Value at Risk (VaR)

The Value at Risk is the standard risk metric in institutional finance. It answers a precise question: "What is the maximum loss I should expect over one trading day, at a given confidence level?"

Method 1 — Historical Simulation (used here)

Ranks all observed daily returns from worst to best and reads the percentile directly. No distributional assumptions. The most honest method when tail events actually occurred in the historical window.

Method 2 — Parametric (Normal Distribution)

Assumes returns are normally distributed. Faster computation but systematically underestimates risk for stocks with fat tails or skew.

How to Interpret VaR

• VaR 95% = 2.3%: On 95% of trading days, the loss should not exceed 2.3%. On 1 day in 20, it can.
• VaR 99% = 4.1%: On 99% of trading days, loss should stay below 4.1%. On 1 day in 100, it can exceed this.
• Dollar impact: If VaR 95% = 2.3% and stock = $200 → max expected 1-day loss = $4.60/share.
• Parametric > Historical VaR: Suggests the distribution is thinner than observed history (possible in low-vol regime). The reverse (historical > parametric) signals fat tails.

CVaR — Expected Shortfall (ES)

The Conditional VaR (CVaR), also called Expected Shortfall, measures the average loss in the worst X% of scenarios — what you actually lose when losses go beyond VaR.

CVaR vs VaR Interpretation

• CVaR/VaR ≈ 1.1–1.3: Thin tails — normal distribution is a reasonable approximation
• CVaR/VaR ≈ 1.3–1.8: Moderate fat tails — common for individual equities
• CVaR/VaR > 1.8: Very fat tails — extreme events are significantly worse than VaR suggests. Increase risk buffers.

Ulcer Index

The Ulcer Index (Peter Martin, 1987) measures the depth AND duration of drawdowns. Unlike standard deviation, it only penalizes downside moves and rewards stocks that recover quickly.

Ulcer Index Interpretation

• UI < 5: Low stress — consistent returns, minimal drawdown pain
• UI 5–10: Moderate — typical for diversified growth equities
• UI 10–20: High — significant drawdown exposure, consider reducing size
• UI > 20: Severe — avoid or use only with strict stop-loss discipline

Sharpe Ratio

The Sharpe Ratio (William Sharpe, 1966) is the most universal risk-adjusted return metric. It measures how much excess return you earn per unit of total risk.

Interpretation Thresholds

• Sharpe < 0: Underperforming risk-free cash — not worth the risk
• Sharpe 0–0.5: Poor — barely justifies risk taken
• Sharpe 0.5–1.0: Average — acceptable for individual stocks
• Sharpe 1.0–2.0: Good — institutional quality returns
• Sharpe > 2.0: Excellent — world-class hedge fund territory
• Sharpe > 3.0: Exceptional — rare, often indicates a specific alpha source

Sortino Ratio

The Sortino Ratio is a refinement of Sharpe that only penalizes downside deviation — negative returns below a minimum acceptable return (MAR, here = 0).

Calmar Ratio

The Calmar Ratio (Terry Young, 1991) relates annual return to maximum drawdown. Preferred by CTAs (Commodity Trading Advisors) and trend-following funds.

Omega Ratio

The Omega Ratio (Keating & Shadwick, 2002) is the most complete single-metric representation of a return distribution. It captures all moments (mean, variance, skewness, kurtosis) simultaneously.

Win Rate & Profit Factor

Information Ratio

Return per unit of total risk without subtracting the risk-free rate. Useful for assessing return consistency regardless of market environment.

What is Geometric Brownian Motion?

The Geometric Brownian Motion (GBM) is the mathematical foundation of the Black-Scholes model and most quantitative finance. It models stock prices as a continuous random process where percentage changes (not absolute changes) follow a normal distribution — ensuring prices can never go negative.

The (μ − σ²/2) Itô Correction

This is the most commonly misunderstood element. In GBM, the drift is adjusted downward by σ²/2. This "Itô correction" accounts for the mathematical difference between arithmetic and geometric returns. Without it, simulated paths would systematically overestimate prices.

Drift Model Options

• Historical (default): Uses the actual mean daily return observed in the data as μ. Most realistic but backward-looking.
• Zero Drift: Sets μ = 0. Pure volatility-driven simulation. Best for asking "what's the range of outcomes with no directional edge?"
• Risk-Free (5.25%): Uses the current Fed Funds rate as μ. Shows what the distribution looks like under risk-neutral pricing — the foundation of options theory.

Reading the Fan Chart

• Dark outer band (P5–P95): 90% confidence interval — 90% of simulated paths end within this range
• Light inner band (P25–P75): Interquartile range — the "most likely" 50% of outcomes
• Median line (P50): The central trajectory — 50% of paths above, 50% below
• Bull line (P95): The optimistic case — only 5% of paths end higher
• Bear line (P5): The pessimistic case — only 5% of paths end lower

Probability Statistics Explained

What is a Drawdown?

A drawdown measures the decline from a portfolio or stock's peak value to any subsequent trough. It is the primary metric used by professional investors to assess downside risk in practical terms — more intuitive than standard deviation because it directly answers: "How much did I lose from my highest point?"

The Underwater Chart Explained

The underwater chart shows the drawdown at every point in time. When the line is at 0%, the stock is at an all-time high. The deeper the line goes, the further the stock is from its last peak. The chart reveals:

• Depth: How far did the stock fall? (the vertical axis)
• Duration: How long did it stay underwater? (the horizontal span)
• Recovery: How quickly did it return to new highs? (return to 0%)

Drawdown Classification

• Minor (<5%): Normal market noise — present in virtually all stocks
• Moderate (5–10%): Standard correction — occurs several times per year in bull markets
• Significant (10–20%): Official "correction" territory — often triggers technical signals
• Severe (20–35%): Bear market territory — typical in sector rotation or macro shocks
• Catastrophic (>35%): Crisis-level — company-specific events or systemic crash

Key Drawdown Metrics

Connection to Other Metrics

• Calmar Ratio: Annual Return / MDD → the higher, the better
• Ulcer Index: Captures both depth and duration of drawdowns
• CVaR: Drawdown-equivalent for single-day distribution tail
• Recovery factor: Total return / MDD → used in managed futures

Why Distribution Analysis Matters

Almost all classical finance models (VaR, Black-Scholes, Sharpe) assume returns follow a normal distribution. In reality, stock returns are non-normal: they have fat tails, are negatively skewed, and exhibit time-varying volatility. Understanding how a stock deviates from normality is critical for proper risk management.

Skewness

Skewness measures the asymmetry of the return distribution relative to its mean.

Investment Implications

• Positive skew (>0): More small losses, occasional large gains — momentum/growth characteristics. Sortino will be higher than Sharpe.
• Negative skew (<0): More small gains, occasional large losses — typical of options selling or defensive stocks. VaR underestimates true tail risk. This is the more dangerous case.

Excess Kurtosis (Fat Tails)

Kurtosis measures the weight of the tails relative to a normal distribution. The normal distribution has kurtosis = 3; we display excess kurtosis = kurtosis − 3.

Jarque-Bera Normality Test

The Jarque-Bera test formally tests whether the distribution of returns is statistically different from a normal distribution, combining both skewness and kurtosis.

Autocorrelation (Return Predictability)

Autocorrelation measures whether today's return predicts tomorrow's return. The Efficient Market Hypothesis (EMH) states autocorrelation should be zero at all lags.

Significant Autocorrelation Patterns

• Significant positive AC at lag 1–5: Short-term momentum — price tends to continue in same direction. Can be exploited with momentum strategies.
• Significant negative AC at lag 1: Short-term mean reversion — common in very liquid large-caps. Contrarian signals at short horizon.
• Significant positive AC at lag 5 or 20: Weekly/monthly seasonality effects. Well-documented in academic literature.

The Black-Scholes Model

Published by Fischer Black and Myron Scholes in 1973 (Nobel Prize 1997), the Black-Scholes model revolutionized finance by providing an analytical formula to price European options. It assumes the underlying follows GBM with constant volatility and interest rates.

The 5 Key Assumptions

• The underlying follows GBM with constant drift μ and volatility σ
• No dividends during the option's life
• Constant risk-free interest rate r
• No transaction costs or taxes
• European option only (exercisable at expiry only)

Put-Call Parity

An iron law of derivatives pricing: call price minus put price must equal the discounted difference between spot and strike. Any deviation creates an arbitrage opportunity.

Moneyness

• ITM (In The Money): Call: S > K · Put: S < K → has intrinsic value
• ATM (At The Money): S ≈ K → maximum time value, maximum gamma
• OTM (Out of The Money): Call: S < K · Put: S > K → pure time value only

The Option Greeks — Complete Guide

Delta (Δ) — Directional Exposure

Gamma (Γ) — Rate of Change of Delta

Theta (Θ) — Time Decay

Vega (ν) — Volatility Sensitivity

Rho (ρ) — Interest Rate Sensitivity

Why Rolling Metrics?

A single static metric (e.g., Sharpe = 1.2) hides critical information: when was performance good or bad? Rolling metrics reveal regime changes, deteriorating quality, and time-varying risk — essential for dynamic portfolio management.

Rolling 30-Day Sharpe Ratio

How to Interpret Rolling Sharpe

• Consistently above 1.0: Persistently strong risk-adjusted returns — excellent sign
• Crossing from positive to negative: Regime change — quality of returns deteriorating
• Spikes above 3.0: Often precede mean reversion — unsustainably high short-term performance
• Extended periods below 0: The stock is destroying risk-adjusted value — reevaluate thesis
• Increasing trend: Alpha accumulation — stock gaining strength relative to volatility

Rolling 30-Day Annualized Volatility

Volatility Regimes to Watch For

• Volatility compression (declining vol): Classic breakout setup — Bollinger Bands squeeze, ADX building. Often precedes a major directional move.
• Volatility expansion (spike): Typically coincides with news events, earnings, macro shocks. Widen stops to 2–3× ATR.
• Sustained high vol: Risk-off environment — reduce position size. VaR estimates become unreliable as the distribution changes faster than the historical window captures.

Rolling 60-Day Momentum Beta

This is an internal factor-based beta — not a classical benchmark beta (which requires SPY or another index). It measures the stock's sensitivity to its own trend momentum factor.

Autocorrelation Chart

Significant Patterns

• Significant positive lag-1: Short-term momentum (daily returns continue) — common in trending stocks. Momentum strategies work.
• Significant negative lag-1: Daily mean reversion — common in very high-liquidity large caps. Market makers providing liquidity cause short-term oscillation.
• Significant positive lag-5 or lag-20: Weekly or monthly seasonality — documented in academic literature, often linked to options expiry cycles and fund rebalancing.
• All lags insignificant: Consistent with weak-form efficiency — no short-term predictability from past prices alone.

What is Factor Investing?

Factor investing is the systematic exploitation of return premia — structural sources of excess return that persist over long periods. Originally formalized by Fama & French (1992), factor models show that a stock's return can be decomposed into market exposure plus specific factor exposures (betas).

This platform computes 6 proprietary factor scores from price and volume data alone, calibrated 0–100 (50 = neutral).

Factor 1 — Momentum (WML)

The Momentum factor (Winners Minus Losers) is one of the most robust and documented market anomalies, persisting across 200 years of data and 40+ global markets (Asness, Moskowitz, Pedersen).

• Score > 70: Strong momentum — stock has been outperforming over 11 months (excl. last month). Statistically associated with continued outperformance over next 3–12 months.
• Score < 30: Negative momentum — consider avoiding or short. Momentum crashes during market reversals — always combine with trend and regime filter.

Factor 2 — Quality

Quality stocks have consistent, stable returns with low volatility of returns. Academically linked to profitability, earnings quality, and low financial leverage.

• Score > 70: Highly consistent returns — institutional favourite characteristics
• Score < 35: Inconsistent returns — high risk of extended poor performance

Factor 3 — Low Volatility

The Low Volatility anomaly (discovered by Black 1972) is the paradox that low-risk stocks have historically outperformed high-risk stocks on a risk-adjusted basis — contradicting CAPM theory. Explained by leverage constraints, benchmark hugging, and lottery preferences.

Factor 4 — Value

Value investing (Graham & Dodd, Fama-French HML) selects stocks trading below their intrinsic value. Our proxy uses the price vs SMA200 relationship as a mean-reversion value indicator.

Factor 5 — Size (Large Cap)

Fama-French SMB (Small Minus Big) shows small caps historically outperform large caps. Our size proxy uses ATR as a percentage of price — lower percentage indicates larger, more liquid companies.

Factor 6 — Trend Strength

A directional factor scoring the combination of ADX strength and +DI/-DI direction. Captures the momentum of the trend rather than return performance.

Combining Factors for Stock Selection

What is Stress Testing?

Stress testing evaluates how a portfolio or position would perform under extreme but historically plausible scenarios. Unlike VaR (which uses probability weighting), stress tests apply specific, often catastrophic, scenarios directly — regardless of their probability. It is mandated by regulators (Basel, Dodd-Frank, EBA) and is standard practice at all institutional investment managers.

Why Historical Scenarios?

Historical scenarios have a crucial advantage: they actually happened. They incorporate the full complexity of market dynamics during stress — liquidity crises, correlation breakdowns, and contagion effects — that models like VaR or GBM cannot capture. They serve as a reality check against purely mathematical models.

Scenario Library Explained

COVID-19 Crash (Feb–Mar 2020)

2008 Financial Crisis (Sep 2008 – Mar 2009)

Dot-com Crash (Mar 2000 – Oct 2002)

Flash Crash (May 6, 2010)

2022 Rate Shock

How to Use Stress Tests Practically

1. Position Sizing

2. Hedging Requirements

3. Concentration Risk Assessment

Apply the same scenario across all your positions simultaneously. If correlations spike to 1.0 (as they do in systemic crises), total portfolio loss = sum of all individual scenario impacts. This reveals dangerous concentration that diversification normally hides.