In the first post of this series about the Sharpe ratio considered as a statistical estimator, I introduced a probabilistic framework to answer the question What is the probability that an estimated Sharpe ratio is statistically significantly greater than a reference Sharpe ratio? In this second

The Sharpe ratio1 is one of the most commonly used measure of financial portfolio performance, but because it is deeply rooted in mean-variance theory, its usage with return distributions deviating from normality (e.g. hedge funds, cryptocurrencies) is frequently questioned2. One solution to this

In statistics, a bootstrap method, also called bootstrapping, is a compute-intensive procedure that allows to estimate the distribution of a statistic through repeated resampling from a single observed sample of data1. Bootstrapping has several applications in quantitative finance, for example to

In this short post, I will provide an overview of the TIC algorithm1 introduced by Marcos Lopez de Prado in his paper Estimation of Theory-Implied Correlation Matrices2, which aims to compute a forward-looking asset correlation matrix blending both empirical and theoretical inputs. I will also

Many different measures of portfolio diversification have been developed in the financial literature, from asset weights-based diversification measures like the Herfindahl Index1 to risk-based diversification measures like the Diversification Ratio of Choueifaty and Coignard2 to other more complex

I previously described on this blog an intuitive way of performing stress tests on a correlation matrix, which consists in shrinking a baseline correlation matrix toward an equicorrelation matrix12. A limitation of this method, though, is that it alters all the correlation coefficients of the

In a previous post, I introduced near efficient portfolios, which are portfolios equivalent to mean-variance efficient portfolios in terms of risk-return but more diversified in terms of asset weights. Such near efficient portfolios might be used to moderate the tendency of efficient portfolios to

I am sometimes asked if I recommend any stock market data (web) API for a personal use, especially because I mention Alpha Vantage in a couple of previous posts1. I will describe in this post part of the thought process and of the due diligence which led me to select this financial market data

In the previous post, I reviewed the turbulence index, an indicator of financial market stress periods based on the Mahalanobis distance, introduced by Chow and al.1 and Kritzman and Li2. In this post, I will review the absorption ratio, a measure of financial market fragility based on principal

One of the challenges in portfolio management is the timely detection of financial market stress periods, typically characterized by an increase in volatility and a breakdown in asset correlations1. Chow and al.2 propose to detect such periods through the usage of the caste distance, a measure

One well-known stylized fact of the Markowitz’s mean-variance framework is that, irrespective of the quality of the estimates of asset returns and (co)variances, efficient portfolios are concentrated in a very few assets1. From a practitioner’s perspective, this has always been a problem12. In

The Ulcer Performance Index1 (UPI) is a portfolio reward-risk measure introduced by G. Martin2 similar in spirit to the Sharpe Ratio, but using the Ulcer Index (UI) as a risk measure instead of the standard deviation. In this blog post, I will present the mathematics behind the Ulcer Performance

Quantifying how diversified is a universe of assets is an open problem in quantitative finance, partly because there is no definite formula for diversification1. Let’s make the (reasonable) assumption that the way assets are moving together within a universe is important for its diversification.

In a previous post, I introduced the Hierarchical Risk Parity portfolio optimization algorithm1. In this post, I will present one of its variations, called Hierarchical Clustering-Based Risk Parity, first described in Papenbrock2 and then generalized in Raffinot34 and in Lohre et al.5, from which

In this short post, I will introduce the Hierarchical Risk Parity portfolio optimization algorithm, initially described by Marcos Lopez de Prado1, and recently implemented in Portfolio Optimizer. I will not go into the details of this algorithm, though, but simply describe some of its general ideas

Financial research has consistently shown that correlations between assets tend to increase during crises and tend to decrease during recoveries1. The recent COVID-19 market crash was no exception, as illustrated on Alvarez Quant Trading blog post Correlations go to One for both the individual

The most common approach to measuring portfolio (risk) factor exposures is linear regression analysis, which describes the relationship between a dependent variable - portfolio returns - and explanatory variables - factors - as linear. One of the outputs of this analysis are the partial regression

The J.P. Morgan Efficiente 5 Index is a tactical asset allocation strategy designed by J.P. Morgan based on a broad universe of 13 ETFs. This post will illustrate how to replicate this strategy with Google Sheets. Notes: A fully functional spreadsheet corresponding to this post is available here.

Estimating how individual assets are moving together is an important part of many financial applications1 and the most commonly used measure for this is the Pearson correlation. Unfortunately, for a variety of reasons, what sometimes appears to be a correlation matrix is actually not a valid