📋 What You'll Learn
- The exact portfolio risk formula and what each variable means
- How to calculate portfolio risk for a 2-asset and multi-asset portfolio
- Why correlation is the most important variable in the equation
- The relationship between portfolio risk and expected return
- How to interpret your portfolio's risk level in plain terms
What Is Portfolio Risk?
Portfolio risk measures how much your portfolio's value could fluctuate over time. Specifically, it is the standard deviation of portfolio returns — a statistical measure of how widely actual returns are likely to deviate from the expected return.
A portfolio with high risk (high standard deviation) can swing dramatically up or down. A low-risk portfolio has more stable, predictable returns. Portfolio risk depends not just on the riskiness of individual holdings, but critically on how those holdings move relative to each other — this is what the portfolio risk formula captures.
💡 Why portfolio risk ≠ average individual risk
If you hold two stocks, each with 20% standard deviation, your portfolio risk is NOT necessarily 20%. It depends on their correlation. If they move in opposite directions, the risks partially cancel out. This is the mathematical basis for diversification — and why the portfolio risk formula matters.
The Portfolio Risk Formula
Two-Asset Portfolio Formula
For a portfolio containing two assets, portfolio risk (standard deviation) is calculated as:
| w₁, w₂ | Weight of Asset 1 and Asset 2 in the portfolio (must sum to 1.0) |
| σ₁, σ₂ | Standard deviation of returns for Asset 1 and Asset 2 (individual risk) |
| ρ₁₂ | Correlation coefficient between Asset 1 and Asset 2 (ranges from −1 to +1) |
| σp | Portfolio standard deviation — the overall risk of the combined portfolio |
Portfolio Variance Formula
The formula above calculates the square root of portfolio variance (σp²). You can also express it as:
Multi-Asset Portfolio Formula (Matrix Form)
For portfolios with three or more assets, the formula extends to a variance-covariance matrix:
| wᵀ | Transpose of the weight vector (a row vector of all asset weights) |
| Σ | The variance-covariance matrix — diagonal elements are individual variances (σᵢ²), off-diagonal elements are covariances (σᵢσⱼρᵢⱼ) between each pair of assets |
| w | Column vector of asset weights |
In plain terms: for every pair of assets in your portfolio, multiply their weights and their covariance, then sum everything up. For 10 assets, that means 10 variance terms and 45 covariance terms — which is why portfolio risk calculation quickly becomes impractical to do by hand for real portfolios.
How to Calculate Portfolio Risk: Step-by-Step Example
Let's calculate the risk of a simple two-asset portfolio using real numbers.
📊 Worked Example: 60% Stocks / 40% Bonds
Asset 1: US Stock ETF — Weight (w₁) = 0.60, Standard Deviation (σ₁) = 18%
Asset 2: US Bond ETF — Weight (w₂) = 0.40, Standard Deviation (σ₂) = 6%
Correlation (ρ₁₂) = −0.20 (stocks and bonds tend to move slightly opposite)
w₁²σ₁² = (0.60)² × (0.18)² = 0.36 × 0.0324 = 0.011664
w₂²σ₂² = (0.40)² × (0.06)² = 0.16 × 0.0036 = 0.000576
2w₁w₂σ₁σ₂ρ₁₂ = 2 × 0.60 × 0.40 × 0.18 × 0.06 × (−0.20) = −0.001296
σp² = 0.011664 + 0.000576 + (−0.001296)
σp = √0.010944
The portfolio has a standard deviation of 10.46%. This means in a typical year, returns will be within roughly ±10.46% of the expected return about 68% of the time.
Note the diversification benefit: The weighted average of individual risks would be (0.60 × 18%) + (0.40 × 6%) = 13.2%. The actual portfolio risk is only 10.46% — 2.74 percentage points lower because the negative correlation between stocks and bonds reduces overall volatility.
How Correlation Changes Portfolio Risk
Correlation is the single most powerful variable in the portfolio risk equation. Using the same 60/40 portfolio above, here's how different correlation assumptions change the result:
| Correlation (ρ) | Scenario | Portfolio Risk (σp) | Diversification Benefit |
|---|---|---|---|
| ρ = +1.0 | Assets move perfectly together | 13.20% | None — no benefit from diversification |
| ρ = +0.5 | Moderate positive correlation | 11.85% | 1.35% reduction |
| ρ = 0 | No correlation (independent) | 10.91% | 2.29% reduction |
| ρ = −0.20 | Slight negative (stocks/bonds typical) | 10.46% | 2.74% reduction |
| ρ = −1.0 | Perfect negative correlation | 6.40% | 6.80% reduction — maximum diversification |
🔑 The key insight
The portfolio risk formula reveals that you can reduce portfolio risk below the weighted average of individual risks whenever assets have a correlation below +1. This is the mathematical foundation of diversification — and why holding 20 stocks in the same sector provides almost no risk reduction (they're all highly correlated), while holding stocks and bonds does.
Portfolio Risk and Return Formula
Portfolio risk and return are calculated differently. Portfolio return is simply the weighted average of individual asset returns. Portfolio risk is not.
This asymmetry is fundamental: you can reduce portfolio risk through diversification (by choosing low-correlation assets) without necessarily reducing expected return. This is why building a diversified portfolio is not about choosing worse assets — it's about choosing assets that don't all fail at the same time.
Risk by Investor Profile (Annual Portfolio Standard Deviation)
Target annualized standard deviation by investor risk profile. Higher risk = wider range of potential outcomes.
Why Manual Portfolio Risk Calculation Has Limits
The two-asset formula is straightforward. But real portfolios have 10, 20, or 50+ positions. For a portfolio of n assets, the variance-covariance matrix requires:
- n variance terms (one per asset)
- n(n−1)/2 unique covariance terms (one per pair of assets)
For a 20-asset portfolio: 20 variances + 190 covariances = 210 calculations. For 50 assets: 1,275 covariance terms. This is why portfolio risk for real-world portfolios is calculated by software, not spreadsheets.
⚠️ Correlations change over time
The correlation coefficients in the portfolio risk formula are not fixed. Stocks and bonds, for example, were negatively correlated for most of the 2000s-2010s, but both fell sharply together in 2022. Using historical correlations without monitoring for changes means your risk calculation can become dangerously outdated — which is exactly why continuous portfolio monitoring matters.
How Guardfolio Calculates Portfolio Risk Automatically
Instead of running the portfolio risk formula manually, Guardfolio connects to your brokerage accounts (read-only) and calculates your portfolio's risk metrics continuously — including standard deviation, concentration risk, correlation between all holdings, and an overall risk score.
The free portfolio risk report at Guardfolio takes about 2 minutes, requires no account, and shows:
- Your portfolio risk score (0–100)
- Concentration risk — which positions dominate your risk
- Diversification analysis — whether your holdings are truly uncorrelated
- Volatility level vs your target risk profile
Guardfolio is an informational tool only. It does not provide financial advice or manage assets. All outputs are for educational purposes and should not be treated as investment recommendations.
Frequently Asked Questions
2. Find the standard deviation of each individual asset's historical returns.
3. Calculate the correlation coefficient between each pair of assets.
4. Plug into: σp² = w₁²σ₁² + w₂²σ₂² + 2w₁w₂σ₁σ₂ρ₁₂
5. Take the square root of the variance to get portfolio standard deviation.
For example, a 60/40 portfolio with stock σ = 18%, bond σ = 6%, and ρ = −0.20 gives a portfolio risk of approximately 10.46%.