CVaR-based Portfolio Optimization

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

Furthermore, various portfolio optimization strategies for a given dispersion measure can be chosen. Examples are:

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:

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:

The so-called mixture CVaR which is a risk management technique that extends the concept of CVaR to scenarios where the underlying distribution is not known with certainty. Instead, it assumes that the distribution is a mixture of multiple sub-distributions, each with its own parameters. By accounting for this heterogeneity, mixture CVaR provides a more comprehensive assessment of risk compared to traditional CVaR, which assumes a single underlying distribution.

Advantages of Mixture CVaR

In the sequel we shall first explain each optimization strategy before showing the differences between the optimization strategies and their implications.

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.

Diversification has been defined in the 2008 paper from Yves Choueifaty (Head of Quantitative Asset Management, Europe, at Lehman Brothers) and Yves Coignard (co-deputy Head of Quantitative Asset Management, Europe, at Lehman Brothers):

https://www.tobam.fr/wp-content/uploads/2014/12/TOBAM-JoPM-Maximum-Div-2008.pd

This optimization strategy ensures that a diversified portfolio is sought while still minimizing the overall dispersion measure. Therefore, a typical solution will have a more diversified solution (aka more instruments in a portfolio) but with a higher risk than a pure risk minimization.

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.

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.

The Sharpe ratio is defined as

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.