minimum variance portfolio vs optimal portfolio

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, ). 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). Lastly, the individual asset returns should be far from equal, since the higher risk assets should have higher returns. At the point where all assets are uncorrelated (zero pairwise correlations), every asset in the portfolio represents an independent bet. 3.2 Using the Solver function in Excel, next calculate the exact weight in WFC and MSFT for the minimum variance portfolio, rounded to the nearest tenth decimal point. We will address shrinkage methods in a later article when we discuss more robust optimization methods. When the Quality Ratio is high we would expect optimization methods to dominate naive methods. Sergio Torrico diana Marwan Benjamin anonymous We find that the 10 industry portfolios; 25 factor portfolios; 38 sub-industry portfolios; and 49 sub-industry portfolios produce 1.4, 1.9, 2.9, and 3.7 unique sources of risk, respectively. 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. In the second step, investor risk preferences enable us to find a unique optimal investor portfolio for each investor. Recall that (Choueifaty and Coignard 2008) showed that the Diversification Ratio of a portfolio is the ratio of the weighted sum of asset volatilities to the portfolio volatility after accounting for diversification. 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. We then compared the Sharpe ratio of each sample from equal weight returns to the Sharpe ratio of the sample of optimal weight returns. 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. At monthly granularity, this would require 42 years of data, while we would need 10 years of weekly data. The formula for portfolio variance in a two-asset portfolio is as follows: Portfolio variance = w 12 12 + w 22 22 + 2w 1 w 2 Cov 1,2 Where: w 1 = the portfolio weight of the first asset. The choices invoked by the Portfolio Optimization Machine produced superior results. (Lopez de Prado 2016) asserts that the correlation structure contains ordinal information, which can be exploited by organizing the assets into a hierarchy. When you diversify a portfolio, you are seeking to reduce volatility. Or in theory, the investor can optimize the results based on a function of risk aversion. 1991. Compare these to the Quality Ratio of our asset class universe at 0.42. This information is not an offer to buy or a solicitation to sell any security or investment product. (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. Our belief that diversified asset classes should have equal long-term Sharpe ratios led us to hypothesize that the Maximum Diversification portfolio should dominate in the asset class universe. Stocks and bonds don't often move in opposite directions, but they have a very low correlation in terms of performance. . The slope of the line, Sp, is called the Sharpe ratio, or reward-to-risk ratio. Efficient Frontier: The efficient frontier is the set of optimal portfolios that offer the highest expected return for a defined level of risk or the lowest risk for a given level of expected return . In contrast, about two-thirds and four-fifths of the risk in the asset class and futures universe, respectively, are derived from sources other than the first principal component. (expected return/volatility) or the minimum -variance portfolioboth of which are asset optimization techniques. 1993. http://www.tobam.fr/inc/uploads/2014/12/TOBAM-JoPM-Maximum-Div-2008.pdf: 4051. This is where we maximize the risk adjusted ratio of return to risk. https://ssrn.com/abstract=2297383. 2016. The number of positive weight assets is five.Moreover, we can obtain that is equal to That optimal risky portfolio provides us the greatest return per unit of risk. ReSolve sub-advises a mutual fund for Horizons that applies the Adaptive Asset Allocation methodology at an 8% volatility target. In fact, they observed a negative relationship between returns and volatility. There are always the same number of eigenvalues and eigenvectors as investments, so a universe of ten investments will be decomposed into ten eigenvectors with associated eigenvalues. The values in Table 2 represent the proportion of samples where the Sharpe ratio for samples of equal weight returns exceeded the Sharpe ratio for samples of optimal strategy returns. 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. 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. Choueifaty, Yves, Tristan Froidure, and Julien Reynier. Finance questions and answers. The global minimum variance portfolio lies to the right of the efficient frontier Solution The correct answer is B. For each approach to portfolio choice we examined the conditions that would render the choice mean-variance optimal. To use this tactic to its fullest extent, you can combine risky assets. 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). q T 1 = 1 where V stands for the covariance matrix of risky asset returns, and is non-singular. 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). Chopra, Vijay K., and William T. Ziemba. The conversion between correlation and covariance is given as: (R1, R2) = Cov(R1, R2)/ 12. You can see that about three-quarters of the variance in the industry and factor sort universes is explained by the first principal component, which represents U.S. equity beta. 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. A complete portfolio is defined as a combination of a risky asset portfolio, with return Rp, and the risk-free asset, with return Rf. Even though it seems that gold is dominated by stocks, gold might still be an attractive asset to hold as a part of a portfolio. How Does a Minimum Variance Portfolio Work? You can stick with low-volatility investments, or you can choose a few volatile investments with low correlation to each other. Since variance represents risk, the portfolio risk is lower when its asset components possess negative covariance. A minimum variance portfolio is an investing method that helps you maximize returns and minimize risk. One is left to wonder why the authors used data at monthly frequency when daily data were available. Financial Modeling & Valuation Analyst (FMVA), Commercial Banking & Credit Analyst (CBCA), Capital Markets & Securities Analyst (CMSA), Certified Business Intelligence & Data Analyst (BIDA), Financial Planning & Wealth Management (FPWM), Capital Allocation Line (CAL) and Optimal Portfolio, are the respective weights for the two assets, and E(R. ) represents the covariance of the two asset returns. When it is low, investors should expect only a very small boost in risk-adjusted performance from using more sophisticated techniques. From equities, fixed income to derivatives, the CMSA certification bridges the gap from where you are now to where you want to be a world-class capital markets analyst. document.getElementById( "ak_js_1" ).setAttribute( "value", ( new Date() ).getTime() ); Thank you for reading CFIs guide on Capital Allocation Line (CAL) and Optimal Portfolio. Maximum Decorrelation described by (Christoffersen et al. Given that the empirical relationship between risk and return has been negative, we might expect optimizations that are optimal when the relationship is positive to produce the worst results. Source: Calculations by ReSolve Asset Management. When investments have similar expected returns (independent of volatility) and unknown correlations, the Inverse Variance portfolio is mean-variance optimal. For more information please click on the link below. ReSolve uses cookies to enhance your browsing experience, analyze and measure your engagement with our content, and provide more relevant content on our website. If returns over the past 5-10 years are mean-reverting over the horizon chosen by (DeMiguel, Garlappi, and Uppal 2007) to estimate portfolio means, we shoud expect performance of optimal portfolios to be disappointing, as the return forecasts for portfolio optimization would be above average for periods that should actually produce below-average returns, and vice versa. Question 1: Refer to Step 3.3. An optimal portfolio is a mean-variance efficient portfolio. To provide what may be more practical insights, we also run simulations on a universe of global asset classes that derive their returns from diverse risk sources, such as regional equity indexes, global bonds, and commodities. For this reason, we describe the method as Hierarchical Minimum Variance. Learn more about the cookies we use. This doesnt mean investors should always eschew optimizations with active views on returns; rather, that investors should take steps to minimize the error term in general. 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 ($\sum_i^N w=1$). Specifically, we make predictions using the Optimization Machine about which portfolio methods are theoretically optimal based on what weve learned about observed historical relationships between risk and return. We should therefore not be surprised to learn that Maximum Diversification underperformed the equal weight portfolio when applied in some equity oriented universes. The Structured Query Language (SQL) comprises several different data types that allow it to store different types of information What is Structured Query Language (SQL)? The process to construct the CAL for a collection of portfolios is described below. After all, the performance boosts observed for the best optimization methods are not very large. In future articles we will explore more challenging problems involving lower quality investment universes with more typical constraints. We can use it to arrive at the answer. Sparsity in this case means reducing the number of assets in our portfolio. 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. The minimum variance allocation is already calculated in previous step. 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. Market factor plus the risk free rate from Ken French database. We will follow a similar process, but we will impose long-only, sum-to-one constraints for all optimizations, and use rolling 252 day (i.e. Optimal portfolio is a term used in portfolio theory to refer to the one portfolio on the Efficient Frontier with the highest return-to-risk combination given the specific investor's tolerance for risk. A graph depicting the risk-and-reward profile of risky assets. These papers conclude that for period lengths of between 3 and 5 years (i.e. The answers to these questions lead directly to an appropriate method of portfolio formation. This is curious for a number of reasons. Document Information click to expand document information. Minimum Variance Portfolio: the portfolio that bears the least risk given the constrains that the example spreadsheet used in the solver add-on . 1 is an N -column vector with all elements equal to one. Figure 4: Quality Ratio: Number of independent bets / number of assets. Optimization based methods dominated naive equal weighted methods in most cases, except where an optimization expressed relationships between risk and return that were precisely converse to what was observed in the historical record. When investments have similar expected Sharpe ratios, and an investor cannot reliably estimate correlations (or we can assume correlations are homogeneous), the optimal portfolio would be weighted in proportion to the inverse of the assets volatilities. They won't rise and fall based on what the S&P 500 does. It can hold investment types that are volatile on their own but when combined create a diversified portfolio with lower volatility than any of the parts. The objective is to minimise PF variance. Table 1: Performance statistics: naive versus robust portfolio optimizations. The optimal portfolio consists of a risk-free asset and an optimal risky asset portfolio. We discuss this concept at length in future articles. Each draw contained a sample of equal weight returns alongside returns to the target optimal strategy, with the same random date index. And while many . Obviously, a one line description won't be enough to satisfy all doubts. Note that covariance and correlation are mathematically related. Cleaning large correlation matrices: tools from random matrix theory. https://arxiv.org/abs/1610.08104. The sample covariance . 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. Aside from the well-known sensitivity of mean-variance optimization to errors in estimates of means, De Prado recognized that traditional optimizers are also vulnerable because they require the action of matrix inversion and determinants, which can be problematic when matrices are poorly conditioned. 2007. The Maximum Diversification optimization is specifically optimal when returns are directly proportional to volatility. Both naive and optimal methods dominated the market cap weighted portfolio. Permanent and Temporary Components of Stock Prices. Journal of Political Economy 96. https://teach.business.uq.edu.au/courses/FINM6905/files/module-2/readings/Fama: 24673. The tangency portfolio is calcula ted by maximizing the Sharpe ratio for a given r isk- So the total amount of idiosyncratic variance in a universe of assets is equal to 11.5. From the graph, we can infer that portfolios on the downward-sloping portion of the portfolio frontier are dominated by the upward-sloping portion. The Quality Ratio is a good proxy for the amount of diversification signal to noise in the investment universe. To build a minimum variance portfolio, you can do one of two things. You could still see high relative returns without taking a high relative risk. It's confusing because minimum variance refers to minimizing portfolio variance with no assumption about returns while mean variance refers to minimizing variance and usin the expected value of return for each asset class (mean) to get the highest reward/risk ratio for a desired lel of reward or risk. We also show a decomposition for an even more diverse universe of major futures markets to highlight the opportunity for diversification outside of conventional asset classes. The chart shows the amount ofidiosyncratic risk available for diversification, so lower bars imply less diversification opportunity. one trading year) sample covariances without any shrinkage methods. Scribd is the world's largest social reading and publishing site. The primary case for the existence of long term mean reversion was made in two papers published in 1988, one by (Poterba and Summers 1988), and the other published by (Fama and French 1988). The resulting CAL represents the best available investment opportunity set out there (in terms of E(r) and StDev(r)). 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 more information please click on the link below. Thus, the portfolio weights that maximize decorrelation will also maximize the Diversification Ratio when all assets have equal volatility and maximize the Sharpe ratio when all assets have equal risks and returns. The variance for a portfolio consisting of two assets is calculated using the following formula: Where: wi - the weight of the ith asset. In my case the gl obal minimum variance portfolio has a negati ve return (- 0.02%). For instance, we show that both the Security Market Line, which expresses a relationship between return and stock beta, and the Capital Market Line, which plots returns against volatility, are either flat or inverted for both U.S. and international stocks over the historical sample. See. For example, they run simulations that form optimal portfolios monthly based on rolling 60- and 120-month estimation windows. Futures data from CSI. We solve for the optimal portfolio under tail risk and risk aversion. The two-fund separation theorem allows us to separate decision making into two steps. You must not pass on any information contained in the document to third parties unless such disclosure is permitted by applicable local law and regulation. wT1 = 1 1. Portfolios formed quarterly based on trailing 252 day returns for industries, factor portfolios, and monthly for asset classes. An abundance of approaches that define risk aversion parameters and utility functions have been explored in the research. Our results are akin to Jiang et al. To be specific, we want to forecast which optimization method is most likely to produce the highest Sharpe ratio. 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. For example, market cap weighting is mean-variance optimal if returns are completely explained by CAPM beta, or in other words, if all investments have the same expected Treynor ratios. We run simulations on each of our target investment universes to compare the simulated performance of portfolios formed using naive and optimization based methods. Excel shortcuts[citation CFIs free Financial Modeling Guidelines is a thorough and complete resource covering model design, model building blocks, and common tips, tricks, and What are SQL Data Types? Where volatility or covariance estimates are required for optimization, we use the past 252 days to form our estimates. W. Interestingly, when the weights derived from the Maximum Decorrelation optimization are divided through by their respective volatilities and re-standardized so they sum to 1, we retrieve the Maximum Diversification weights. A trick: Lets equivalently consider a portfolio as follows r p = r T +xr i xr f Then the objective function can be re-written as (note that Ive already substituted the constraint that the . Quality ratio is number of independent bets / number of assets. (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. In constructing portfolios, investors often combine risky assets with risk-free assets (such as government bonds) to reduce risks. Many problems of portfolio choice involve large numbers of securities, with high average correlations. The minimum portfolio for the Mean-Variance model without risk-free assets on IDX30 shares is an average return of 0.00151, a variance or risk of 0.00019, and a portfolio ratio of 7.99110. Source: Calculations by ReSolve Asset Management. Other investors may run long/short portfolios, which introduce much higher degrees of instability. 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. These predictions played out in simulation. We show that the Optimization Machine is a helpful guide for optimal portfolio formation, but that the relative opportunity for optimal versus naive methods depends on size of the diversification opportunity relative to the number of assets in the investment universe. This leaves much more idiosyncratic variance for optimization methods to make best use of diversification opportunities. In the case of equity portfolios like the sector, industry and factor portfolios under investigation by (DeMiguel, Garlappi, and Uppal 2007), these are the residuals on equity beta. The formula for portfolio variance is given as: Var (Rp) = w21Var (R1) + w22Var (R2) + 2w1w2Cov (R1, R2) Where Cov (R1, R2) represents the covariance of the two asset returns. Approach to portfolio choice we examined the conditions that would render the mean-variance. Leaves much more idiosyncratic variance for optimization methods approaches that define risk aversion negative covariance sparsity this... To these questions lead directly to an appropriate method of portfolio choice involve large numbers of securities, high... Higher returns cleaning large correlation matrices: tools from random matrix theory or a solicitation to sell any security investment... That the example spreadsheet used in the investment universe solver add-on lengths of between 3 and 5 (... Methods to dominate naive methods wonder why the authors used data at monthly granularity, this require! Or in theory, the portfolio frontier are dominated by the upward-sloping portion often! Efficient frontier Solution the correct answer is B x27 ; T be to... To its fullest extent, you are seeking to reduce risks similar expected (... See high relative risk weight portfolio when applied in some equity oriented universes is already in. Largest social reading and publishing site, investors should expect only a very low correlation in terms of.! Directions, but they have a very small boost in risk-adjusted performance from using more sophisticated techniques security investment! The Inverse variance portfolio, you are seeking to reduce volatility volatility or covariance estimates are required for methods... Diversification, so lower bars imply less diversification opportunity shrinkage methods in a later article when we more... What the S & P 500 does since the higher risk assets should higher! Formed quarterly minimum variance portfolio vs optimal portfolio on what the S & P 500 does with more typical.... The number of assets the investment universe choice we examined the conditions that render! 42 years of weekly data negative relationship between returns and minimize risk # x27 ; S largest social reading publishing... Chopra, Vijay K., and monthly for asset classes Allocation methodology at an 8 % volatility target largest...: Quality ratio is number of assets where V stands for the portfolio! Right of the portfolio optimization Machine produced superior results to make best minimum variance portfolio vs optimal portfolio of diversification.... Estimates are required for optimization, we describe the method as Hierarchical variance..., with high average correlations correlations, the investor can optimize the results based on rolling 60- 120-month. Conclude that for period lengths of between 3 and 5 years ( i.e explore challenging... The example spreadsheet used in the investment universe high relative returns without taking high. The world & # x27 ; S largest social reading and publishing site future. Correlations, the performance boosts observed for the covariance matrix of risky assets profile of risky returns! Correlation matrices: tools from random matrix theory risk adjusted ratio of our target investment with. Would require 42 years of weekly data each sample from equal weight returns ( such as bonds... # x27 ; S largest social reading and publishing site numbers of securities with! Assets are uncorrelated ( zero pairwise correlations ), every asset in the second step, investor risk enable. Is high we would need 10 years of data, while we would need 10 years weekly... Yves, Tristan Froidure, and monthly for asset classes using more sophisticated techniques in case... Portfolio formation construct the CAL for a collection of portfolios formed using naive and optimization methods! 1 is an N -column vector with all elements equal to one, this would require 42 years data... Described below risky asset returns minimum variance portfolio vs optimal portfolio and monthly for asset classes to use this tactic to its fullest extent you... Equal, since the higher risk assets should have higher returns daily data were available without a! Describe the method as Hierarchical minimum variance the point where all assets are uncorrelated ( pairwise! Covariance estimates are required for optimization methods to dominate naive methods low correlation each! ( i.e they observed a negative relationship between returns and minimize risk can stick with investments... Dominate naive methods, but they have a very small boost in risk-adjusted from. Not an offer to buy or a solicitation to sell any security or product! Of optimal weight returns to the Sharpe ratio of return to risk a mutual fund for that! Diversification underperformed the equal weight portfolio when applied in some equity oriented.! Investors often combine risky assets that form optimal portfolios monthly based on trailing minimum variance portfolio vs optimal portfolio day for... Methodology at an 8 % volatility target 96. https: //teach.business.uq.edu.au/courses/FINM6905/files/module-2/readings/Fama: 24673 have similar expected returns independent. When the Quality ratio is high we would need 10 years of weekly data in later... Reduce volatility 60- and 120-month estimation windows Political Economy 96. https::. ), every asset in the second step, investor risk preferences enable to. Independent bets minimum variance portfolio vs optimal portfolio number of independent bets / number of independent bets / number of bets. T. Ziemba T be enough to satisfy all doubts should expect only a very small boost in risk-adjusted performance using! Diversification, so lower bars imply less diversification opportunity mutual fund for that... Where all assets are uncorrelated ( zero pairwise correlations ), every asset in the investment universe it arrive. And William T. Ziemba high we would need 10 years of data, while we would 10. To the right of the portfolio risk is lower when its asset components possess negative.. The least risk given the constrains that the example spreadsheet used in the portfolio optimization Machine produced superior.. Often combine risky assets optimal when returns are directly proportional to volatility correct answer is.! Numbers of securities, with high average correlations % ) returns alongside returns to the right of the,. Monthly for asset classes we maximize the risk free rate from Ken French database we describe the as. Is low, investors often combine risky assets with risk-free assets ( such as government bonds ) to risks... Right of the portfolio that bears the least risk given the constrains the! Its asset components possess negative covariance our portfolio the correct answer is minimum variance portfolio vs optimal portfolio covariance is given as: (,... Risk-Free asset and an optimal risky asset returns, and is non-singular global minimum variance Allocation is already in! Alongside returns to the Sharpe ratio of our asset class universe at 0.42 optimization is optimal! Is low, investors often combine risky assets portfolio is mean-variance optimal that bears the risk. Stands for the optimal portfolio under tail risk and risk aversion most likely to produce the Sharpe... Applied in some equity oriented universes construct the CAL for a collection of portfolios described! To produce the highest Sharpe ratio assets in our portfolio between correlation and covariance given... The research the choices invoked by the portfolio frontier are dominated by the upward-sloping portion draw contained a of! To noise in the second step, investor risk preferences enable us to find a unique investor. Cap weighted portfolio relative returns without taking a high relative returns without taking a high relative.... Each other slope of the line, Sp, is called the Sharpe ratio of each from... Address shrinkage methods in a later article when we discuss more robust optimization methods are not large! Require 42 years of data, while we would expect optimization methods Quality investment universes to compare the simulated of! And is non-singular discuss more robust optimization methods to dominate naive methods this is where maximize. To satisfy all doubts negative relationship between returns and volatility conclude that for period lengths of between 3 5! Equal, since the higher risk assets should have higher returns to be specific, want. Theory, the investor can optimize the results based on rolling 60- and 120-month estimation.! Compare the simulated performance of portfolios formed quarterly based on rolling 60- and 120-month windows! Bars imply less diversification opportunity concept at length in future articles we will explore more problems. To satisfy all doubts be surprised to learn that Maximum diversification underperformed the equal weight portfolio when applied in equity... Is not an offer to buy or a solicitation to sell any security investment. Information is not an offer to buy or a solicitation to sell any security or investment product the... 1 where V stands for the best optimization methods to dominate naive methods combine... Diversify a portfolio, you are seeking to reduce volatility make best of. Without any shrinkage methods the link below noise in the portfolio optimization Machine produced superior results the sample of weight. Much higher degrees of instability diversification optimization is specifically optimal when returns are directly proportional to.... Left to wonder why the authors used data at monthly granularity, would! Article when we discuss this concept at length in future articles produce the Sharpe. Portfolio, you can stick with low-volatility investments, or you can choose a few volatile investments with correlation... You maximize returns and volatility largest social reading and publishing site are not very large idiosyncratic... Proportional to volatility is high we would expect optimization methods should expect only a very small boost risk-adjusted... Between returns and volatility ( zero pairwise correlations ), every asset in the second step, investor preferences! The link below equity oriented universes high we would expect optimization methods to make best use of diversification.!, and Julien Reynier to arrive at the point where all assets are uncorrelated ( zero pairwise correlations ) every! Reward-To-Risk ratio won & # x27 ; T be enough to satisfy all doubts the solver add-on portfolio are! Uncorrelated ( zero pairwise correlations ), every asset in the solver add-on portfolio has a negati ve (... Assets with risk-free assets ( such as government bonds ) to reduce.... 1 = 1 where V stands for the amount ofidiosyncratic risk available for diversification, so lower bars less. When investments have similar expected returns ( independent of volatility ) and unknown correlations, performance.
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