Equity Risk Factor Model

Statistical Methods – Final Project

A linear regression analysis predicting annualized return volatility for S&P 500 stocks using fundamental accounting ratios, market metrics, and sector classification. Built in RStudio as part of a Statistical Methods course.

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Project Overview

Research question: Can firm-level fundamentals and sector membership predict how volatile a stock's returns will be over the next year?

Why this project: Volatility is the foundational input to options pricing (Black-Scholes), Value-at-Risk models, and portfolio construction. Understanding which firm characteristics drive risk is core work in quantitative finance and asset management.

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Repository Structure

equity-risk-factor-model/
│
├── README.md
│
├── data/
│   ├── raw/
│   │   └── sp500_raw.csv          # Original dataset (untouched)
│   └── processed/
│       └── sp500_clean.csv        # After cleaning & feature engineering
│
├── R/
│   ├── 01_data_prep.R             # Load, clean, log-transform market cap
│   ├── 02_exploration.R           # Summary stats, EDA plots
│   ├── 03_models.R                # All four lm() model definitions
│   └── 04_diagnostics.R          # Residuals, AIC comparison, model selection
│
├── plots/
│   ├── volatility_by_sector.png
│   ├── correlation_matrix.png
│   ├── residuals_model3.png
│   └── model_comparison.png
│
├── report/
│   └── equity_risk_factor_model.Rmd        # R Markdown source -> knits to final PDF
│
└── output/
    └── equity_risk_factor_model.pdf        # Final submission

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Dataset

Source: Live data pulled programmatically via tidyquant

• Constituents & sectors: tq_index("SP500") -> current S&P 500 members and GICS sector classifications sourced from Wikipedia
• Price history: tq_get(get = "stock.prices") -> 1 year of daily adjusted closing prices from Yahoo Finance
• Fundamentals: tq_get(get = "key.stats") -> PE ratio, beta, market cap, debt/equity, revenue growth from Yahoo Finance

Data reflects the trailing 12 months as of the date 01_data_prep.R is run. Raw API responses are cached to data/raw/ - the fetch only runs once.

Variables

| volatility_1yr | Continuous | Response (Y) | Annualized std dev of daily returns over trailing 12 months | | log_market_cap | Continuous | Predictor | Log-transformed market capitalization (reduces skew) | | beta_1yr | Continuous | Predictor | 1-year rolling beta vs. S&P 500 | | debt_to_equity | Continuous | Predictor | Total debt / shareholders' equity | | pe_ratio | Continuous | Predictor | Price-to-earnings ratio | | revenue_growth | Continuous | Predictor | Year-over-year revenue growth (%) | | sector | Factor | Predictor | GICS sector classification (11 levels) |

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Models

Four models are built progressively, each motivated by a specific analytical question:

| 1 | volatility ~ log_market_cap + beta_1yr | Baseline: market-level signals only | | 2 | volatility ~ log_market_cap + beta_1yr + debt_to_equity + pe_ratio | Add fundamental accounting ratios | | 3 | volatility ~ log_market_cap + beta_1yr + debt_to_equity + pe_ratio + sector | Add sector as a factor variable | | 4 | volatility ~ log_market_cap + beta_1yr + debt_to_equity * sector | Interaction: does leverage effect differ by sector? |

Model selection uses Adjusted R², AIC, and residual diagnostics (Q-Q plot, scale-location, Cook's distance).

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How to Reproduce

Requirements

• R (≥ 4.2.0)
• RStudio
• Packages: tidyverse, tidyquant, GGally, car, broom

Install all dependencies at once:

install.packages(c("tidyverse", "tidyquant", "GGally", "car", "broom",
                   "cowplot", "scales", "knitr", "rmarkdown"))

Run order

Execute scripts in numbered order from the R/ directory:

source("R/01-data_prep.R")
source("R/02-exploration.R")
source("R/03-models.R")
source("R/04-diagnostics.R")

Render the report

Open report/equity_risk_factor_model.Rmd in RStudio and knit to PDF:

rmarkdown::render("report/equity_risk_factor_model.Rmd", output_format = "pdf_document")

The output will be saved to output/equity_risk_factor_model.pdf.

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Key Findings (Summary)

• Sector membership is the single strongest categorical predictor - Technology and Consumer Discretionary stocks carry significantly higher baseline volatility than Utilities and Consumer Staples, holding other factors constant.
• Leverage (D/E ratio) amplifies volatility, but the magnitude differs meaningfully by sector — the interaction term in Model 4 captures this.
• Beta, as expected, is strongly significant across all models, but fundamental variables add incremental explanatory power beyond it.
• Model 4 (~ log_market_cap + beta + D/E + P/E + sector) is selected as the best model based on lowest AIC and highest Adjusted R², with well-behaved residuals and no severe multicollinearity (VIF < 5 for all terms).

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Analytical Framework

This project decomposes stock volatility into three components:

1. Market-wide (systematic) risk - captured by beta and market cap
2. Firm-specific (idiosyncratic) risk - captured by leverage and valuation ratios
3. Sector-level structural risk - captured by the GICS sector factor

This maps directly to the systematic vs. idiosyncratic risk framework used in quantitative portfolio management.

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Author

Herman Kaufman
Statistical Methods - Final Project