Value at Risk for a high-dimensional equity portfolio : A comparative study investigating computational complexity and accuracy for different methods

Abstract: Risk management is practiced in many financial institutions and one of the most commonly used risk measures is Value at Risk. This measure represents how much a portfolio of assets could lose over a pre-specified time horizon to a cer- tain probability. Value at Risk is often utilized to calculate capital requirements and margins, which work as collateral to cover potential losses that might occur due to market turbulence. It is important that the calculation of Value at Risk is accurate which require complex and time demanding models but many financial institutions also wishes to calculate Value at Risk continuously throughout the day, which requires computational speed. Today’s most commonly used method for calculating Value at Risk is his- torical simulation which is a simple but often inaccurate method. It is criticized by many scholars since it heavily depends on the assumption that history will repeat itself. A substitute method to historical simulation is the Monte Carlo simulation which is seen as a more accurate and robust method. However, for a high-dimensional portfolio, Monte Carlo simulated Value at Risk is very com- putationally demanding and in many cases it is not possible to use it due to time constraints. The study investigates alternative methods for calculating Value at Risk with the purpose of finding a method that could be used as a substitute to the Monte Carlo method. The portfolio used in this thesis is a high-dimensional equity portfolio containing 2520 equities with 10 years of observations. I find that by first using a clustering algorithm to divide the equities in to groups based on their correlation, and then applying principal component analysis to achieve a lower dimensional problem, computational time can be reduced by approxi- mately 99% and still provide an accurate result.