Value-at-Risk for a Multi-Asset Portfolio : A Backtesting Comparison of Historical, Parametric, and Monte Carlo Methods

Sammanfattning

Value at Risk (VaR) is a widely applied risk metric in finance for estimating the potential loss in the value of a portfolio over a certain time horizon at a specified confidence level. It is a fundamental tool in market risk management, regulatory capital calculations, and internal risk governance. While widely used, different VaR methods can produce quite different results depending on portfolio structure and risk profile. Hence, this thesis implements and evaluates three standard VaR models for a multi-asset portfolio. The portfolio contains equities, bonds, interest rate swaps, OTC European options, and swaptions. The empirical objective is to study how Historical Simulation, Parametric (Variance-Covariance), and Monte Carlo Simulation behave for this portfolio under a common backtesting framework. In addition to VaR, the thesis also reports Conditional Value-at-Risk (CVaR) as a complementary measure of tail-loss severity. A practical contribution of the study is a custom Python application, developed from scratch, for data collection, instrument modeling, pricing, risk simulation, and backtesting. Each asset class was modeled using pricing methods that capture its distinct risk factors and market behavior. To ensure realistic modeling, several key features were incorporated, including Cholesky decomposition to simulate correlated asset returns and explicit FX risk. Data preprocessing was handled carefully, including estimating missing historical option prices using several methods. A backtesting module was then added to the application to check how well the models performed over rolling periods. Each model was tested on the same portfolio over a rolling 252-day window, with 1-day 99% VaR forecasts evaluated through out-of-sample backtesting. Breaches, where actual losses exceeded the predicted VaR, were recorded to assess forecast performance, while CVaR was used to summarize the severity of losses beyond the VaR threshold. For the portfolio and sample period studied in this thesis, the Monte Carlo specification produced the strongest backtesting outcome, Historical Simulation performed somewhat less well, and the normal Parametric model produced the weakest result. The reported CVaR estimates support the same general interpretation by showing that the normal Parametric specification delivered the least conservative tail-risk assessment, while Historical Simulation and Monte Carlo implied more severe losses beyond the 99% VaR threshold. Keywords: Risk management, Value-at-Risk, Conditional Value-at-Risk, Monte Carlo simulation, Financial modeling, Basel regulation, Backtesting, Multi-asset portfolio, Historical Simulation, Parametric VaR.