Methodology & Transparency

The Sharpe ratio, developed by Nobel laureate William Sharpe, is the most widely used measure of risk-adjusted return in finance. It answers the question: "How much return am I getting for each unit of risk I'm taking?"

The numerator (Rₚ - Rf) represents the "excess return" - the return above what you could earn with a risk-free investment. The denominator (σₚ) is the portfolio's standard deviation, measuring total volatility.

A key insight is that the Sharpe ratio penalizes both upside and downside volatility equally. This can be a limitation for asymmetric return distributions, but for typical portfolio analysis it provides an excellent standardized comparison metric.

In FactorIQ, we use a "hurdle rate" (default 4%) rather than the traditional risk-free rate. This allows comparison against your personal required rate of return, which may be higher than Treasury yields.

Expected return is a fundamental concept from Modern Portfolio Theory. For a portfolio, it's simply the weighted average of the expected returns of its constituent assets.

The calculation uses arithmetic annualization: we compute the average daily return and multiply by 252 (trading days per year). This method is standard for short-to-medium term projections.

For long-term projections, geometric annualization would be more conservative because it accounts for compounding effects. However, arithmetic returns are more commonly used in practice because they better represent the expected value of future wealth.

When historical data is unavailable or insufficient for a stock, we use a default expected return of 8%, which approximates long-term equity market averages.

Volatility (standard deviation) is the foundational risk measure in Modern Portfolio Theory. It captures the total dispersion of returns - both upside and downside.

For a portfolio, we use the Markowitz formula which accounts for correlations between assets. This is crucial because diversification reduces portfolio volatility below the weighted average of individual volatilities.

The formula involves a double summation over all asset pairs: σₚ² = Σᵢ Σⱼ wᵢwⱼσᵢσⱼρᵢⱼ

Where wᵢ is weight, σᵢ is individual volatility, and ρᵢⱼ is the correlation between assets i and j.

When historical volatility data is unavailable, we estimate individual stock volatility using: σ_stock ≈ β × σ_market × 1.3, where the 1.3 factor accounts for idiosyncratic risk.

Risk classification translates the technical volatility measure into an intuitive 1-5 scale that matches common investment profile questionnaires.

The thresholds are based on historical asset class volatilities: - Level 1 (Conservative): <6% - typical of short-term bonds - Level 2 (Moderate): 6-12% - typical of diversified bond portfolios - Level 3 (Balanced): 12-18% - typical of balanced stock/bond portfolios - Level 4 (Aggressive): 18-25% - typical of equity portfolios - Level 5 (Very Aggressive): >25% - typical of concentrated stock positions

This classification helps match portfolio risk to investor risk tolerance and time horizon.

Marginal Contribution to Risk (MCR) is a powerful tool for understanding risk sources in a portfolio. It decomposes total portfolio volatility into contributions from each holding.

The key insight is the Euler decomposition property: the sum of all MCRs equals exactly the portfolio volatility. This allows us to attribute 100% of portfolio risk to individual holdings.

MCR accounts for both: 1. The holding's own volatility 2. How it correlates with other holdings

A high-volatility stock that's uncorrelated with the rest of your portfolio may contribute less risk than a medium-volatility stock that's highly correlated with other holdings.

The risk-to-value ratio (MCR weight ÷ portfolio weight) identifies inefficient holdings: - Ratio > 1.3: Disproportionate risk contributor - Ratio < 0.8: Risk-efficient holding - Ratio ≈ 1.0: Proportional risk contribution

Diversification benefit is one of the key insights from Modern Portfolio Theory - that combining imperfectly correlated assets reduces overall risk below the weighted average of individual risks.

The calculation compares: 1. Undiversified volatility: Σᵢ wᵢσᵢ (assuming perfect correlation) 2. Actual portfolio volatility: √(Σᵢ Σⱼ wᵢwⱼσᵢσⱼρᵢⱼ)

A diversification benefit of 20% means your portfolio is 20% less volatile than it would be if all your holdings moved in perfect sync.

Factors that increase diversification benefit: - Holdings in different sectors - Mix of domestic and international assets - Combination of stocks and bonds - Assets with low or negative correlation

Historical stress testing answers the question: "What would have happened to my portfolio during past market crises?"

The methodology uses a CAPM-style approach with sector adjustments:

1. Start with the market impact (e.g., -57% for 2008 crisis) 2. Adjust by the stock's beta (higher beta = amplified move) 3. Apply sector-specific multipliers based on historical sector performance

For example, during the 2008 crisis: - Financial stocks: 1.5× multiplier (banks hit hardest) - Healthcare stocks: 0.6× multiplier (defensive sector) - Technology stocks: 1.1× multiplier (slightly worse than market)

The scenarios are based on S&P 500 peak-to-trough declines, providing a realistic view of downside risk. The "worst case" shown is the most severe scenario for your specific portfolio, accounting for your sector allocation.

Beta is a measure of systematic (market) risk from the Capital Asset Pricing Model (CAPM). It quantifies how much a stock moves relative to the overall market.

- β = 1.0: Stock moves with the market - β > 1.0: Stock is more volatile than market (amplified moves) - β < 1.0: Stock is less volatile than market (dampened moves)

In stress testing, beta adjusts the base market impact: - A stock with β = 1.5 would experience a 1.5× the market decline - A stock with β = 0.7 would only experience 0.7× the market decline

This captures the empirical observation that high-beta stocks fall more in crashes and rise more in recoveries.

Sector multipliers capture the differential performance of industry sectors during specific market events.

Examples from historical scenarios:

2008 Financial Crisis: - Financials: 1.5× (banks at epicenter) - Healthcare: 0.6× (defensive) - Consumer Defensive: 0.7× (essential goods)

Dot-Com Crash (2000): - Technology: 2.0× (bubble center) - Healthcare: 0.6× (defensive) - Energy: 0.7× (unrelated sector)

COVID Crash (2020): - Energy: 1.8× (oil price collapse) - Consumer Cyclical: 1.5× (travel, retail) - Technology: 0.7× (benefited from work-from-home)

A multiplier >1 means the sector performed worse than the market; <1 means it held up better.

Geometric Brownian Motion (GBM) is the standard model for simulating asset price movements. It has two components:

1. Drift term (μ × dt): The expected directional movement 2. Diffusion term (σ × √dt × Z): Random fluctuations

The critical detail is the Ito correction (-σ²/2). Under GBM, the expected value of future wealth is exp(μ×t), but the median is exp((μ - σ²/2)×t). The Ito correction ensures our simulation produces the correct distribution.

Without this correction, simulations would systematically overestimate future wealth, especially for high-volatility portfolios over long time horizons.

GBM assumes: - Returns are log-normally distributed - Volatility is constant over time - Markets are efficient (no predictable patterns)

These assumptions are imperfect but provide a reasonable baseline for long-term projections.

The Ito correction is a subtle but crucial detail in Monte Carlo simulation. It arises from Ito's Lemma, which describes how functions of stochastic processes evolve.

Under GBM, log returns follow: ln(S_t/S_0) ~ N((μ - σ²/2) × t, σ² × t)

This means: - The expected value of S_t is S_0 × exp(μ × t) - But the median of S_t is S_0 × exp((μ - σ²/2) × t)

The correction is especially important for: - High volatility portfolios (larger σ² adjustment) - Long time horizons (effect compounds over time)

For example, with μ = 10% and σ = 20%: - Without correction: median grows at 10%/year - With correction: median grows at 10% - 2% = 8%/year

This ensures the 50th percentile of our simulations matches the mathematical expectation.

Percentile projections provide a probability-weighted view of future outcomes. Rather than a single point estimate, they show the full range of possibilities.

Interpretation: - 5th percentile: Only 5% of outcomes are worse than this - 25th percentile: Conservative estimate (1-in-4 chance of being worse) - 50th percentile: Median outcome (equally likely to be higher or lower) - 75th percentile: Optimistic estimate (3-in-4 chance of being worse) - 95th percentile: Only 5% of outcomes are better than this

The spread between percentiles indicates uncertainty: - Narrow spread: More predictable outcomes - Wide spread: Greater uncertainty (higher volatility)

The "probability cone" visualization shows how uncertainty compounds over time - the spread widens as the projection extends further into the future.

Probability of loss provides an intuitive risk measure - the chance that your portfolio will be worth less at the end of the projection period than at the start.

Key insights: - Longer time horizons generally reduce probability of loss (growth overcomes volatility) - Higher expected return reduces probability of loss - Higher volatility increases probability of loss

For a well-diversified portfolio with 8% expected return and 15% volatility: - 5-year horizon: ~15-20% probability of loss - 10-year horizon: ~10-15% probability of loss - 20-year horizon: ~5-8% probability of loss

This metric helps investors understand whether their time horizon is sufficient to ride out potential downturns.

Correlation measures the strength and direction of the linear relationship between two variables. For sectors:

High correlation (0.7-1.0): - Technology & Consumer Cyclical: 0.7 - Financials & Industrials: 0.6

Moderate correlation (0.4-0.6): - Technology & Financials: 0.6 - Healthcare & Consumer Defensive: 0.5

Lower correlation (0.2-0.4): - Technology & Energy: 0.3 - Healthcare & Energy: 0.2

For portfolio construction, lower correlations between holdings are desirable - they provide diversification benefits. The correlation matrix is used in: 1. Portfolio variance calculation 2. Risk contribution analysis 3. Regime-adjusted stress testing

One of the most important findings in financial research is that correlations increase during market stress. This phenomenon, called "correlation breakdown," means diversification benefits erode exactly when you need them most.

The research by Longin & Solnik (2001) documented that international equity market correlations: - Average around 0.4-0.6 in normal markets - Rise to 0.7-0.9 during extreme market moves

We use VIX as a regime indicator: - VIX < 20: Normal market (baseline correlations) - VIX 20-30: Elevated volatility (correlations × 1.15) - VIX > 30: Crisis mode (correlations × 1.3)

This adjustment provides more realistic risk estimates during stressed markets, preventing overconfidence in diversification benefits when they matter most.

The economic sentiment score aggregates multiple macro indicators to provide a holistic view of economic conditions.

Component weights (default): - VIX (Fear Gauge): 30% - most responsive to market stress - CFNAI (Economic Activity): 25% - broad economic health - Consumer Sentiment: 15% - forward-looking spending indicator - Yield Curve: 15% - recession predictor - OECD Leading Index: 15% - composite leading indicator

Each indicator is normalized to a -100 to +100 scale: - CFNAI: 0 = trend growth, positive = expansion - Consumer Sentiment: 80 = neutral baseline - VIX: Inverted (low VIX = bullish, high VIX = bearish) - Yield Curve: Positive spread = healthy, inverted = recession warning - OECD CLI: 100 = neutral, above = expansion

The composite score provides a quick read on overall market conditions.

The VIX, often called the "fear index," measures the market's expectation of 30-day volatility implied by S&P 500 index option prices.

Historical context: - VIX < 12: Very calm markets, possible complacency - VIX 12-20: Normal volatility range - VIX 20-30: Elevated concern, increased hedging activity - VIX 30-40: High fear, typical of corrections - VIX > 40: Extreme fear, crisis levels (2008: peaked ~80, 2020: peaked ~82)

For sentiment scoring, we invert the VIX because low volatility is generally associated with bullish conditions and high volatility with bearish conditions.

The VIX tends to be mean-reverting - extreme readings typically don't persist. Very low VIX can sometimes be a contrarian warning sign of complacency.

The yield curve shape reflects market expectations about future economic conditions and interest rates.

Normal (positive slope): Long-term rates > short-term rates. This is the typical state because: - Investors demand higher yields for longer-term uncertainty - Economy expected to grow, with higher future interest rates

Flat: Long-term and short-term rates similar. Indicates: - Uncertainty about economic direction - Potential transition period

Inverted (negative slope): Short-term rates > long-term rates. This is a recession warning because: - Markets expect Fed to cut rates in response to economic weakness - Every US recession since 1955 was preceded by an inversion - Typical lead time: 12-18 months before recession

The 10Y-2Y spread is the most watched measure, though other spreads (10Y-3M) are also tracked.

The CFNAI is one of the most comprehensive measures of US economic activity. It combines 85 monthly indicators across four categories:

1. Production & Income (23 indicators) 2. Employment, Unemployment & Hours (24 indicators) 3. Personal Consumption & Housing (15 indicators) 4. Sales, Orders & Inventories (23 indicators)

Interpretation: - CFNAI = 0: Economy growing at historical trend - CFNAI > 0: Above-trend growth (expansion) - CFNAI < 0: Below-trend growth (contraction risk) - CFNAI < -0.7 (3-month average): High recession probability

The CFNAI is designed to be centered at zero, making it easy to identify whether current conditions are above or below trend. Its breadth makes it less susceptible to noise from any single indicator.

Consumer sentiment is a forward-looking indicator because consumer spending drives ~70% of US GDP. When consumers feel confident, they spend more, which drives economic growth.

Historical context: - Index range: roughly 50-110 - 80 = approximate neutral point - Above 90: High confidence, strong spending likely - Below 70: Low confidence, potential spending pullback

The index is derived from a monthly survey asking about: 1. Personal finances (current and expected) 2. Business conditions (12-month and 5-year outlook) 3. Buying conditions for major purchases

Consumer sentiment can be both a leading and coincident indicator: - Leading: Sentiment drops before recessions as consumers anticipate problems - Coincident: Sentiment reflects current economic conditions

Extreme readings often revert to mean, but persistent low readings can indicate sustained economic weakness.

The OECD Composite Leading Indicator (CLI) is designed to anticipate turning points in economic activity 6-9 months ahead.

The CLI aggregates several leading indicators: - Building permits - Stock prices - Money supply - Interest rate spread - Manufacturing orders - Consumer expectations

Index interpretation: - CLI = 100: Economy at long-term trend - CLI > 100: Expected expansion - CLI < 100: Expected contraction - CLI turning down from >100: Peak ahead - CLI turning up from <100: Trough ahead

The OECD designs the CLI to have roughly balanced leads across different business cycles. It's particularly useful for identifying the direction of change rather than the level of activity.

Historical performance: The CLI has successfully signaled major turning points including the 2008 recession and 2020 pandemic shock.