Back in 2018, I got interested in capital market. Understanding the fundamental theories of capital market is critical to accumulation of wealth. I started reading all the classics such as The Intelligent Investor and Security Analysis. In this series of posts, I’d like to share with my reader the journey of understanding financial theories in the context of Python programming language. In the first big part of the posts, we will focus on the linearity aspect of financial models, Capital Asset Pricing Model (CAPM), Arbitrage Pricing Theory (APT) and the linear optimization. Later sections will cover the non-linear models.
Linearity is a simplified version of the real world. It is conceptual easy to understand as well as computationally tractable. It is a good starting point to understand the financial theories. e theories.
What is Linearity?
In mathematics, linearity refers to relationships that satisfy two key properties:
- Additivity: f(x + y) = f(x) + f(y)
- Homogeneity: f(αx) = αf(x)
In finance, linear relationships manifest as:
- Portfolio returns are linear combinations of individual asset returns
- Risk factors combine linearly to explain asset returns
- Optimization problems can be formulated as linear or quadratic programs
1. Capital Asset Pricing Model (CAPM)
Theoretical Foundation
The Capital Asset Pricing Model represents one of the most influential theories in finance. CAPM provides a linear relationship between an asset’s expected return and its systematic risk. CAPM assumes that the investors are rational amongst a few other factors.
The CAPM Formula
The Security Market Line equation expresses expected return as:
\[E[R_i] = R_f + \beta_i (E[R_m] - R_f)\]Where:
- $E[R_i]$ = Expected return of asset i
- $R_f$ = Risk-free rate (e.g., Treasury bill rate)
- $\beta_i$ = Beta of asset i (systematic risk measure)
- $E[R_m]$ = Expected market return
- $(E[R_m] - R_f)$ = Market risk premium
Understanding Beta
Beta measures an asset’s sensitivity to market movements:
\[\beta_i = \frac{Cov(R_i, R_m)}{Var(R_m)} = \rho_{i,m} \frac{\sigma_i}{\sigma_m}\]Interpretation:
- $\beta = 1$: Asset moves exactly with the market
- $\beta > 1$: Asset is more volatile than the market (aggressive)
- $\beta < 1$: Asset is less volatile than the market (defensive)
- $\beta < 0$: Asset moves opposite to the market (rare, e.g., gold)
Economic Intuition
CAPM’s linear relationship captures a fundamental trade-off:
- Higher systematic risk (beta) requires higher expected return
- Diversifiable risk is not compensated (can be eliminated)
- The market portfolio is mean-variance efficient
- Risk premium is proportional to beta
# Python Implementation
from scipy import stats
# Sample data
stock_returns = [0.065, 0.0265, -0.0593, -0.001, 0.0346]
market_returns = [0.055, -0.09, -0.041, 0.045, 0.022]
# Perform linear regression
beta, alpha, r_value, p_value, std_err = stats.linregress(stock_returns, market_returns)
print(f"Beta: {beta}, Alpha: {alpha}")
Beta: 0.5077431878770808, Alpha: -0.008481900352462384
CAPM Applications
1. Cost of Equity Capital
- $r_e = R_f + \beta_e (R_m - R_f)$
- Used in DCF valuations and WACC calculations
2. Performance Evaluation
- Jensen’s Alpha: $\alpha = R_p - [R_f + \beta_p(R_m - R_f)]$
- Positive alpha indicates superior performance
3. Portfolio Management
- Beta targeting for desired risk levels
- Market timing strategies based on beta adjustments
2. Arbitrage Pricing Theory (APT)
Theoretical Foundation
Arbitrage Pricing Theory extends CAPM by acknowledging that **multiple factors drive asset returns. APT is based on the law of one price and assumes that **arbitrage opportunities** are quickly eliminated, which means that APT is less restrictive than CAPM.
APT Assumptions
- Asset returns follow a linear factor model
- No arbitrage opportunities exist
- Perfect competition in capital markets
- Investors prefer more wealth to less
The Multi-Factor Model
APT expresses returns as a linear combination of multiple factors:
\[R*i = \alpha_i + \beta*{i,1}F*1 + \beta*{i,2}F*2 + \ldots + \beta*{i,k}F_k + \epsilon_i\]Where:
- $R_i$ = Return on asset i
- $\alpha_i$ = Asset-specific return (should be zero in equilibrium)
- $F_j$ = Factor j (e.g., GDP growth, inflation, interest rates)
- $\beta_{i,j}$ = Sensitivity of asset i to factor j
- $\epsilon_i$ = Idiosyncratic error term
Economic Intuition
APT’s multi-factor approach recognizes that:
- Multiple sources of systematic risk affect asset returns
- Factor diversification can reduce portfolio risk
- Different assets have different factor exposures
- Risk premiums exist for non-diversifiable factors
# Python Implementation
import numpy as np
import statsmodels.api as sm
# Generate sample data
num_periods = 9
all_values = np.array([np.random.random(8) for _ in range(num_periods)])
y_values = all_values[:, 0]
x_values = all_values[:, 1:]
x_values = sm.add_constant(x_values)
# Perform OLS regression
results = sm.OLS(y_values, x_values).fit()
print(results.summary())
OLS Regression Results
===============================================================================
Dep. Variable: y R-squared: 0.968
Model: OLS Adj. R-squared: 0.747
Method: Least Squares F-statistic: 4.377
Date: Sat, 31 May 2025 Prob (F-statistic): 0.353
Time: 15:50:56 Log-Likelihood: 14.758
No. Observations: 9 AIC: -13.52
Df Residuals: 1 BIC: -11.94
Df Model: 7
Covariance Type: nonrobust
===============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
const -0.3433 0.213 -1.612 0.353 -3.049 2.362
x1 0.2754 0.201 1.371 0.401 -2.278 2.828
x2 0.5616 0.262 2.144 0.278 -2.767 3.890
x3 -0.2459 0.345 -0.713 0.606 -4.631 4.139
x4 0.4624 0.248 1.862 0.314 -2.693 3.618
x5 0.8224 0.349 2.357 0.255 -3.612 5.257
x6 -1.3773 0.413 -3.332 0.186 -6.629 3.875
x7 1.1494 0.337 3.411 0.182 -3.133 5.432
===============================================================================
Omnibus: 0.037 Durbin-Watson: 2.092
Prob(Omnibus): 0.982 Jarque-Bera (JB): 0.268
Skew: -0.022 Prob(JB): 0.875
Kurtosis: 2.156 Cond. No. 23.6
===============================================================================
Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
APT vs. CAPM Comparison
| Aspect | CAPM | APT |
|---|---|---|
| Factors | Single (market) | Multiple |
| Assumptions | Restrictive | More flexible |
| Theoretical Base | Mean-variance optimization | Arbitrage arguments |
| Risk Measures | Beta only | Multiple betas |
| Applications | Cost of capital | Risk management, attribution |
Practical Applications
1. Risk Management
- Multi-factor risk models for portfolio monitoring
- Stress testing across different economic scenarios
- Hedging strategies targeting specific risk factors
2. Performance Attribution
- Decomposing returns by factor contributions
- Identifying sources of outperformance/underperformance
- Style analysis of investment strategies
3. Portfolio Construction
- Factor-based investing (smart beta strategies)
- Risk budgeting across multiple dimensions
- Tactical asset allocation based on factor views
The Linearity Advantage: Why These Models Work
Mathematical Benefits
1. Superposition Principle Linear models satisfy additivity: the effect of multiple factors equals the sum of individual effects.
2. Scalability Linear relationships are scale-invariant: doubling inputs doubles outputs proportionally.
3. Computational Efficiency Linear systems can be solved using matrix algebra, enabling:
- Real-time portfolio optimization
- Large-scale risk calculations
- Monte Carlo simulations
Economic Benefits
1. Intuitive Interpretation
- Coefficients represent sensitivities
- Risk contributions are additive
- Marginal effects are constant
2. Portfolio Construction
- Diversification benefits through linear combinations
- Risk budgeting across factors or assets
- Rebalancing strategies based on linear rules
3. Regulatory Compliance
- Value-at-Risk (VaR) calculations
- Capital requirements based on linear risk measures
- Stress testing using linear factor models
Next: Linear Optimization in Finance →