CVaR-based Portfolio Optimization

introduction a risk based optimal portfolio seeks to maximize expected returns while minimizing the variability of potential outcomes by analyzing the expected return and the dispersion of possible returns, investors can identify portfolios that align with their risk tolerance and financial goals the dispersion of potential outcomes is often referred to as risk, and the metric used to quantify this dispersion is called a risk measure throughout this document, we will use the terms "dispersion" and "risk measure" interchangeably the dispersion measure can be defined in many ways well known examples are volatility (standard deviation) variance conditional value at risk ( cvar ) furthermore, various portfolio optimization strategies for a given dispersion measure can be chosen examples are minimum risk portfolio ( min risk ) efficient portfolio at the most left of the efficient frontier maximum diversification portfolio ( max diverse ) minimum risk portfolio under the constraint of a maximization of the diversification maximum of sharpe ratio ( max sharpe ) portfolio with the highest expected excess rate of return per unit of risk minimization of risk for a targeted portfolio return ( risk ) portfolio on the efficient frontier corresponding to a given expected portfolio return diversified portfolio for targeted expected rate of return ( diverse ) diversified portfolio targeting an expected portfolio return while trying to stay on the efficient frontier dispersion measures cvar conditional value at risk (cvar) optimization is a risk management technique used to minimize the potential for large losses in a portfolio of investments cvar, also called the expected shortfall, is a measure of the average loss that can be expected in the worst case scenario it is a more sophisticated measure of risk than traditional value at risk (var), which only considers the probability of a loss exceeding a certain threshold in mathematical terms, if x is the payoff of the portfolio, then the cvar of a given level q is defined as \mathrm{cvar} q = \frac{1}{q}\int 0^q \mathrm{var} \gamma (x) d\gamma cvar takes into account the magnitude of potential losses and focuses on reducing the expected loss in the worst case scenario as a result, the cvar optimizer will tend to pick instruments that have a lower probability of large losses this may mean that the optimizer picks instruments that are less volatile in this graph, the x axis represents the potential loss, and the y axis represents the probability of that loss occurring the cvar is the average of the losses that exceed the var such that the cvar is greater than the var this is because the cvar takes into account the magnitude of potential losses, not just the probability of a loss exceeding a certain threshold this means that the cvar is a more conservative measure of risk than the var the advantages of cvar optimization it is a more accurate measure of tail risk cvar considers the entire distribution of potential losses, not just the probability of a loss exceeding a certain threshold this makes it a more accurate measure of risk, especially for portfolios with a high probability of large losses it is more coherent with risk aversion cvar is a coherent risk measure , which means that it is consistent with the principle of risk aversion this means that portfolios with lower cvar are always considered to be less risky than portfolios with higher cvar it is more robust to changes in the distribution of returns cvar is less sensitive to changes in the distribution of returns than var this makes it a more reliable measure of risk in situations where the distribution of returns is unknown or uncertain optimization strategies in the sequel we shall first explain each optimization strategy before showing the differences between the optimization strategies and their implications minimization of risk (min risk) this optimization strategy represents the efficient frontier's leftmost extreme, offering the minimum possible risk among all efficient portfolios it is a frequently employed strategy by professional investors who prioritize risk mitigation over maximizing returns maximum diversification (mdp) in their seminal 2008 paper "toward maximum diversification", yves choueifaty and yves coignard introduced a novel approach to portfolio diversification that challenges traditional methods they argue that maximizing diversification should involve minimizing the contribution of the portfolio's least diversified holdings, rather than simply focusing on the average volatility of the assets https //www tobam fr/wp content/uploads/2014/12/tobam jopm maximum div 2008 pd https //www tobam fr/wp content/uploads/2014/12/tobam jopm maximum div 2008 pdf the central idea of their approach is encapsulated in the following formula \mathrm{dr} = \frac{w^t\cdot\sigma}{\sqrt{w^t\cdot\sigma\cdot w}} where w is the portfolio weight (and t denotes its transpose) ∑ is the covariance matrix this formula essentially measures the ratio of the average volatility of the individual assets to the overall portfolio volatility a higher diversification ratio indicates a more diversified portfolio by maximizing this ratio, the authors propose a method to construct a "most diversified portfolio" (mdp) the key insight of choueifaty and coignard's work is that traditional diversification methods, like minimizing portfolio variance, can lead to concentrated portfolios if some assets have very low volatility this is because these low volatility assets may appear to reduce portfolio risk even if they are highly correlated with other holdings the mdp approach, on the other hand, explicitly focuses on reducing the contribution of the least diversified assets, leading to a more balanced and robust portfolio the paper demonstrates that the mdp exhibits several desirable properties, including superior risk adjusted returns compared to traditional methods, the mdp often leads to higher sharpe ratios reduced concentration risk the mdp avoids excessive concentration in a few assets, even if they have low volatility robustness to estimation errors the mdp is less sensitive to errors in estimating asset correlations and volatilities diversified min risk this strategy combines the minimization of risk while maximizing the diversification ratio and is thus a compromise between a diversified portfolio (aka assets which a decoupled from each other from a volatility perspective) and a portfolio which minimizes risk minimization of risk for targeted expected rate of return (risk) this strategy corresponds to 'risk' and is widely used for portfolio optimization and generates portfolios that fall along the efficient frontier the user needs to specify the desired expected rate of return this value must be within the range of expected returns, from the minimum risk portfolio to the highest return portfolio among the components if the input value falls outside this range, it will automatically be adjusted to the nearest limit diversified portfolio for targeted expected rate of return (diverse) this strategy corresponds to 'diverse' and looks for the optimal diversified portfolio for targeted a given expected rate of return the optimized portfolio will be close to the one found by strategy 'risk' but with an improved diversity maximization of sharpe ratio (max sharpe) the sharpe ratio is defined as \gamma = \frac{r r f}{\rho}, where r is portfolio expected rate of return, r f is the risk free rate and \rho is the dispersion of portfolio rate of return this strategy corresponding to 'sharpe' identifies the portfolio that has the highest expected excess rate of return relative to the amount of risk the investor assumes it is a widely preferred strategy among investors seeking to maximize their returns while managing risk metric value mean mean 4 67% 4 67% maximum 24 5% 90th percentile 7 98% third quartile 6 03% median 4 66% first quartile 2 78% 10th percentile 0 71% minimum 5 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