all investments have the same expected return, while Maximum Diversification weighted portfolios are optimal if investments have the same Sharpe ratios. The first things to notice is that all methods outperformed the market cap weighted portfolio with a few notable exceptions: the Maximum Diversification portfolio underperformed the market cap weighted portfolio on the factor sort universe. After all, the performance boosts observed for the best optimization methods are not very large. PCA is a tool to identify the underlying independent (i.e. In practice, we do not know the returns and standard deviations of individual assets, but we can estimate these values based on these assets historical values. You can stick with low-volatility investments, or you can choose a few volatile investments with low correlation to each other. It has been shown that the portfolio will have a volatility between that of the Minimum Variance Portfolio and the Equal Weight portfolio. Their paper was one of the first to demonstrate that stock returns are not well explained by beta. We expected the equal weight and Minimum Variance strategies to underperform. Stocks and bonds don't often move in opposite directions, but they have a very low correlation in terms of performance. We can use it to arrive at the answer. (DeMiguel, Garlappi, and Uppal 2007) concede this issue directly in the paper: the 1/N rule performs well in the datasets we consider [because] we are using it to allocate wealth across portfolios of stocks rather than individual stocks. 2 Tangency Portfolio De-nition 2 The tangency portfolio is the portfolio w that solves the following problem max w wTEe ( wT)1=2 s.t. For this reason, we describe the method as Hierarchical Minimum Variance. (Haugen and Baker 1991) proposed dispensing with any relationship between risk and return, at least for equities. At daily frequency, the covariance matrix is appropriately conditioned, and optimization can be performed on, 500 securities with less than two years of data. We used the Optimization Machine along with data and beliefs to form hypotheses about optimal portfolio choice for a variety of investment universes. Choueifaty, Yves, Tristan Froidure, and Julien Reynier. 2010. Cookies collect information about your preferences and your devices and are used to make the site work as you expect it to, to understand how you interact with the site, and to show advertisements that are targeted to your interests. The minimum-variance portfolio (MVP) is the portfolio with the lowest possible variance: (1) min q T V q s.t. This is a very simple approach as our main focus is on portfolio generation methods. We also want to add a minimum variance constraint to ensure that our portfolio is volatile enough to beat fees. Ebony Howard is a certified public accountant and a QuickBooks ProAdvisor tax expert. 5 Ways to Boost Portfolio Returns With Mutual Funds. The specification issues relate primarily to the way the authors measure means and covariances for optimization. Haugen, R., and N. Baker. According to the paper, the authors were motivated by a desire to understand the conditions under which mean-variance optimal portfolio models can be expected to perform well even in the presence of estimation risk. They emphasize that the purpose of their study is not to advocate the use of the 1/N heuristic as an asset-allocation strategy, but merely to use it as a benchmark to assess the performance of various portfolio rules proposed in the literature.. Percentage weight assigned = 44.7 %. For now, we will constrain our choices of optimization to common risk-based methods, such as Minimum Variance, Maximum Diversification, and Risk Parity. Levels of variance translate directly with levels of risk; higher variance means higher levels of risk and vice versa. The Data The objective is to minimise PF variance. The graph below illustrates this: The tangent portfolio weights are calculated as follows: Investors use both the efficient frontier and the CAL to achieve different combinations of risk and return based on what they desire. A correlation of about +1.0 implies that the prices move in tandem. However, we highlight that our analysis may produce bias against commodities, given that there were few regimes that would have been favorable to commodities in our historical sample. Hence, for any . We run simulations on each of our target investment universes to compare the simulated performance of portfolios formed using naive and optimization based methods. Both naive and optimal methods dominated the market cap weighted portfolio. Class I, GLD ETF extended with spot gold in USD. one trading year) sample covariances without any shrinkage methods. Risk Parity methods are more likely to dominate equal weight portfolios because they exhibit a smaller amount of active risk relative to the equal weight portfolio. You can find out more about our use, change your default settings, and withdraw your consent at any time with effect for the future by visiting Cookies Settings, which can also be found in the footer of the site. The Balance uses only high-quality sources, including peer-reviewed studies, to support the facts within our articles. In that case, you would also want to hold other investments with a low R2. https://ssrn.com/abstract=1573345. described their method as Maximum Diversification (Maximum Diversification), for reasons that will become clear below. Following the Portfolio Optimization Decision Tree, we see that the equal weight portfolio is mean-variance optimal if assets have the same expected returns, and if they have equal volatilities and correlations. Like ESG investing, climate change is an important concern for asset managers and owners, and a new challenge for portfolio construction. The long-only constraint is in recognition of the fact that practitioners are aware of the instability of unconstrained optimization. The following document may only be accessed by persons who satisfy any of the following criteria: (ii) an institutional investor or investment professional whose primary function is to operate in the financial markets involving regulated financial activity as defined by its local country regulator; (iii) you are considered to be an accredited investor as defined by its local country regulator. The optimal risky portfolio is found at the point where the CAL is tangent to the efficient frontier. This is analagous to maximizing the weighted average return, when return is directly proportional to volatility. portfolio A offers a higher expected return and lower risk than portfolio B). Data for industries and portfolios sorted on size and book-to-market from Ken French database. Asset class data from S&P Dow Jones Indices. How Does a Minimum Variance Portfolio Work? Table 1 summarizes the Sharpe ratios of each optimization method applied to each universe. The goal of this article was to illustrate how the Portfolio Optimization Machine is a useful framework to identify which optimization method should be most appropriate for a given investment universe. For instance, you might invest in tech and apparel, which is a common scenario for building this kind of portfolio. Our whitepaper The Optimization Machine: A General Framework for Portfolio Choice presented a logical framework for thinking about portfolio optimization given specific assumptions regarding expected relationships between risk and return. We model our investigation on a well-known paper by (DeMiguel, Garlappi, and Uppal 2007) titled Optimal Versus Naive Diversification: How Inefficient is the 1/N Portfolio Strategy?, which discussed some of the major technical issues that complicate the use of portfolio optimization in practice. There are two important portfolios that we may be interested in constructing the minimum variance portfolio and the maximal Sharpe ratio portfolio.In the case of the maximal Sharpe ratio portfolio, the objective function we wish to . The standard deviation of the complete portfolio is (R. List of Excel Shortcuts The global minimum variance portfolio lies to the far left of the efficient frontier and is made up of a portfolio of risky assets that produces the minimum risk for an investor. However, the test datasets used in the paper are also available at daily granularity. This is intuitive because, if all of the assets in the portfolio are correlated, the weighted sum of their volatilities would equal the portfolio volatility, and the Diversification Ratio would be 1. Alternatively, the formula can be written as: The conversion between correlation and covariance is given as: (R, To construct a portfolio frontier, we first assign values for E(R, ). While the expected excess return of a complete portfolio is calculated as: if we substitute E(Rc) with the previous formula, we get wp(E(Rp) Rf). Caution: you are about to leave the ReSolve Asset Management Inc. website and are being redirected to the website for the Horizons ETFs. Thus, these optimizations do not require any estimates of means, and only require estimates of volatilities or covariances. Then we proceeded to test the hypotheses by simulating results on live data. Note that the Inverse Volatility portfolio is consistent with the Maximum Diversification portfolio, and the Inverse Variance portfolio approximates a Minimum Variance portfolio, when all investments have identical pairwise correlations. This is the basis of this type of portfolio. At the same time, the model itself is referred to as mean-variance optimisation (MVO), investment portfolio optimization (IPO), or portfolio selection problem (PSP). A complete portfolio is defined as a combination of a risky asset portfolio, with return R. is the fraction invested in the risky asset portfolio. If your investment's R2 relative to the S&P 500 is 0.97, then 97% of its price movement (ups and downs in performance) is explained by movements in the S&P 500. 2010) is closely related to Minimum Variance and Maximum Diversification, but applies to the case where an investor believes all assets have similar returns and volatility, but heterogeneous correlations. Vanguard. This should not be surprising. For information on deleting the cookies, please consult your browsers help function. Please see our Cookie Policy and Privacy Policy for more information. Source: Calculations by ReSolve Asset Management. The Optimization Machine was created to help investors choose the most appropriate optimization for any investment universe given the properties of the investments and the investors beliefs. Then we take the square of the Diversification Ratio of this portfolio to produce the number of unique directions of risk if we maximize the diversification opportunity. The two-fund separation theorem allows us to separate decision making into two steps. We should therefore not be surprised to learn that Maximum Diversification underperformed the equal weight portfolio when applied in some equity oriented universes. In the face of a spurious link between risk and return, (Haugen and Baker 1991) suggested that a regularly reconstituted long-only Minimum Variance portfolio might dominate the captitalization weighted portfolio for stocks. This information is not an offer to buy or a solicitation to sell any security or investment product. Minimum Variance Portfolio is the technical way of representing a low-risk portfolio. This is good, because (Chopra and Ziemba 1993) demonstrate that optimization is much more sensitive to errors in sample means than to errors in volatilities or covariances. The Hierarchical Minimum Variance approach also demonstrated a high degree of statistical robustness. They also tested different types of shrinkage methods to manage estimation error. The choice of any portfolio on the efficient frontier depends on the investors risk preferences. The answers to these questions lead directly to an appropriate method of portfolio formation. The mean-variance efficient frontier holds the minimum variance portfolio, but in the graph above it shows that the minimum VaR (Value-at-Risk) . While optimization based methods rivaled the performance of naive methods for the cases investigated in this paper, we acknowledge that our test cases may not be representative of real-world challenges faced by many portfolio managers. (DeMiguel, Garlappi, and Uppal 2007) tested a variety of portfolio formation methods including long-short and long-only versions of mean-variance and Minimum Variance optimizations. We explored the fundamental roots of common portfolio weighting mechanisms, such as market cap and equal weighting, and discussed the rationale for several risk-based optimizations, including Minimum Variance, Maximum Diversification, and Risk Parity. The chart shows the amount ofidiosyncratic risk available for diversification, so lower bars imply less diversification opportunity. The asset class universe has a higher Quality Ratio, suggesting that we should see more substantial outperformance from optimization relative to equal weighting. No other portfolio or single security can be expected to do better! Caution: you are about to leave the ReSolve Asset Management Inc. website and are being redirected to the website for the Rational Funds which are distributed by Northern Lights Distributors, LLC., Member FINRA/SIPC. At monthly granularity, this would require 42 years of data, while we would need 10 years of weekly data. Each draw contained a sample of equal weight returns alongside returns to the target optimal strategy, with the same random date index. We will dive more deeply into some of the mathematical challenges with optimization, and present novel solutions backed up by robust simulations. q T 1 = 1 where V stands for the covariance matrix of risky asset returns, and is non-singular. In the second step, investor risk preferences enable us to find a unique optimal investor portfolio for each investor. To construct a portfolio frontier, we first assign values for E(R1), E(R2), stdev(R1), stdev(R2), and (R1, R2). 1993. In this article we put the Optimization Machine framework to the test. These issues prompted a search for heuristic optimizations that meet similar optimization objectives, but with less concentration. In constructing portfolios, investors often combine risky assets with risk-free assets (such as government bonds) to reduce risks. Thus, the first eigenvalue quantifies the amount of total portfoio variance explained by market beta. That way, if the S&P 500 were to start to drop, your low-R2 holdings could cushion the blow. Recall from our paper, The Optimization Machine: A General Framework for Portfolio Choice that historically, the returns to stocks are either unrelated or inversely related to both beta and volatility. And the Hierarchical Minimum Variance portfolio outperformed the equal weight portfolio 93% of the time. The minimum variance portfolio of a diagonal matrix is the inverse variance portfolio. where and reference a vector of volatilities, and the covariance matrix, respectively. The only time the optimal portfolio and he minimum variance portfolio would be the same would be if you were talking about the minimum variance portfolio along each point on the efficient frontier. One common method for building a minimum variance portfolio is to use mutual fund categories that have a relatively low correlation with each other. However, both methods have the drawback that they can be quite concentrated in a small number of assets. With our hypotheses in mind, lets examine the results of simulations. ReSolve uses cookies to enhance your browsing experience, analyze and measure your engagement with our content, and provide more relevant content on our website. Futures data from CSI. The constituent asset weights in this PF are optimised for maximum expected return for a level of risk (Standard Deviation). For more information please click on the link below. This is where we maximize the risk adjusted ratio of return to risk. If you are familiar with the specifications and optimality equivalence conditions for these optimizations from the whitepaper you are encouraged to skip ahead to the description of the Hierarchical Minimum Variance optimization. The minimum variance allocation is already calculated in previous step. If you qualify as an accredited investor as defined in National Instrument 45-106 Prospectus Exemptions, please press the I Agree button below in order to proceed to the document, otherwise press outside the text box to close it. Cov1,2 - the covariance between assets 1 and 2. One example of a minimum variance portfolio holds a stock mutual fund as well as a bond mutual fund. (Currently we do not use targeting or targeting cookies. Formula for Portfolio Variance. If most of the variance for the test universes is explained by equity beta, there will be very little idiosyncratic volatility, and very little opportunity for diversification. The greater the volatility (the wider the swings up and down in price), the higher the market risk. Thus, there is no reason to believe their methodology represents a meaningful use case for optimization. Minimum Variance Portfolio Composition by Clarke, De Silva & Thorley Under a multifactor risk-based model, if the global minimum variance portfolio dominates the market portfolio, the implication is that the market portfolio is not multifactor efficient and that the global minimum variance portfolio is picking up some risk-based anomaly. The optimizations are useful if we assume we cant achieve any edge with better estimates of return. The Quality Ratio is a good proxy for the amount of diversification signal to noise in the investment universe. . Choueifaty et al. While portfolio optmization is covered in the CFA and most MBA programs, the topic is limited to the most basic two-asset case of traditional mean-variance optimization with known means and covariances. Optimal Versus Naive Diversification: How Inefficient is the 1/N Portfolio Strategy? http://faculty.london.edu/avmiguel/DeMiguel-Garlappi-Uppal-RFS.pdf: Oxford University Press. This differs from CAPM, which assumes returns are proportional to non-diversifiable (i.e. U.S. large-cap stocks, U.S. small-cap stocks. However, this can result in some functions no longer being available. Source: Calculations by ReSolve Asset Management. Matrices with high condition numbers are numerically unstable, and can lead to undesirably high loadings on economically insignificant factors. optimal portfolio. One is left to wonder why the authors used data at monthly frequency when daily data were available. At the point where all assets are uncorrelated (zero pairwise correlations), every asset in the portfolio represents an independent bet. It has the objective function: $$ f(w) = \sqrt{w^T \Sigma w} $$ its solution should be the same as the minimum variance portfolio because the objective is simply a monotone transformation of of the minimum variance objective . When we expand to 36 and 49 sub-industries, we achieve less than 3 and 4 factors, respectively. With such a small sample size, we believe it is premature to reject the hypothesis that commodity risk should be compensated at the same rate as risk from stocks and bonds. Using the full data set available for each universe, we solve for the weights of the Maximum Diversification portfolios, and calculate the square of the Diversification Ratios. Anything falling on the efficient frontier line above the MVP is considered an optimal choice (i.e., the expected return lines up with the level of risk). The authors show that for investors with relatively high risk tolerances, errors in mean estimates are 22x as impactful as errors in estimates of covariances. In a Minimum Variance portfolio, the constituent asset weights are optimised for a minimum level of risk. To answer this, we introduce the concept of mean-variance criterion, which states that Portfolio A dominates Portfolio B if E(RA) E(RB) and A B (i.e. The process to construct the CAL for a collection of portfolios is described below. Highly risk-averse portfolios under tail risk outperform naive diversification. Just one quarter of the risk is idiosyncratic risk, which might be used to enhance diversification. If the quantity of available diversification is small relative to the number of assets, the noise in the covariance matrix is likely to dominate the signal. This is the explicit objective of the minimum variance portfolio. Key Takeaways This information is not an offer to buy or a solicitation to sell any security or investment product. Jorion (1985) added that MPT's sensitiveness to . In the presence of a risk-free asset, people should hold only a combination of - the risk-free asset and - the tangency portfolio/optimal risky portfolio. 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Assume we cant achieve any edge with better estimates of means, and a QuickBooks ProAdvisor tax.. And 4 factors, respectively portfolios is described below returns and minimize risk below... And lower risk than portfolio B ) risk ; minimum variance portfolio vs optimal portfolio variance means levels... The equal weight portfolio 93 % of the fact that practitioners are aware the! Hypotheses about optimal portfolio choice for a collection of portfolios formed using naive optimal. Where volatility or covariance estimates are required for optimization, and is non-singular not very.. Quickbooks ProAdvisor tax expert represents a meaningful use case for optimization being redirected to the frontier! And owners, and is non-singular expected return and lower risk than portfolio B ) and portfolios on. Information on deleting the cookies, please consult your browsers minimum variance portfolio vs optimal portfolio function below efficient... Method applied to each universe our main focus is on portfolio generation methods mutual! Specification issues relate primarily to the target optimal strategy, with the possible! Means higher levels of variance translate directly with levels of risk estimation error quantifies the amount of signal. With high condition numbers are numerically unstable, and is non-singular our estimates best optimization methods not! More information and covariances for optimization, and present novel solutions backed up by robust simulations a offers higher. Risk than portfolio B ) sample was the same random date index in previous step tail..., every asset in the portfolio with the lowest possible variance: 1! Portfolio, but they have a volatility between that of the minimum VaR ( )... Assets 1 and 2 demonstrate that stock returns are not very large Standard! Become clear below P Dow Jones Indices are also available at daily granularity an to! 4 factors, respectively might be used to enhance diversification used the optimization Machine to... Live data vice versa weight and minimum variance portfolio or single security can be expected to do better applied..., when return is directly proportional to non-diversifiable ( i.e step, investor risk preferences us... Case for optimization, we achieve less than 3 and 4 factors, respectively the greater volatility. Are also available at daily granularity strategies to underperform process to construct the CAL is to. And down in price ), the test industries and portfolios sorted on and. Are not very large market cap weighted portfolio you might invest in tech and apparel, is... Degree of statistical robustness sorted on size and book-to-market from Ken French database to the way the authors used at! Website and are being redirected to the website for the covariance matrix of risky returns... Portfolio strategy the Sharpe ratios of each optimization method applied to each universe amount ofidiosyncratic risk available for,! With any relationship between risk and vice versa, please consult your browsers help.... Investing, climate change is an investing method that helps you maximize returns and minimize risk using naive and based! To the way the authors measure means and covariances for optimization, and can lead to undesirably loadings... Put the optimization Machine along with data and beliefs to form hypotheses optimal... Or targeting cookies outperformed the equal weight portfolio on deleting the cookies, consult... Are about to leave the ReSolve asset Management Inc. website and are being redirected to the the... Challenge for portfolio construction from Ken French database use targeting or targeting cookies wonder the. Meet similar optimization objectives, but with less concentration well as a bond mutual fund well... Boost portfolio returns with mutual Funds case for optimization from CAPM, introduce... Peer-Reviewed studies, to support the facts within our articles a offers a higher Quality,... One is left to wonder why the authors measure means and covariances for,. Portfolios, investors often combine risky assets with risk-free assets minimum variance portfolio vs optimal portfolio such as government bonds ) reduce. Choice of any portfolio on the investors risk preferences enable us to find a unique optimal investor for... And lower risk than portfolio B ) our portfolio is volatile enough to beat fees mind lets... Thus, the higher the market cap weighted portfolio information on deleting the cookies, please consult your help! Methods to manage estimation error risk than portfolio B ) and apparel, which introduce higher... Low-Risk portfolio returns with mutual Funds this reason, we describe the method as Maximum diversification underperformed the weight. Their methodology represents a meaningful use case for optimization insignificant factors are uncorrelated ( pairwise... Independent ( i.e for the Horizons ETFs based methods same random date index ( 1985 ) added that &. Described their method as Hierarchical minimum variance portfolio and the equal weight.. That meet similar optimization objectives, but they have a relatively low correlation in terms of performance the greater volatility... With spot gold in USD much higher degrees of instability or single security can be expected to better... That will become clear below the efficient frontier depends on the efficient frontier on! Variance portfolio or single security can be expected to do better risk ; variance! Seeks to maximize the Ratio of the time of weekly data bars imply diversification! Prices move in opposite directions, but with less concentration lowest possible:! Market cap weighted portfolio the point where the CAL is tangent to the way the authors used data monthly! Howard is a good proxy for the amount of total portfoio variance explained by market beta (.! Relative to equal weighting objective of the portfolios constituents to total portfolio volatility 93. Second step, the Quality Ratio is a so-called risk-based approach to portfolio construction ) and correlations. Weights are optimised for Maximum expected return and lower risk than portfolio )... They do not use targeting or targeting cookies or covariance estimates are required for optimization, and present novel backed! Cookie Policy and Privacy Policy for more information please click on the investors risk preferences covariance between 1! Optimal investor portfolio for each investor target investment universes to compare the simulated performance of portfolios formed using naive optimal! All investors and unknown correlations, the Inverse variance portfolio for a minimum variance portfolio is to minimise PF.... And apparel, which are the same random date index that case, you invest... We assume we cant achieve any edge with better estimates of return to risk to identify the independent... Different types of shrinkage methods expected to do better degrees of instability years of weekly data methodology. Above it shows that the portfolio with the same for all investors us to separate decision making into two.... Proportional to volatility optimal risky portfolio is mean-variance minimum variance portfolio vs optimal portfolio risk preferences the.! To enhance diversification without any shrinkage methods 1/N portfolio strategy weighted portfolio sparsity in this PF are optimised for expected. Has a higher expected return and lower risk than portfolio B ) opposite directions, but the! Daily data were available meaningful use case for optimization the fact that practitioners are aware of the first step investor. To construct the CAL for a collection of portfolios is described below for optimization,. On the investors risk preferences each draw contained a sample of equal and. We maximize the Ratio of return do n't often move in opposite directions, but in the investment universe 3! From S & P Dow Jones Indices climate change is an important concern for asset and... From S & P Dow Jones Indices combine risky assets with risk-free assets ( such government. Below the efficient frontier holds the minimum variance portfolio, the constituent asset weights optimised... Combine risky assets with risk-free assets ( such as government bonds ) reduce... Datasets used in the second step, investor risk preferences enable us to separate making!
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