# Posts by Tag

## The Gerber Statistic: A Robust Co-Movement Measure for Correlation Matrix Estimation

The Gerber statistic is a measure of co-movement similar in spirit to the Kendall’s Tau coefficient that has been introduced in Gerber et al.1 to estimate c...

## The Informativeness: Measuring the Homogeneity of a Universe of Assets

In this post, I will describe a measure of the homogeneity of a universe of assets, called the informativeness, introduced by Brockmeier et al.1 in their pap...

## Correlation Matrices Denoising: Results from Random Matrix Theory

The estimation of empirical correlation matrices in finance is known to be affected by noise, in the form of measurement error, due in part to the short leng...

## Computation of Theory-Implied Correlation Matrices: Overview and Example

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 Correlatio...

## The Effective Number of Bets: Measuring Portfolio Diversification

Many different measures of portfolio diversification have been developed in the financial literature, from asset weights-based diversification measures like ...

## Correlation Matrix Stress Testing: Shrinkage Toward the Lower and Upper Bounds of a Correlation Matrix

I previously described on this blog an intuitive way of performing stress tests on a correlation matrix, which consists in shrinking a baseline correlation ...

## The Matrix Effective Rank: Measuring the Dimensionality of a Universe of Assets

Quantifying how diversified is a universe of assets is an open problem in quantitative finance, partly because there is no definite formula for diversificati...

## Correlation Matrix Stress Testing: Shrinkage Toward an Equicorrelation Matrix

Financial research has consistently shown that correlations between assets tend to increase during crises and tend to decrease during recoveries1. ...

## When a Correlation Matrix is not a Correlation Matrix: the Nearest Correlation Matrix Problem

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...

## The Turbulence Index: Regime-based Partitioning of Asset Returns

The turbulence index, introduced in the previous blog post, is a measure of statistical unusualness of asset returns popularized by Kritzman and Li1. It pro...

## The Diversification Ratio: Measuring Portfolio Diversification

Continuing the series of blog posts on diversification indicators, I describe in this post a correlation-based measure of portfolio diversification called th...

## Mean-Variance Optimization in Practice: Subset Resampling-based Efficient Portfolios

In a previous post, I introduced near efficient portfolios, which are portfolios equivalent to mean-variance efficient portfolios in terms of risk-return bu...

## Mean-Variance Optimization in Practice: Well Diversified (Near) Efficient Portfolios

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)varian...

## Ulcer Performance Index Portfolio Optimization

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 In...

## Beyond Hierarchical Risk Parity: Hierarchical Clustering-Based Risk Parity

In a previous post, I introduced the Hierarchical Risk Parity portfolio optimization algorithm1. See Lopez de Prado, M. (2016). Building dive...

## Hierarchical Risk Parity: Introducing Graph Theory and Machine Learning in Portfolio Optimizer

In this short post, I will introduce the Hierarchical Risk Parity portfolio optimization algorithm, initially described by Marcos Lopez de Prado1, and recen...

## The Portfolio Optimization Machine

In 2018, guys at ReSolve Asset Management published the paper Portfolio Optimization: A General Framework for Portfolio Choice in which they describe

## The Gerber Statistic: A Robust Co-Movement Measure for Correlation Matrix Estimation

The Gerber statistic is a measure of co-movement similar in spirit to the Kendall’s Tau coefficient that has been introduced in Gerber et al.1 to estimate c...

## The Turbulence Index: Regime-based Partitioning of Asset Returns

The turbulence index, introduced in the previous blog post, is a measure of statistical unusualness of asset returns popularized by Kritzman and Li1. It pro...

## The Diversification Ratio: Measuring Portfolio Diversification

Continuing the series of blog posts on diversification indicators, I describe in this post a correlation-based measure of portfolio diversification called th...

## The Effective Number of Bets: Measuring Portfolio Diversification

Many different measures of portfolio diversification have been developed in the financial literature, from asset weights-based diversification measures like ...

## The Absorption Ratio: Measuring Financial Risk, Part 2

In the previous post, I reviewed the turbulence index, an indicator of financial market stress periods based on the Mahalanobis distance, introduced by Chow...

## The Turbulence Index: Measuring Financial Risk

One of the challenges in portfolio management is the timely detection of financial market stress periods, typically characterized by an increase in volatilit...

## The Matrix Effective Rank: Measuring the Dimensionality of a Universe of Assets

Quantifying how diversified is a universe of assets is an open problem in quantitative finance, partly because there is no definite formula for diversificati...

## The Turbulence Index: Regime-based Partitioning of Asset Returns

The turbulence index, introduced in the previous blog post, is a measure of statistical unusualness of asset returns popularized by Kritzman and Li1. It pro...

## The Informativeness: Measuring the Homogeneity of a Universe of Assets

In this post, I will describe a measure of the homogeneity of a universe of assets, called the informativeness, introduced by Brockmeier et al.1 in their pap...

## The Absorption Ratio: Measuring Financial Risk, Part 2

In the previous post, I reviewed the turbulence index, an indicator of financial market stress periods based on the Mahalanobis distance, introduced by Chow...

## The Turbulence Index: Measuring Financial Risk

One of the challenges in portfolio management is the timely detection of financial market stress periods, typically characterized by an increase in volatilit...

## The Matrix Effective Rank: Measuring the Dimensionality of a Universe of Assets

Quantifying how diversified is a universe of assets is an open problem in quantitative finance, partly because there is no definite formula for diversificati...

## Residualization of Risk Factors: Examples and Pitfalls

The most common approach to measuring portfolio (risk) factor exposures is linear regression analysis, which describes the relationship between a dependent ...

## Replicating the J.P. Morgan Efficiente Index

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.

Crypto-assets (Bitcoin, Ethereum…) have recently been attracting the attention of more and more investors, with for example JPMorgan Chase & Co. suggesti...

In this post, I will show you how to integrate the Portfolio Optimizer Web API in Google Sheets.

## Computation of Theory-Implied Correlation Matrices: Overview and Example

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 Correlatio...

## Beyond Hierarchical Risk Parity: Hierarchical Clustering-Based Risk Parity

In a previous post, I introduced the Hierarchical Risk Parity portfolio optimization algorithm1. See Lopez de Prado, M. (2016). Building dive...

## Hierarchical Risk Parity: Introducing Graph Theory and Machine Learning in Portfolio Optimizer

In this short post, I will introduce the Hierarchical Risk Parity portfolio optimization algorithm, initially described by Marcos Lopez de Prado1, and recen...

## The Effective Number of Bets: Measuring Portfolio Diversification

Many different measures of portfolio diversification have been developed in the financial literature, from asset weights-based diversification measures like ...

## The Absorption Ratio: Measuring Financial Risk, Part 2

In the previous post, I reviewed the turbulence index, an indicator of financial market stress periods based on the Mahalanobis distance, introduced by Chow...

## The Matrix Effective Rank: Measuring the Dimensionality of a Universe of Assets

Quantifying how diversified is a universe of assets is an open problem in quantitative finance, partly because there is no definite formula for diversificati...

## Replicating the J.P. Morgan Efficiente Index

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.

## Random Portfolios as Benchmarks for Tactical Asset Allocation Strategies

If you are familiar with tactical asset allocation (TAA) strategies, like the Global Equities Momentum (GEM) TAA strategy of Gary Antonacci, you know how har...

## Mean-Variance Optimization in Practice: Subset Resampling-based Efficient Portfolios

In a previous post, I introduced near efficient portfolios, which are portfolios equivalent to mean-variance efficient portfolios in terms of risk-return bu...

## Mean-Variance Optimization in Practice: Well Diversified (Near) Efficient Portfolios

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)varian...

## The Turbulence Index: Regime-based Partitioning of Asset Returns

The turbulence index, introduced in the previous blog post, is a measure of statistical unusualness of asset returns popularized by Kritzman and Li1. It pro...

## The Turbulence Index: Measuring Financial Risk

One of the challenges in portfolio management is the timely detection of financial market stress periods, typically characterized by an increase in volatilit...

When backtesting an investment strategy, that is, when simulating an investment strategy using historical prices to test how this strategy would have behaved...

## Selecting a Stock Market Data (Web) API: Not So Simple

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 po...

## The Probabilistic Sharpe Ratio: Hypothesis Testing and Minimum Track Record Length for the Difference of Sharpe Ratios

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

## The Probabilistic Sharpe Ratio: Bias-Adjustment, Confidence Intervals, Hypothesis Testing and Minimum Track Record Length

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 ...

## The Probabilistic Sharpe Ratio: Hypothesis Testing and Minimum Track Record Length for the Difference of Sharpe Ratios

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

## The Probabilistic Sharpe Ratio: Bias-Adjustment, Confidence Intervals, Hypothesis Testing and Minimum Track Record Length

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 ...

## Integration in a Web Page

In this post, I will show how to integrate the Portfolio Optimizer Web API in a web page.

## Implementing an Investable Portfolio From an Optimal Portfolio

As an investor, have you ever wondered how to convert an optimal portfolio1, possibly made of real-valued weights with dozens of decimals (e.g., 12.3456789%)...

## Integration in Excel

In this post, I will show you how to integrate the Portfolio Optimizer Web API in Excel.

## Random Portfolios as Benchmarks for Tactical Asset Allocation Strategies

If you are familiar with tactical asset allocation (TAA) strategies, like the Global Equities Momentum (GEM) TAA strategy of Gary Antonacci, you know how har...

## crypto-assets

Crypto-assets (Bitcoin, Ethereum…) have recently been attracting the attention of more and more investors, with for example JPMorgan Chase & Co. suggesti...

## Residualization of Risk Factors: Examples and Pitfalls

The most common approach to measuring portfolio (risk) factor exposures is linear regression analysis, which describes the relationship between a dependent ...

## The Absorption Ratio: Measuring Financial Risk, Part 2

In the previous post, I reviewed the turbulence index, an indicator of financial market stress periods based on the Mahalanobis distance, introduced by Chow...

## Selecting a Stock Market Data (Web) API: Not So Simple

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 po...

## Bootstrap Simulation with Portfolio Optimizer: Usage for Financial Planning

In statistics, a bootstrap method, also called bootstrapping, is a compute-intensive procedure that allows to estimate the distribution of a statistic throu...

## Bootstrap Simulation with Portfolio Optimizer: Usage for Financial Planning

In statistics, a bootstrap method, also called bootstrapping, is a compute-intensive procedure that allows to estimate the distribution of a statistic throu...

## Trading Strategy Monitoring: Modeling the PnL as a Geometric Brownian Motion

Systematic trading strategies have the unfortunate habit of exhibiting worse performances in real-life than in backtests, partially due to backtest overfitti...

## Trading Strategy Monitoring: Modeling the PnL as a Geometric Brownian Motion

Systematic trading strategies have the unfortunate habit of exhibiting worse performances in real-life than in backtests, partially due to backtest overfitti...

## The Informativeness: Measuring the Homogeneity of a Universe of Assets

In this post, I will describe a measure of the homogeneity of a universe of assets, called the informativeness, introduced by Brockmeier et al.1 in their pap...

## The Diversification Ratio: Measuring Portfolio Diversification

Continuing the series of blog posts on diversification indicators, I describe in this post a correlation-based measure of portfolio diversification called th...

## The Diversification Ratio: Measuring Portfolio Diversification

Continuing the series of blog posts on diversification indicators, I describe in this post a correlation-based measure of portfolio diversification called th...

## The Diversification Ratio: Measuring Portfolio Diversification

Continuing the series of blog posts on diversification indicators, I describe in this post a correlation-based measure of portfolio diversification called th...

## The Turbulence Index: Regime-based Partitioning of Asset Returns

The turbulence index, introduced in the previous blog post, is a measure of statistical unusualness of asset returns popularized by Kritzman and Li1. It pro...

## The Turbulence Index: Regime-based Partitioning of Asset Returns

The turbulence index, introduced in the previous blog post, is a measure of statistical unusualness of asset returns popularized by Kritzman and Li1. It pro...

## The Mathematics of Bonds: Simulating the Returns of Constant Maturity Government Bond ETFs

With more than \$1.2 trillion under management in the U.S. as of mid-July 20221, investors are more and more using bond ETFs as building blocks in their asset...

## The Mathematics of Bonds: Simulating the Returns of Constant Maturity Government Bond ETFs

With more than \$1.2 trillion under management in the U.S. as of mid-July 20221, investors are more and more using bond ETFs as building blocks in their asset...

## Corrected Cornish-Fisher Expansion: Improving the Accuracy of Modified Value-at-Risk

Modified Value-at-Risk (mVaR) is a parametric approach to computing Value-at-Risk introduced by Zangari1 that adjusts Gaussian Value-at-Risk for asymmetry a...

## Corrected Cornish-Fisher Expansion: Improving the Accuracy of Modified Value-at-Risk

Modified Value-at-Risk (mVaR) is a parametric approach to computing Value-at-Risk introduced by Zangari1 that adjusts Gaussian Value-at-Risk for asymmetry a...

## Corrected Cornish-Fisher Expansion: Improving the Accuracy of Modified Value-at-Risk

Modified Value-at-Risk (mVaR) is a parametric approach to computing Value-at-Risk introduced by Zangari1 that adjusts Gaussian Value-at-Risk for asymmetry a...

## Corrected Cornish-Fisher Expansion: Improving the Accuracy of Modified Value-at-Risk

Modified Value-at-Risk (mVaR) is a parametric approach to computing Value-at-Risk introduced by Zangari1 that adjusts Gaussian Value-at-Risk for asymmetry a...