Monte Carlo Portfolio Risk Engine

Key Takeaways

Disclaimer: this is an educational project and does not constitute investment advice.

Overview: Why This Portfolio, Why This Model

Most discussions of portfolio risk stop at annualized volatility. It is a useful number, but it tells you nothing about the shape of the distribution: whether tail losses are symmetric or skewed, moderate or catastrophic, and how much the picture changes when market conditions shift. This project was built to answer exactly those questions.

The portfolio chosen for the analysis, 60% S&P 500, 25% Gold and 15% VIX, is deliberately unconventional. Including VIX as a direct portfolio component creates a risk profile that is genuinely difficult to read from static statistics. VIX is mean-reverting, highly non-normal, and dominated by spike behavior during crises. A Monte Carlo simulation run across multiple regimes is one of the few tools that makes this kind of distribution visible in a meaningful way.

The model is built end to end in Python. It starts from raw historical price data, runs a parameter estimation step to calibrate all inputs, and generates 10,000 correlated return paths using a multivariate normal distribution. Both central tendency and tail risk metrics are then derived from the resulting distribution. Scenario analysis tests whether the conclusions hold when market conditions change. As the results show, they do not hold at all.

Part 1: Data and Inputs

The simulation is calibrated on daily historical price data covering January 2015 through late 2024, approximately 2,140 trading days in total. Returns are computed as simple daily percentage changes. Mean returns and the full covariance matrix are estimated from this sample using estimate_params.py, which reads the Excel dataset and exports all parameters to CSV for verification before the simulation runs.

Estimated Daily Parameters

The 2.25% daily mean for VIX is a statistical artifact of large historical spikes, not a genuine return expectation. See Part 4 for a full discussion.

Correlation Structure

The covariance matrix is the most consequential input. It governs how asset returns move together and ultimately shapes the tail risk profile more than any individual asset's mean return. Two figures stand out.

All correlations estimated from the full 2015–2024 sample. Daily figures.

The +0.39 correlation between S&P 500 and Gold is worth pausing on. Many investors treat Gold as a near-zero or negative correlation asset relative to equities. Over this sample period, that assumption does not hold. The relationship is meaningfully positive, which reduces the diversification benefit from the Gold allocation compared to what a naive assumption would suggest. The VIX correlation behaves exactly as expected: strongly negative and consistent throughout.

Part 2: Methodology

2.1 Multivariate Normal Simulation

The simulation draws 10,000 joint return scenarios from a multivariate normal distribution parameterized by the estimated mean vector and covariance matrix. For each scenario, 252 daily return vectors are generated, one per trading day, preserving the full correlation structure at every step. Portfolio daily returns are the weighted sum of asset returns, and portfolio value paths are built by compounding from a normalized starting value of 1.0.

The final portfolio value at day 252 is recorded for each of the 10,000 paths, producing the distribution from which all risk metrics are derived. Using 10,000 paths ensures that even the 1st-percentile estimates are based on 100 data points, making VaR 99% numerically stable. This is a meaningful improvement over the 100-path Excel prototype built earlier in the project, where a single outlier path could move the VaR 99% figure by several percentage points.

2.2 Scenario Construction

Three scenarios are tested by modifying the simulation inputs. Rather than imposing arbitrary price shocks, the approach models changes to the statistical properties of returns directly, which is more consistent with how risk regimes actually shift in financial markets.

Scaling the covariance matrix by k increases variances by k, so volatilities increase by √k. A 1.5× covariance scale means roughly +22% volatility, not +50%.

The mean return adjustment in the stress scenario (−0.2 pp/day) is aggressive by design. Over 252 trading days it represents roughly −40 percentage points of annualized expected return, consistent with a prolonged deterioration in fundamentals rather than a short-term shock. Combined with doubled covariances, this is what produces the near-total drawdown scenarios visible at the left tail of the stress distribution.

2.3 Risk Metrics

Two families of tail risk metrics are computed at both 95% and 99% confidence levels.

Value at Risk (VaR) is the percentile threshold of the final return distribution. VaR 95% is the 5th percentile, the level below which only 5% of simulated outcomes fall. It answers one question: what is the minimum loss in the worst 5% of scenarios? It says nothing about how bad things get once you are inside that tail.

Expected Shortfall (ES), also called Conditional VaR, is the average return across all simulations that fall below the VaR threshold. Given that you are in the worst 5% of scenarios, ES tells you how bad it is on average. It is the more conservative and more informative measure for portfolios with fat-tailed distributions. A large gap between VaR and ES is itself a risk signal: the tail is not just long, it is heavy.

Part 3: Results

3.1 Full Results Table

All figures represent final portfolio returns after 252 trading days. Stress scenario values highlighted in red.

3.2 Base Scenario

The base scenario produces a mean final return of +217.4% and a median of +164.9% over the year. The gap between these two figures is the first diagnostic signal: the distribution is right-skewed, with a small number of extreme upside paths pulling the mean well above the central outcome. The standard deviation of 213.1% reflects an exceptionally wide distribution. It does not look like a bell curve, even though the simulation draws from a normal at the daily level. A year of compounding transforms daily normality into annual non-normality.

Annualized portfolio volatility implied by the simulation is approximately 60.8%, well above the 12–20% range typical of diversified equity portfolios. The driver is VIX. With a daily standard deviation of roughly 25.5%, it dominates the portfolio's variance even at a 15% weight because its volatility is nearly six times that of the equity component.

The base case VaR 95% of −4.1% sounds reassuring: 95% of simulations end without losing more than 4.1% over the year. The ES 95% of −23.9% immediately qualifies that reading. Once you are in the worst 5% of outcomes, the average loss is nearly 24%. The average tail loss is nearly six times the entry point of the tail. That gap is the clearest sign of a fat-tailed distribution, and it is a direct consequence of holding VIX.

3.3 High Volatility and Stress Scenarios

The high-volatility scenario makes the regime-dependence of VaR impossible to ignore. Scaling covariances by 1.5× without touching expected returns pushes VaR 95% from −4.1% to −29.4%, more than seven times larger. ES 95% moves from −23.9% to −45.0%. The base-case risk figure offers almost no useful guidance once market volatility rises. This is the core limitation of VaR as a risk measure: it is calibrated to a specific regime and breaks down the moment that regime changes.

The stress scenario compounds the problem further. Adding a daily return penalty of −0.2 pp on top of doubled covariances pushes VaR 95% to −68.9% and ES 95% to −77.1%. The worst simulated outcome across 10,000 paths is −94.8%, near-total capital loss. Mean final return falls to +93% and the median collapses to +33%. Even the central outcome becomes modest and uncertain. The maximum of +2,808% confirms the distribution remains right-skewed; the issue is that the entire left tail has shifted dramatically to the downside.

The progression from base to stress illustrates the core principle this model was built to demonstrate. VaR is not a fixed number. It is a function of the market environment. A portfolio that looks reasonably hedged under normal conditions can carry extreme tail exposure once volatility rises, especially when it holds assets whose own volatility is regime-dependent.

Part 4: The Role of VIX

VIX is the unconventional element of this model and deserves its own discussion. It is not a conventional asset. You cannot buy it directly, its returns are mean-reverting, it exhibits strong jump behavior, and its historical mean return of +2.25% per day is a statistical artifact of large crisis spikes rather than a genuine drift. Knowing why it is included and what it actually contributes is essential to reading the results correctly.

The structural role of VIX in the portfolio is that of a convex hedge. It spikes sharply when equity markets fall, providing positive returns precisely when the 60% equity allocation is losing value. The −0.54 correlation with S&P 500 captures this relationship. In calm markets, VIX mean-reverts downward and acts as a drag on returns. In a crisis, it can more than offset equity losses. That asymmetric payoff structure is what makes the allocation defensible from a tail risk management perspective.

The practical limitation is significant. Modeling VIX returns with a multivariate normal distribution understates how extreme real-world spikes can be. During March 2020, VIX moved from roughly 15 to 85 in a matter of weeks, a move of more than 20 historical standard deviations. A normal simulation simply cannot generate that. The model produces plausible tail scenarios under ordinary stress, but will underestimate the severity of genuine market dislocations, which is exactly when the VIX allocation is most active. A Student-t copula or a regime-switching model for VIX would be the most impactful methodological extension of this framework.

Despite this limitation, the model correctly captures the structural effect. VIX is the primary source of both the portfolio's extreme upside scenarios (max final return of +2,716% in the base case) and its heavy-tailed downside distribution. The combination of a high observed mean and high variance produces the right-skewed, fat-tailed return distribution that appears consistently across all three scenarios.

Part 5: Project Files

The project is structured as a pipeline: parameter estimation feeds the simulation scripts, and unit tests verify the core risk functions independently of the simulation results. All scripts run from the same directory as the dataset.

Recommended execution order: estimate_params.py → portfolio_monte_carlo_simulation.py → portfolio_scenarios_simulation.py. Unit tests can be run at any point independently.

Part 6: Limitations and Extensions

Three assumptions in the current model are meaningful simplifications of real market dynamics. They do not undermine the model's core purpose, but they are worth understanding clearly before extending the framework.

These are real constraints, but they do not undermine what the model is designed to show. The simulation makes visible how correlation structure shapes diversification, how tail metrics differ from central tendency measures, and how dramatically risk estimates change across volatility regimes. The specific numbers matter less than the structural relationships they reveal.

Downloads

All files used to build the project: