Capital Asset Pricing Model (CAPM)

SUMMARY

The Capital Asset Pricing Model (CAPM) is a foundational theory in modern finance that describes the relationship between systematic risk and expected return for assets, particularly stocks. CAPM provides a theoretical framework for calculating the required rate of return for an asset based on its sensitivity to market risk (beta) and the market risk premium.

Understanding CAPM

The Capital Asset Pricing Model expresses the expected return of an asset as a function of the risk-free rate, the asset's correlation with market returns (beta), and the market risk premium. The model builds on portfolio optimization theory and introduces the concept of systematic and unsystematic risk.

The CAPM formula

The fundamental CAPM equation is:

$E(R_i) = R_f + \beta_i(E(R_m) - R_f)$

Where:

$E(R_i)$ = Expected return of asset i
$R_f$ = Risk-free rate
$\beta_i$ = Beta of asset i
$E(R_m)$ = Expected return of the market
$(E(R_m) - R_f)$ = Market risk premium

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Beta calculation and interpretation

Beta ( $\beta$ ) measures an asset's sensitivity to market movements and is calculated as:

$\beta_i = \frac{Cov(R_i, R_m)}{Var(R_m)}$

Where:

$Cov(R_i, R_m)$ = Covariance between asset returns and market returns
$Var(R_m)$ = Variance of market returns

Beta interpretation:

β > 1: Asset is more volatile than the market
β = 1: Asset moves in line with the market
β < 1: Asset is less volatile than the market
β = 0: Asset returns are uncorrelated with market returns

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Market risk premium

The market risk premium $(E(R_m) - R_f)$ represents the additional return investors expect to earn for taking on market risk. This component is crucial for:

Asset pricing
Investment decision-making
Portfolio rebalancing algorithms
Risk-adjusted return calculations

Applications in modern finance

Portfolio management

CAPM serves as a foundation for:

Security selection
Performance evaluation
Risk-adjusted return analysis
Portfolio optimization

Risk assessment

The model helps quantify:

Systematic risk exposure
Required risk premiums
Cost of equity capital
Investment hurdle rates

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Extensions and limitations

Modern adaptations

Fama-French Three-Factor Model
Smart beta strategies
Multi-factor models
Conditional CAPM

Key limitations

Assumes perfect market conditions
Single-period model
Only considers systematic risk
Linear risk-return relationship
Static beta assumption

Integration with trading systems

Modern trading platforms incorporate CAPM in:

Risk management systems
Portfolio optimization algorithms
Real-time risk assessment
Performance attribution analysis

CAPM in quantitative analysis

Time-series implementation

# Beta calculation example
import numpy as np

def calculate_beta(asset_returns, market_returns):
    covariance = np.cov(asset_returns, market_returns)[0][1]
    market_variance = np.var(market_returns)
    return covariance / market_variance

Risk decomposition

CAPM helps decompose total risk into:

Systematic risk (market risk)
Unsystematic risk (diversifiable risk)

This decomposition is crucial for:

Portfolio construction
Risk budgeting
Asset allocation decisions