Capital Asset Pricing Model (CAPM) for Indian Markets

Introduction

The Capital Asset Pricing Model (CAPM) is a cornerstone of modern financial theory developed independently by William Sharpe, John Lintner, and Jan Mossin in the 1960s, building on Harry Markowitz's work on portfolio theory. For this contribution, Sharpe shared the 1990 Nobel Prize in Economics with Markowitz and Merton Miller.

CAPM provides a framework to understand the relationship between systematic risk and expected return for assets, particularly stocks. It's widely used in finance for pricing risky securities, generating expected returns, and evaluating investment performance. The model introduces the concept that investors need to be compensated in two ways: the time value of money and risk.

Despite its theoretical elegance and practical utility, CAPM has been challenged by empirical tests, leading to the development of more complex models. Nevertheless, it remains widely taught and used due to its simplicity and powerful insights into the nature of risk and return.

Big-Picture Intuition

CAPM is often described as the finance world's "one-factor model." At its core, it presents a revolutionary idea: investors are only compensated for taking systematic (market-wide) risk, not for specific risks that can be eliminated through diversification.

In essence, CAPM distinguishes between two types of risk:

Systematic Risk (Market Risk)

• Cannot be eliminated through diversification

• Affects the entire market (e.g., economic cycles, interest rates)

• Investors are rewarded with a risk premium for bearing this risk

• Measured by beta (β)

Unsystematic Risk (Idiosyncratic Risk)

• Can be eliminated through diversification

• Specific to individual stocks or sectors

• No expected premium for bearing this risk

• Examples: company scandals, management changes, product failures

This distinction leads to the powerful conclusion that diversification is the "free lunch" of investing—it eliminates specific risks without reducing expected returns. What remains is systematic risk, which becomes the only relevant risk factor in pricing assets.

The Core Equation

The CAPM equation elegantly expresses the expected return of an asset as a function of the risk-free rate plus a risk premium:

\boxed{\;\mathbb{E}[R_i] \;=\; R_f \;+\;\beta_i \,\bigl(\mathbb{E}[R_m] - R_f\bigr)\;}

Components of the CAPM Formula

Term	Meaning	Practical Interpretation
$\mathbb{E}[R_i]$	Expected return of asset/portfolio i	The return investors should require for investing in the asset
$R_f$	Risk-free rate	Typically yields from T-bills, government securities, or overnight repo rates
$\mathbb{E}[R_m]$	Expected return of the market portfolio	Return of a broad market index like NIFTY 50 or S&P 500
$\beta_i$	Asset's market beta (sensitivity to market movements)	Measure of systematic risk; how much the asset's returns move with the market
$\mathbb{E}[R_m] - R_f$	Market risk premium	Extra return investors expect for taking on market risk rather than investing in risk-free assets

This straight-line relationship between beta and expected return is known as the Security Market Line (SML). It represents the central insight of CAPM: higher beta (more systematic risk) should lead to higher expected returns.

Beta Definition

Beta is calculated as:

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

Or equivalently:

\beta_i = \rho_{i,m} \frac{\sigma_i}{\sigma_m}

Where:

•

\operatorname{Cov}(R_i,R_m)

is the covariance between the asset's returns and market returns

•

\operatorname{Var}(R_m)

is the variance of market returns

•

\rho_{i,m}

is the correlation coefficient between the asset and market

•

\sigma_i

is the standard deviation of the asset's returns

•

\sigma_m

is the standard deviation of market returns

Assumptions

The classical CAPM rests on several key assumptions that simplify the complexities of real markets:

Classical CAPM Assumptions

1.

Mean-variance investors — Investors care only about expected return and variance of their portfolios.

2.

Single-period horizon — All investors have the same evaluation period.

3.

Homogeneous expectations — All investors foresee the same expected returns and covariance matrix.

4.

Perfect markets — No taxes, trading frictions, or short-sale constraints.

5.

Unlimited lending/borrowing at $R_f$ — Everyone can leverage or de-leverage at the same risk-free rate.

Real financial markets violate these assumptions to varying degrees, which has led to the development of extensions like the Black CAPM (zero-beta CAPM), multi-factor models (Fama-French), and models incorporating liquidity premiums.

Deriving the Formula

The derivation of CAPM follows from Markowitz's Modern Portfolio Theory and reveals why only systematic risk matters:

•

Start with the Efficient Frontier under Markowitz optimization.

•

Add a risk-free asset to obtain the Capital Market Line (CML), which represents all efficient portfolios combining the risk-free asset and a risky portfolio.

•

The market portfolio sits at the tangency point of the CML with the efficient frontier. This represents all investable assets, weighted by their market value.

•

Any efficient portfolio is a mix of the risk-free asset and the market portfolio.

From these insights, we can derive the CAPM equation for any asset $i$ :

\mathbb{E}[R_i] - R_f \;=\; \beta_i \bigl(\mathbb{E}[R_m] - R_f\bigr)

This shows that expected excess return is proportional to covariance with the market, not to the asset's total variance. This is the fundamental insight of CAPM—only systematic risk is priced in equilibrium.

Estimating CAPM in Practice

OLS Regression Approach

In practice, CAPM parameters are typically estimated using Ordinary Least Squares (OLS) regression:

R_{i,t}-R_{f,t} \;=\; \alpha_i \;+\; \beta_i(R_{m,t}-R_{f,t}) \;+\; \varepsilon_t

Where:

•

R_{i,t}-R_{f,t}

is the excess return of asset

i

at time

t

•

R_{m,t}-R_{f,t}

is the excess return of the market at time

t

•

\alpha_i

is the intercept, representing Jensen's Alpha (abnormal return)

•

\beta_i

is the slope coefficient, representing the asset's beta

•

\varepsilon_t

is the error term

In matrix form, the OLS estimator is $\hat{\theta}=(X^{\!\top}X)^{-1}X^{\!\top}y$ , which produces:

•

\hat{\beta}_i

— market risk estimate

•

\hat{\alpha}_i

— abnormal return estimate (Jensen's Alpha)

•

R^2

— fraction of variance explained by the market

Required Inputs

Component	Typical Data Source
Risk-free rate ( $R_f$ )	3-month T-bill or 10-year G-Sec yield (annualized)
Market return ( $R_m$ )	Broad index (NIFTY 50, S&P 500, MSCI World, etc.)
Return frequency	Daily or monthly; market and asset returns should use the same frequency

Note: Implementation in Our Portfolio Optimizer

Our portfolio optimization backend automatically:

1.

Aligns dates between the portfolio and benchmark returns

2.

Builds daily excess return series

3.

Runs OLS regression using statsmodels

4.

Produces rolling year-by-year betas to analyze beta stability

5.

Calculates confidence intervals and p-values for statistical significance

The Security Market Line

The Security Market Line (SML) is a graphical representation of the CAPM relationship. It plots expected excess returns against beta:

Security Market Line

Asset A: β=0.8, E(R)=10% (Undervalued)

Asset B: β=1.2, E(R)=7% (Overvalued)

Asset C: β=1.5, E(R)=10.5% (Correctly Priced)

The SML has several key interpretations:

Assets On the Line

Correctly priced assets should fall exactly on the SML, indicating they offer returns commensurate with their systematic risk.

Assets Above the Line

Assets plotting above the SML have positive alpha (α), suggesting they're undervalued and offer excess returns beyond what their systematic risk would predict.

Assets Below the Line

Assets plotting below the SML have negative alpha, suggesting they're overpriced and don't adequately compensate investors for the systematic risk taken.

Common Applications

Use Case	How CAPM Helps
Cost of equity (DCF, WACC)	$k_e = R_f + \beta (MRP)$ , where MRP is the Market Risk Premium. This estimates the required return for equity investors.
Performance attribution	Decompose portfolio return into alpha (skill) + beta × market (exposure to market movements).
Risk budgeting / hedging	Size positions in a portfolio to achieve a desired target beta, managing overall market exposure.
Factor models baseline	CAPM serves as the "1-factor" benchmark before adding size, value, momentum, and other factors.
Asset allocation	Helps determine expected returns for different asset classes based on their systematic risk.

Limitations & Extensions

While elegant in theory, CAPM has several empirical challenges. Various extensions have been developed to address these limitations:

Limitation	Extension/Solution
Beta instability — Betas tend to vary over time	Rolling betas, regime-switching models, Kalman filter estimation
Non-normal returns — Asset returns often exhibit fat tails and skewness	Downside beta, conditional value-at-risk (CVaR) models, quantile regressions
Multiple priced risks — Market beta alone doesn't explain all variation in returns	Fama-French 3/5-factor models, Carhart 4-factor model, APT (Arbitrage Pricing Theory)
No risk-free borrowing — Not all investors can borrow at the risk-free rate	Black CAPM (Zero-beta CAPM) replaces the risk-free rate with a zero-beta portfolio
Liquidity concerns — CAPM doesn't account for transaction costs or liquidity differences	Liquidity-adjusted CAPM, Acharya-Pedersen model

Despite these limitations, CAPM remains a fundamental model in finance due to its simplicity, intuitive appeal, and practical insights into the relationship between risk and return.

Key References

Sharpe, W. F. (1964). "Capital Asset Prices: A Theory of Market Equilibrium Under Conditions of Risk." Journal of Finance, 19(3), 425-442.Access the paper

Lintner, J. (1965). "The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets." Review of Economics & Statistics, 47(1), 13-37.

Mossin, J. (1966). "Equilibrium in a Capital Asset Market." Econometrica, 34(4), 768-783.

Black, F. (1972). "Capital Market Equilibrium with Restricted Borrowing." Journal of Business, 45(3), 444-455.

Bodie, Kane & Marcus. Investments (12 ed.), McGraw-Hill, 2021 – Ch. 9-10.

Fama, E. F., & French, K. R. (2004). "The Capital Asset Pricing Model: Theory and Evidence." Journal of Economic Perspectives, 18(3), 25-46.Access the paper

Capital Asset Pricing Model (CAPM)

Understanding the relationship between risk and expected return