## 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...

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

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 ...

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 ...

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...

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...

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...

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

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...

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...

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

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...

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 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...

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...

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

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

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...

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

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...

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 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 ...

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

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...

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

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...

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

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 ...

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

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

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 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...

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

In a previous post, I introduced the Hierarchical Risk Parity portfolio optimization algorithm1. In this post, I will present one of its variations, called ...

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

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

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

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...

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...

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

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)...

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%)...

In this post, I will show how to integrate the Portfolio Optimizer Web API in a web page. As a working example, I will display the mean-variance minimum var...

In this post, I will show you how to integrate the Portfolio Optimizer Web API in Google Sheets. As a working example, I will display the mean-variance mini...