Portfolio Standard Deviation Calculator
Calculate Portfolio Risk (Standard Deviation)
Analyze investment portfolio risk with correlation effects. Calculate diversification benefits, optimal allocation, and risk-adjusted returns.
Portfolio Risk Analysis
12.25%
Diversification Benefit
Correlation Impact
Step-by-Step Calculation:
Risk-Return Visualization:
Portfolio Analysis:
Portfolio standard deviation measures total risk, considering individual asset risks and their correlations.
What is Portfolio Standard Deviation?
Portfolio standard deviation is a statistical measure of the total risk or volatility of an investment portfolio. It quantifies how much the portfolio's returns are expected to fluctuate around their mean. Unlike measuring individual asset risk, portfolio standard deviation accounts for diversification benefits through correlation between assets.
Portfolio Risk Formulas
2-Asset Portfolio
Two assets only
Exact formula
N-Asset Portfolio
General formula
Covariance matrix
Covariance
Relationship measure
From correlation
Diversification
Risk reduction
Diversification effect
Mathematical Formulation
1. Two-Asset Portfolio Variance
2. General N-Asset Portfolio
3. Matrix Notation
Risk Measures Comparison
| Risk Measure | Formula | Interpretation | Application |
|---|---|---|---|
| Standard Deviation | σ = √Variance | Total volatility | Overall portfolio risk |
| Beta | β = Cov(rᵢ,rₘ)/σₘ² | Systematic risk | Market risk exposure |
| Value at Risk (VaR) | VaR = μ - zσ | Worst-case loss | Risk management |
| Sharpe Ratio | S = (rₚ - r_f)/σₚ | Risk-adjusted return | Performance evaluation |
Real-World Applications
Investment Management
- Portfolio construction: Building efficient portfolios with optimal risk-return tradeoff
- Asset allocation: Determining optimal mix of stocks, bonds, and alternatives
- Risk management: Setting risk limits and monitoring portfolio volatility
- Performance attribution: Analyzing sources of portfolio risk and return
Corporate Finance
- Capital budgeting: Evaluating project risk in portfolio context
- Mergers & acquisitions: Assessing diversification benefits of combined entities
- Risk hedging: Using negatively correlated assets to reduce overall risk
- Capital structure: Considering business risk in financing decisions
Personal Finance
- Retirement planning: Building diversified portfolios for long-term goals
- Education savings: Age-appropriate asset allocation
- Risk tolerance assessment: Matching portfolio risk to investor profile
- Estate planning: Risk management for wealth preservation
Institutional Investing
- Pension funds: Liability-driven investing with risk control
- Endowments: Long-term portfolio management with diversification
- Hedge funds: Absolute return strategies with risk management
- Insurance companies: Asset-liability matching
Correlation Effects on Portfolio Risk
| Correlation | Risk Reduction | Example Portfolio | Practical Implication |
|---|---|---|---|
| ρ = -1 | Maximum (can achieve zero risk) | Long stock + Put option | Perfect hedge possible |
| ρ = -0.5 | Significant reduction | Stocks + Gold | Good diversification |
| ρ = 0 | Moderate reduction | Stocks + Real Estate | Some diversification |
| ρ = +0.5 | Limited reduction | Tech stocks + Growth stocks | Limited diversification |
| ρ = +1 | No reduction | S&P 500 + Total stock market | No diversification |
Step-by-Step Calculation Examples
Example 1: Two-Asset Portfolio
Given: w₁ = 0.6, σ₁ = 15%, w₂ = 0.4, σ₂ = 25%, ρ = 0.3
- Calculate weighted variances: w₁²σ₁² = 0.6² × 0.15² = 0.36 × 0.0225 = 0.0081
- w₂²σ₂² = 0.4² × 0.25² = 0.16 × 0.0625 = 0.01
- Calculate covariance term: 2w₁w₂σ₁σ₂ρ = 2 × 0.6 × 0.4 × 0.15 × 0.25 × 0.3 = 0.0054
- Sum for variance: σₚ² = 0.0081 + 0.01 + 0.0054 = 0.0235
- Take square root: σₚ = √0.0235 = 0.1533 = 15.33%
- Weighted average volatility: 0.6×15% + 0.4×25% = 19%
- Diversification benefit: 19% - 15.33% = 3.67% risk reduction
Example 2: Three-Asset Portfolio
Given: w = [0.5, 0.3, 0.2], σ = [12%, 18%, 22%], ρ matrix with all correlations = 0.2
- Calculate variance terms: Σ wᵢ²σᵢ² = 0.5²×0.12² + 0.3²×0.18² + 0.2²×0.22² = 0.0036 + 0.002916 + 0.001936 = 0.008452
- Calculate covariance terms: For each pair i≠j: wᵢwⱼσᵢσⱼρᵢⱼ
- Sum all covariance terms: 2×(0.5×0.3×0.12×0.18×0.2 + 0.5×0.2×0.12×0.22×0.2 + 0.3×0.2×0.18×0.22×0.2) = 0.002016
- Total variance: 0.008452 + 0.002016 = 0.010468
- Portfolio standard deviation: √0.010468 = 0.1023 = 10.23%
Example 3: Perfect Hedge (ρ = -1)
Given: w₁ = 0.6, σ₁ = 20%, w₂ = 0.4, σ₂ = 30%, ρ = -1
- Variance: σₚ² = 0.6²×0.2² + 0.4²×0.3² + 2×0.6×0.4×0.2×0.3×(-1) = 0.0144 + 0.0144 - 0.0288 = 0
- Standard deviation: σₚ = √0 = 0% (zero risk!)
- Condition for zero risk: w₁/w₂ = σ₂/σ₁ = 30%/20% = 1.5, which matches 0.6/0.4 = 1.5
- Interpretation: Perfect negative correlation allows complete risk elimination with proper weights
Modern Portfolio Theory Concepts
Related Calculators
Frequently Asked Questions (FAQs)
Q: What's the difference between standard deviation and beta?
A: Standard deviation measures total risk (both systematic and unsystematic). Beta measures only systematic risk (market risk). Diversification eliminates unsystematic risk, so for well-diversified portfolios, beta is more relevant.
Q: How does correlation affect portfolio risk?
A: Lower correlation reduces portfolio risk through diversification. Negative correlation provides maximum diversification benefits. Positive correlation provides little diversification. Correlation of +1 gives no diversification benefit.
Q: What is a "good" portfolio standard deviation?
A: It depends on investor risk tolerance: Conservative: 5-10%, Moderate: 10-15%, Aggressive: 15-20%+. Compare to benchmark: S&P 500 historical σ ≈ 15%. Lower is better for same return (higher Sharpe ratio).
Q: Can portfolio standard deviation be lower than all individual assets?
A: Yes! Through diversification with assets having correlation less than +1, portfolio standard deviation can be lower than the weighted average of individual volatilities, and can even be lower than the least volatile asset in the portfolio.
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