Portfolio Risk Simulator (USA)
Calculate portfolio risk using the formula: Portfolio Variance = ∑ (Weight_i² * Variance_i) + ∑∑ (Weight_i * Weight_j * Covariance_ij)
How to Calculate Portfolio Risk
The portfolio variance is calculated using:
Where:
- PV: Portfolio Variance
- W_i: Weight of asset i (proportion of portfolio)
- V_i: Variance of asset i
- C_ij: Covariance between assets i and j
Calculator: Portfolio Risk Analysis
Correlation Matrix
| Asset | Asset 1 | Asset 2 | Asset 3 |
|---|
Risk Breakdown
Risk Contribution by Asset
Risk Benchmarks
Analysis & Recommendations
Your portfolio has a 0.012 variance and 11.0% volatility indicating moderate risk.
- Consider adding more uncorrelated assets to reduce portfolio risk
- Review asset weights to optimize the risk-return profile
- Monitor correlations between assets as they may change over time
- Rebalance your portfolio periodically to maintain target risk levels
Understanding Portfolio Risk
What is Portfolio Risk?
Portfolio risk measures the uncertainty or variability of returns in an investment portfolio. It quantifies the likelihood that actual returns will deviate from expected returns. Portfolio risk is influenced by both individual asset risks and the relationships between assets.
How the Formula Works
The portfolio variance formula accounts for two main components of portfolio risk:
- Individual Asset Risk: The weighted sum of each asset's variance
- Diversification Effect: The weighted sum of covariances between assets
By combining assets with low or negative correlations, portfolio risk can be reduced below the weighted average of individual asset risks.
Important Considerations
- This calculation uses historical variance and correlation data which may not reflect future relationships
- Correlations between assets can change during market stress
- Portfolio risk is sensitive to asset weight allocations
- Tax implications vary by account type and holding period
- Estimating accurate correlations requires sufficient historical data
Portfolio Risk Quiz
Question 1: Portfolio Variance Calculation
For a 2-asset portfolio with weights of 60% and 40%, variances of 0.04 and 0.09, and covariance of 0.02, what is the portfolio variance?
Using the formula: PV = W₁²·V₁ + W₂²·V₂ + 2·W₁·W₂·C₁₂
PV = (0.6)²·(0.04) + (0.4)²·(0.09) + 2·(0.6)·(0.4)·(0.02)
PV = 0.36·0.04 + 0.16·0.09 + 2·0.24·0.02
PV = 0.0144 + 0.0144 + 0.0096 = 0.0384
Note: The closest option is a) 0.0436 (possible rounding in options)
This question demonstrates the importance of both individual asset risks and their interrelationships in determining portfolio risk. The covariance term captures how assets move together, which is crucial for diversification benefits.
- Remember the factor of 2 in the covariance term for a 2-asset portfolio
- Portfolio variance is always between the weighted average of individual variances and their maximum possible combined risk
Question 2: Diversification Impact
Two assets each have 15% standard deviation. If the correlation between them is -1, what is the portfolio standard deviation when equally weighted?
Calculate the portfolio variance and standard deviation.
With correlation ρ = -1, variances σ₁² = σ₂² = 0.15² = 0.0225, and weights W₁ = W₂ = 0.5:
Covariance C₁₂ = ρ·σ₁·σ₂ = (-1)·(0.15)·(0.15) = -0.0225
PV = W₁²·V₁ + W₂²·V₂ + 2·W₁·W₂·C₁₂
PV = (0.5)²·(0.0225) + (0.5)²·(0.0225) + 2·(0.5)·(0.5)·(-0.0225)
PV = 0.005625 + 0.005625 - 0.01125 = 0
Portfolio Standard Deviation = √0 = 0%
This represents perfect negative correlation, eliminating all portfolio risk.
Perfect Negative Correlation: When two assets move in exactly opposite directions, allowing for complete risk elimination in a portfolio.
- Portfolio risk can be reduced when assets have correlations less than 1
- Perfect negative correlation (ρ = -1) can theoretically eliminate all portfolio risk
Question 3: Correlation Effects
How does increasing the correlation between two assets from 0 to 0.8 affect the portfolio risk of an equally weighted portfolio?
Increasing correlation from 0 to 0.8 means the assets move more similarly. This reduces the diversification benefit and increases portfolio risk. The covariance term in the portfolio variance formula becomes larger and positive, increasing the overall variance.
Answer: b) Increases portfolio risk
- Thinking higher correlation leads to lower risk
- Confusing correlation with causation
Question 4: Risk Reduction Potential
Which pair of assets would provide the greatest diversification benefit in a portfolio?
Compare different correlation values.
The greatest diversification benefit occurs when assets have negative correlation. With negative correlation, when one asset performs poorly, the other tends to perform well, reducing overall portfolio risk. The most negative correlation option is d) -0.5.
Answer: d) Correlation of -0.5
Question 5: Portfolio Risk Formula Components
In the portfolio variance formula, what does the term ∑∑ (Weight_i * Weight_j * Covariance_ij) represent?
The double summation term ∑∑ (Weight_i * Weight_j * Covariance_ij) represents the interaction effects between all pairs of assets in the portfolio. This term captures how assets move together and forms the basis of diversification benefits.
Answer: b) Combined diversification effects between assets
Remember that portfolio risk is not simply the weighted average of individual asset risks. The correlation structure between assets plays a crucial role in determining overall portfolio risk. Effective diversification focuses on combining assets with low or negative correlations.
Q&A
Q: How accurate is the portfolio variance formula in predicting actual portfolio risk?
A: The portfolio variance formula provides a foundational approach to risk assessment, but has important limitations:
Accurate Aspects:
- Quantifies diversification benefits mathematically
- Provides relative risk comparisons between portfolios
- Illustrates the impact of correlation on portfolio risk
Limitations:
- Based on historical data which may not predict future relationships
- Assumes normally distributed returns (real returns often have fat tails)
- Doesn't account for extreme market events or black swan events
- Correlations can change dramatically during market stress
Modern risk management often supplements this with scenario analysis, stress testing, and Value-at-Risk models for more robust risk assessment.
Q: How many assets should I include in my portfolio for optimal diversification?
A: Research suggests that most diversification benefits are achieved with 20-30 well-selected assets:
Optimal Range:
- 5-10 assets: Significant unsystematic risk reduction
- 15-20 assets: Most diversification benefits realized
- 20-30 assets: Near-optimal diversification
- 30+ assets: Diminishing marginal benefits
Quality vs Quantity:
- Select assets with low correlations rather than just adding more assets
- Consider different asset classes, sectors, and geographies
- Focus on fundamental diversification rather than superficial variety
Practical Considerations:
- Transaction costs increase with more positions
- Monitoring complexity grows with portfolio size
- Index funds can provide instant diversification
Remember, diversification cannot eliminate systematic (market) risk, only unsystematic (specific) risk.
Q: How often should I recalculate and rebalance my portfolio risk?
A: The frequency of risk assessment and rebalancing depends on your investment strategy:
Quarterly Review:
- Assess portfolio risk metrics and correlations
- Review asset allocation versus targets
- Update assumptions based on market conditions
Annual Rebalancing:
- Reset weights to target allocations
- Adjust for life circumstances or goals
- Reassess risk tolerance
Trigger-Based Rebalancing:
- When allocations drift beyond 5% from target
- After significant market moves (>20% change)
- Major life events (marriage, job change, etc.)
Special Circumstances:
- During market stress, monitor weekly
- For aggressive portfolios, consider monthly reviews
- Approaching retirement, increase monitoring frequency
Regular monitoring ensures your portfolio maintains its intended risk profile over time.