# Posts by Year

## Capital Market Assumptions: Combining Institutions’ Forecasts for Improved Accuracy

Capital market assumptions1 (CMAs) are forecasts of future risk/return characteristics for broad asset classes over the next 5 to 20 years produced by leadi...

## Volatility Forecasting: HAR Model

Among the different members of the family of volatility forecasting models by weighted moving average1 like the simple and the exponentially weighted moving...

## Combating Volatility Laundering: Unsmoothing Artificially Smoothed Returns

Introduction It is common knowledge that returns to hedge funds and other alternative investments [like private equity or real estate] are often highly seri...

## Bootstrap Simulations with Exact Sample Mean Vector and Sample Covariance Matrix

Bootstrapping is a statistical method which consists in sampling with replacement from an original data set to compute the distribution of a desired statist...

## Cluster Risk Parity: Equalizing Risk Contributions Between and Within Asset Classes

The equal risk contribution (ERC) portfolio, introduced in Maillard et al.1, is a portfolio aiming to equalize the risk contributions from [its] different co...

## Volatility Forecasting: GARCH(1,1) Model

In the previous post of this series on volatility forecasting, I described the simple and the exponentially weighted moving average volatility forecasting m...

## Random Portfolio Benchmarking: Simulation-based Performance Evaluation in Finance

As noted in Surz1, the question “Is [a mutual fund’s]2 performance good?” can only be answered relative to something1, typically by comparing that fund to a ...

## Sparse Index Tracking: Limiting the Number of Assets in an Index Tracking Portfolio

In the previous post, I introduced the index tracking problem1, which consists in finding a portfolio that tracks as closely as possible2 a given financial ...

## Beyond Modified Value-at-Risk: Application of Gaussian Mixtures to the Computation of Value-at-Risk

In a previous post, I described a parametric approach to computing Value-at-Risk (VaR) - called modified VaR12 - that adjusts Gaussian VaR for asymmetry and...

## Index Tracking: Reproducing the Performance of a Financial Market Index (and more)

An index tracking portfolio1 is a portfolio designed to track as closely2 as possible a financial market index when its exact replication3 is either impracti...

## Volatility Forecasting: Simple and Exponentially Weighted Moving Average Models

One of the simplest and most pragmatic approach to volatility forecasting is to model the volatility of an asset as a weighted moving average of its past sq...

## Range-Based Volatility Estimators: Overview and Examples of Usage

Volatility estimation and forecasting plays a crucial role in many areas of finance. For example, standard risk-based portfolio allocation methods (minimum ...

## Correlation Matrix Stress Testing: Random Perturbations of a Correlation Matrix

In the previous posts of this series, I detailed a methodology to perform stress tests on a correlation matrix by linearly shrinking a baseline correlation m...

## Managing Missing Asset Returns in Portfolio Analysis and Optimization: Backfilling through Residuals Recycling

In a multi-asset portfolio, it is usual that some assets have shorter return histories than others1. Problem is, the presence of assets whose return histori...

## Simulation from a Multivariate Normal Distribution with Exact Sample Mean Vector and Sample Covariance Matrix

In the research report Random rotations and multivariate normal simulation1, Robert Wedderburn introduced an algorithm to simulate i.i.d. samples from a mu...

## The Bogle Model for Bonds: Predicting the Returns of Constant Maturity Government Bond ETFs

In his original 1991 article Investing in the 1990s1, John Bogle described a simple model to help investors setting reasonable expectations for long-term U.S...

## The Single Greatest Predictor of Future Stock Market Returns, Ten Years After

In his 2013 post The Single Greatest Predictor of Future Stock Market Returns, Jesse Livermore1 from the blog Philosophical Economics introduced an indicato...

## 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...

## 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...

## 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 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 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...

## 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...

## 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...

## 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 ...

## 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...

## 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 ...

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

## 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...

## 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 and Tiingo in a couple of ...

## 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...

## 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...

## 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...

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

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 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...

## 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. The recen...

## 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. This post will illu...

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

## 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 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 a s...

## Integration in Excel

In this post, I will show you how to integrate the Portfolio Optimizer Web API in Excel. As a working example, I will display the assets weights \$(w_1, w_2)...

## 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%)...