Mastering Portfolio Risk: Balancing Risk and Reward for Better Investment Outcomes

Introduction: Why Portfolio Risk Matters

Whether you are a seasoned investor or just starting out, one key truth remains: understanding risk is just as important as chasing returns. The best-performing portfolios aren’t just the ones with the highest returns—they’re the ones that balance risk and reward effectively.

To achieve this balance, we need to understand portfolio returns, volatilities, which serve as the foundation for portfolio construction and risk management. In this section, we’ll break these concepts down in a clear, approachable way and demonstrate why they are critical in making informed investment decisions.

Portfolio return can be expressed as:

Yes, you can express the portfolio return as a product of single-day returns. This approach is particularly useful when you want to account for compounding effects over multiple periods. The formula for the cumulative return of a portfolio over multiple days can be written as:

\(R_p = \prod_{t=1}^{T} (1 + R_{p,t}) - 1\)

where \(R_p\) is the cumulative portfolio return over ( T ) days and \(R_{p,t}\) is the portfolio return on day t.

To calculate the daily portfolio return \(R_{p,t}\), you can use the weighted average of the daily returns of the individual assets:

\(R_{p,t} = \sum_{i=1}^{n} w_i \cdot R_{i,t}\)

where wi  is the weight of the ( i )-th asset in the portfolio. - \(R_{i,t}\) is the return of the ( i )-th asset on day t.

Combining these formulas, the cumulative portfolio return over multiple days can be expressed as:

\(R_p = \prod_{t=1}^{T} \left( 1 + \sum_{i=1}^{n} w_i \cdot R_{i,t} \right) - 1\)

This formula accounts for the compounding effect of daily returns over the entire period.

The portfolio volatility can be expressed as follows:

\(\sigma_p = \sqrt{ \sum_{i=1}^{n} \sum_{j=1}^{n} w_i w_j \, \text{Cov}(R_i, R_j) }\)

where \(\sigma_p\) is the portfolio standard deviation., wi is the weight of the ( i )-th asset in the portfolio, wj  is the weight of the ( j )-th asset in the portfolio.
\(\text{Cov}(R_i, R_j)\) is the covariance between the returns of the ( i )-th and ( j )-th assets. - n is the total number of assets in the portfolio.

The covariance matrix is crucial for understanding how different assets in a portfolio interact with each other. It helps in assessing the overall risk of the portfolio by considering not only the individual risks of the assets but also how they co-move. For example, if two assets have a positive covariance, they tend to move together, while a negative covariance indicates they move in opposite directions.

Constructing Sample Portfolios & Evaluating Weight Allocations

Now that we understand returns, volatilities, and the covariance matrix, it’s time to construct a real-world portfolio and assess whether it is properly allocated for risk and return.

Here's a diversified portfolio with 60% equity, 20% bonds, and 20% alternative investments, balancing growth potential, income stability, and diversification.

Equity Positions (60% Allocation - 12 Stocks)

Bond Positions (20% Allocation - 4 ETFs)

Alternative Investments (20% Allocation - 4 ETFs)

Portfolio Breakdown

• 60% Equities: Large-cap global stocks for growth.
• 20% Bonds: Diversified bond exposure for stability.
• 20% Alternatives: Gold, real estate, commodities, and Bitcoin for diversification.

A well-structured portfolio balances return potential while keeping risk exposure under control. But how do we determine if our allocations are effective?
In order to do so, we will construct multiple sample portfolios with different asset weightings and compare their portfolio volatility and expected returns. This allows us to identify whether the portfolio is too concentrated or well-diversified.

Let's analyse above portfolio return and volatility over the past year with an initial investment of 100'000$:

Final Portfolio Value: $118,146.81
Simple Return: 18.15%, Compounded Return: 18.93%

The simple (arithmetic) return is calculated as the percentage change in the value of an investment over a period of time by taking the ending value of the investment, subtract the beginning value, then divide the result by the beginning value. The geometric or compounded return we already introduced before.

Portfolio Volatility: 0.64%

Please note that the portfolio volatility is relatively low due to diversification effects caused by the returns of assets moving in opposite directions or not being correlated.

To determine optimal portfolio weight allocations, we can choose between Mean-Variance Optimisation (MVO) and Monte Carlo simulation. While MVO offers a mathematically precise and efficient solution for maximising returns or Sharpe ratios under linear constraints, we opt for the Monte Carlo approach as it provides greater flexibility for future extensions of our model. Unlike MVO, which is limited to linear and static assumptions, Monte Carlo allows us to easily incorporate more complex or dynamic constraints—such as position limits, rebalancing rules, or scenario-based simulations—as our portfolio strategy evolves.

Optimising Portfolio Weights Using Monte Carlo Simulation

Instead of guessing the best portfolio allocation, we let Monte Carlo simulations do the work by testing thousands of combinations and analysing the results.

• Step 1: Generate random weight allocations for 100'000 of portfolios.
• Step 2: Compute expected return and volatility for each portfolio.
• Step 3: Calculate Sharpe Ratio to rank portfolios.
• Step 4: Identify the portfolio with the best risk-return tradeoff.

The Sharpe Ratio is a measure used to evaluate the risk-adjusted return of an investment portfolio. It helps investors understand how much excess return they are receiving for the extra volatility they endure by holding a riskier asset. A higher Sharpe Ratio indicates a more attractive risk-adjusted return.

The formula for the Sharpe Ratio is: \(\text{Sharpe Ratio} = \frac{R_p - R_f}{\sigma_p}\)

where \(R_p\) is the expected portfolio return, \(R_f\)is the risk-free rate (the return of a risk-free investment). \(\sigma_p\)  is the portfolio standard deviation (a measure of the portfolio's volatility).

Optimal portfolio: Highest Sharpe Ratio Portfolio: The allocation that provides maximum return per unit of risk
MVP: lowest portfolio still on efficient frontier

Please note that a Monte Carlo simulation also allows to analyse robustness of your portfolio by altering assets weights. Below graph shows the distribution of return and volatilty of each simulations outcome:

Disclaimer: This analysis is based exclusively on historical returns and volatilities. The simulation assumes that past asset return distributions—specifically means and covariances—will remain consistent in the future. However, market conditions are inherently dynamic, and relying on historical data introduces a degree of uncertainty. Portfolios that are highly optimised to past performance may be prone to overfitting and could underperform in evolving market environments.

Once the Monte Carlo simulations are complete and stored, the resulting data can be used to derive meaningful statistical insights. For instance, you can estimate the probability of achieving a target return by calculating the proportion of simulations exceeding that threshold—for example, np.mean(df['Return'] > 0.18) estimates the likelihood of returns surpassing 18%. Additionally, percentile analysis provides insight into downside risk or tail events; for example, np.percentile(df['Return'], 5) captures the 5th percentile return, helping quantify potential losses in adverse scenarios.

While Monte Carlo simulations rely on historical parameters, they offer the flexibility to stress test a wide range of risk-return scenarios. By adjusting input assumptions—such as lowering expected returns to model a bear market, increasing volatility, or introducing non-normal return distributions—we can explore how portfolios might behave under various conditions. Portfolios that consistently perform well across thousands of simulations are likely more resilient. Rather than over-optimising for a single, fixed scenario, Monte Carlo simulations help assess portfolio robustness under uncertainty.

Identifying Risk Drivers Using PCA Analysis

Now that we have optimised our portfolio weights using Monte Carlo simulation, it’s time to dive deeper and understand what’s driving our portfolio risk.

Even if we have optimised for maximum return per unit of risk, we still need to ask: Where does the risk in our portfolio come from, are we overly exposed to a single market factor (e.g., equities, interest rates) and how well is our risk diversified across independent drivers?

To explore these questions, we will apply Principal Component Analysis (PCA), a robust statistical method. While traditional thinking suggests that simply holding a variety of assets reduces risk, this doesn’t hold if those assets are strongly correlated.

PCA digs deeper by analysing the portfolio’s covariance structure to identify independent sources of risk. This helps us detect any overexposure to specific factors and assess whether genuine diversification exists. Moreover, PCA allows us to connect these risk factors—principal components—to real-world market influences like broad equity movements, interest rate shifts, or region- and sector-specific trends.

How PCA Works in Portfolio Risk Analysis

PCA transforms the covariance matrix of asset returns into principal components (PCs) by extracting two key elements: eigenvalues and eigenvectors:

• Eigenvalues tell us how much of the total risk (or variance) each principal component explains. The higher the eigenvalue, the more important that component is. If for instance PC1 dominates (e.g., explains >70% variance), the portfolio relies too much on one risk factor, on the other hand if the variance is spread across multiple PCs, the portfolio has better risk diversification.

• Eigenvectors show the weight or influence each asset has on a given principal component. They define the direction of each component in terms of the original assets.

Finally, we look at how much variance each principal component captures and try to relate it to real-world risk drivers. For example, one component might reflect broad market risk, while another could capture interest rate sensitivity or sector-specific exposure.

Mapping Principal Components to Market Risk Drivers

If the first principal component accounts for most of the portfolio’s risk, it may be necessary to adjust the asset weights to reduce reliance on a single risk factor. On the other hand, if the variance is spread more evenly across several components, the portfolio is likely well-diversified, and no major changes are needed. However, if one asset contributes disproportionately to the dominant component, it might be worth considering alternatives with lower correlations, such as commodities or other non-traditional investments.

With our newly gained understanding, we can now revisit the sample portfolio and draw the following insights.

Note: An optimal PCA output shows well-distributed variance across multiple components and balanced, same-direction loadings per component. This means the portfolio benefits from genuine diversification, avoids unnecessary internal volatility, and is resilient to changes in single market factors.

Analysing Internal Portfolio Risk Across Portfolio Variations

Even if overall portfolio volatility is low, that doesn’t always mean the portfolio is well-diversified. If most of the risk comes from just one or two main drivers, the portfolio could still be vulnerable. To explore this, we introduce Internal Portfolio Risk (IPR) - a measure that captures how much of the portfolio’s risk comes from independent sources instead of highly correlated movements.

In our portfolio, overall volatility is 0.64%, while internal portfolio risk is significantly lower at around 0.11%. This gap indicates that most of the risk is concentrated in just a few dominant factors. To address this, we should consider rebalancing the portfolio to distribute risk more evenly, introducing lower-correlated assets like commodities or alternatives, or reducing exposure to heavily weighted market drivers such as those captured by PC1.

Conclusion

This analysis provided a structured, data-driven approach to understanding and optimising portfolio risk. We covered key techniques such as calculating returns and volatility, analysing risk through the covariance matrix, optimising weights with Monte Carlo simulations, and using PCA to uncover hidden sources of risk.

A few important insights emerged. Diversification is not just about holding many assets - it’s about understanding the underlying sources of risk. Monte Carlo simulations allowed us to test thousands of portfolio variations and identify those with the best risk-return trade-offs, visualised on the efficient frontier. PCA helped reveal whether the portfolio is overly dependent on a single risk factor, like PC1, and allowed us to connect these components to real-world market influences. Finally, comparing Internal Portfolio Risk to overall volatility served as a valuable check: if IPR is significantly lower, it indicates risk is concentrated in just a few factors. In such cases, rebalancing the portfolio and adding less correlated assets can help distribute risk more effectively.

Alberto Desiderio is deeply passionate about data analytics, particularly in the contexts of financial investment, sports, and geospatial data. He thrives on projects that blend these domains, uncovering insights that drive smarter financial decisions, optimise athletic performance, or reveal geographic trends.