Risk Glossary: VaR, Beta, Correlation, Volatility

These four risk metrics quantify different exposures. Together they describe what can go wrong and how fast — but each has a blind spot that catches traders who use it in isolation. This glossary covers the formulas, the regime dependence, and the integration rules that turn them into a working position-sizing system.

Core Concepts: Four Metrics, Four Exposures

VaR (Value at Risk) is the maximum expected loss over a given horizon at a confidence level. A 1-day 95% VaR of $10,000 means: on 95% of days, the loss stays under $10,000; on 5% of days (roughly 1 in 20), it can be worse. Calculation methods: Historical reorders past returns and takes the 5th percentile; Variance-covariance assumes a normal distribution (fast but understates tail risk); Monte Carlo simulates thousands of paths (most flexible, most compute-heavy). VaR's blind spot: it says nothing about the size of losses beyond the threshold. The 5% tail can be 1.5× VaR (mild) or 5× VaR (2008-style). Use Conditional VaR (Expected Shortfall) alongside — it averages the tail.

Beta is the sensitivity of an asset's returns to a benchmark. Beta 1.2 vs SPX: when SPX moves +1%, the asset moves +1.2% on average. Beta > 1 amplifies the market; beta < 1 dampens it; beta < 0 moves opposite (rare; put options, some hedges). Beta is regime-dependent — a stock can have beta 0.6 in calm markets and 1.4 in crashes, because correlations spike to 1 in stress. Use rolling 60-day beta, not 3-year.

Correlation is the linear co-movement between two assets, range −1 to +1. +1 is perfectly synchronized; 0 is no linear relationship; −1 is perfectly inverse. Correlation is unstable: the 60-day correlation between equities and bonds has been +0.3 (1970s), −0.4 (2000s), and +0.6 (2022). The dangerous pattern is assets that historically show low correlation suddenly moving to +0.8 in a crash — this is why "diversified" portfolios still lost 30% in 2008.

Volatility is the standard deviation of returns — the size of typical price swings. Two flavors: Realized vol is backward-looking, computed from actual returns and annualized by multiplying daily stdev by √252 (~15.87). Implied vol is forward-looking, extracted from option prices; VIX is 30-day implied vol on SPX, annualized. Implied > realized means options are "expensive" (sell premium); implied < realized means options are "cheap" (buy premium). The spread (variance risk premium) averages 3–5 vol points on SPX.

Concrete example: a $100,000 account holds $50,000 in an SPX-tracking ETF (beta 1.0) and $50,000 in a stock with beta 1.4. Portfolio beta = (0.5 × 1.0) + (0.5 × 1.4) = 1.2. A 1-day SPX move of −2% implies a −2.4% portfolio move, or −$2,400. If 1-day 99% VaR per position must stay under 1% of account ($1,000), the beta-1.4 position is too large and must be cut.

Practical Application: Integrating the Four Metrics

Step 1 — Compute 1-day VaR per position and cap it. Set position size so 1-day 99% VaR per position is under 1% of account. For a $100,000 account, that is $1,000 max VaR per position.

Metric	Target	How to set
1-day 99% VaR / position	≤1% of account	Size by historical vol × 2.33 (z-score)
Portfolio beta	≤ your risk tolerance (typ. 0.8–1.2)	Weighted avg of position betas
Max single-position correlation	Stress at +0.8 across all	Worst-case drawdown test
Realized vs implied vol	Track spread for option trades	VRP = implied − realized

Step 2 — Limit portfolio beta to your risk tolerance. Sum (position weight × position beta) = portfolio beta. A conservative target is 0.8; a market-neutral book targets 0.0; an aggressive long-only book may accept 1.2.

Step 3 — Stress-test correlations at +0.8 across all positions for the worst-case drawdown. If that scenario breaks your max drawdown rule (e.g., 20%), you are over-leveraged or under-diversified. Re-measure correlations during volatility spikes — do not assume the calm-market value holds.

Step 4 — Use the implied/realized vol spread for option decisions. When implied > realized by 3+ points, selling premium is statistically favorable; when implied < realized, buying premium is favored.

Risk integration checklist:

• 1-day 99% VaR per position ≤1% of account
• Portfolio beta within tolerance (0.8–1.2)
• Worst-case drawdown tested at +0.8 correlation
• Rolling 60-day beta used (not 3-year)
• Implied vs realized vol tracked for option trades
• All metrics logged at /journal

Common Mistakes

Mistake 1: Trusting VaR alone without Expected Shortfall. VaR tells you the threshold, not the tail size — a 5% tail can be 5× VaR in a crash. Correction: always report Conditional VaR alongside VaR; it averages losses beyond the threshold.

Mistake 2: Using 3-year beta in a volatile regime. Long-window beta hides regime shifts; a stock can look calm over 3 years and spike to 1.4 in a crash. Correction: use rolling 60-day beta and re-measure during volatility spikes.

Mistake 3: Assuming diversification holds in a crash. Low calm-market correlations jump to +0.8 in stress, collapsing diversification exactly when you need it. Correction: stress-test the portfolio at +0.8 across all positions; if max drawdown breaks, cut size before the crash, not during it.

Advanced Tips

For multi-asset portfolios, use Monte Carlo VaR with fat-tailed distributions (Student's t) to capture tail risk the variance-covariance method misses. Track the equity-bond correlation monthly — it has flipped positive since 2022, breaking the classic 60/40 hedge. Model portfolio risk in /tools and reconcile against realized drawdowns in your journal at /journal quarterly. Validate any new sizing rule on /paper before deploying live capital.

Summary

VaR, beta, correlation, and volatility each measure a different risk exposure, and each has a blind spot. Cap 1-day 99% VaR per position at 1% of account, limit portfolio beta to your tolerance, stress-test correlations at +0.8, and pair VaR with Expected Shortfall. Used together with rolling 60-day windows, these metrics turn risk from a feeling into a system that survives the tails.