2026-02-08

Introduction

Modern portfolio management emphasizes diversification to reduce risk while maintaining returns. This presentation examines:

  • U.S. Equities: S&P 500 Index
  • U.S. Investment-Grade Bonds: Bloomberg Barclays US Aggregate Bond Index (proxied via ETF AGG)

Date range analyzed: January 01, 2025 to December 31, 2025

We will explore:

  1. Correlation between equities and bonds
  2. Linear regression to quantify relationships
  3. Visualizations and statistical interpretation

Goal: Understand how diversification can improve portfolio risk-adjusted returns.

Regression and Correlation Formulas

Linear Regression:

\[R_e = \beta_0 + \beta_1 R_b + \epsilon\]

Where:

  • \(\beta_0\) = intercept
  • \(\beta_1\) = slope coefficient
  • \(\epsilon\) = error term (mean = 0)

Correlation:

\[\rho_{e,b} = \frac{\text{Cov}(R_e, R_b)}{\sigma_e \sigma_b}\]

Where:

  • \(\text{Cov}(R_e, R_b)\) = covariance
  • \(\sigma_e, \sigma_b\) = std. deviations

These formulas quantify the diversification effect in a portfolio.

Hypothesis Testing in Regression

Testing: Do bond returns significantly predict equity returns?

Hypotheses: \(H_0: \beta_1 = 0\) (no relationship) vs. \(H_a: \beta_1 \neq 0\)

Test Statistic: \[t = \frac{\hat{\beta}_1}{\text{SE}(\hat{\beta}_1)} \sim t_{n-2} \quad \text{where} \quad \text{SE}(\hat{\beta}_1) = \sqrt{\frac{\sum(y_i - \hat{y}_i)^2/(n-2)}{\sum(x_i - \bar{x})^2}}\]

Decision: Reject \(H_0\) if \(p < \alpha\) (typically 0.05)

Result: \(p = 0.122 > 0.05\) → Fail to reject \(H_0\). No significant linear relationship between daily bond and equity returns.

Scatter Plot of Equity vs Bond Returns

Distribution of Daily Returns

3D Plot: Equity vs Bond (Lagged)

R Code: Correlation & Regression Analysis

# 1. Correlation between equity and bond returns
correlation <- cor(data$Equity, data$Bond)
correlation

# 2. Simple linear regression: Equity ~ Bond
lm_simple <- lm(Equity ~ Bond, data = data)
summary(lm_simple)

# 3. Multiple regression: Equity ~ Bond + Lagged Bond
data_3d <- data %>% mutate(Bond_lag1 = lag(Bond, 1)) %>% drop_na()
lm_multiple <- lm(Equity ~ Bond + Bond_lag1, data = data_3d)
summary(lm_multiple)

Regression Output

## **Correlation coefficient:** 0.098
## **Simple Regression (Equity ~ Bond)**
Coefficient Estimate Std.Error p.value
(Intercept) Intercept 0.0007 0.0007 0.3754
Bond Bond 0.3931 0.2532 0.1218 
## 
## **Model Fit:** R² = 0.0097 | Adj. R² = 0.0057

Interpretation & Key Findings (1/2)

Correlation:

The correlation between daily equity returns (S&P 500) and bond returns (AGG) is very low: \(\rho = 0.098\)

This suggests that equities and bonds move largely independently, supporting diversification.

Simple Linear Regression (Equity ~ Bond):

  • Slope coefficient: 0.393 (p = 0.122)
  • Intercept: 0.00066 (p = 0.375)
  • R² ≈ 0.01 → only ~1% of equity return variability explained by bond returns
  • Interpretation: No statistically significant relationship between same-day equity and bond returns.

Interpretation & Key Findings (2/2)

Multiple Regression (Equity ~ Bond + Lagged Bond):

  • Current bond return slope: 0.406 (p = 0.111)
  • Previous day bond return slope: -0.314 (p = 0.218)
  • R² ≈ 0.016 → still very low explanatory power

Interpretation:

Including the lagged bond return does not meaningfully improve predictive power, suggesting equities are mostly uncorrelated with past or current bond returns.

Key Takeaway:

The lack of correlation confirms that bonds and equities can effectively diversify portfolio risk.

Conclusion

  • Finance Insight:
    • Low correlation between equities and bonds implies that adding bonds to a portfolio provides diversification benefits, potentially reducing risk without sacrificing expected return.
    • The lack of statistical significance in regression confirms that bond movements do not predict equity returns in this dataset, consistent with modern portfolio theory.
  • Takeaway: Diversification is effective: combining equities and bonds can reduce portfolio volatility, as their returns are largely independent.