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IntermediateOptions, Derivatives & FinancePython

Run this module

cd "CAPM"
python "capm_calculator.py"

View source on GitHub

CAPM — Capital Asset Pricing Model

CAPM is the idea that won a Nobel Prize and still anchors how the industry thinks about return and risk. Its claim is bold and simple: the only risk the market pays you to bear is systematic risk — how much an asset moves with the market (its beta). Diversifiable, asset-specific risk earns you nothing, because you could have diversified it away for free.

Expected return = R_f + beta * (R_m - R_f)

R_f is the risk-free rate and (R_m - R_f) is the market risk premium — the extra return you earn for each unit of market risk (beta) you take on.

Functions

Function Description
capm_expected_return(risk_free_rate, beta, market_return) CAPM fair return (scalar or array)
jensens_alpha(actual_return, risk_free_rate, beta, market_return) Realised return minus CAPM-required return
estimate_beta(asset_returns, market_returns) Beta = Cov(asset, market) / Var(market)
security_market_line(betas, risk_free_rate, market_return) Fair return across a range of betas

The pieces

  • Risk-free rate (R_f) — what you earn with no risk, e.g. a short-dated government bill.
  • Beta (b) — sensitivity to the market. b = 1 moves with the market, b > 1 amplifies it, b < 1 dampens it, b < 0 hedges it.
  • Market risk premium (R_m - R_f) — the extra return investors demand for holding the risky market over the risk-free asset.

Example

from capm_calculator import capm_expected_return, jensens_alpha, estimate_beta

rf, rm = 0.03, 0.09
print(capm_expected_return(rf, beta=1.2, market_return=rm))   # fair return

# Did a 14% return with beta 1.2 represent skill?
print(jensens_alpha(0.14, rf, 1.2, rm))                       # Jensen's alpha

# Estimate beta from return histories
print(estimate_beta(asset_returns, market_returns))

Jensen's alpha — skill or just risk?

A 20% return sounds great until you learn it came from a beta-2 bet in a bull market — CAPM would have expected that. Jensen's alpha strips out the return you were owed for the risk taken and leaves only the unexplained excess:

alpha = actual_return - [ R_f + beta * (R_m - R_f) ]

Persistent positive alpha is the holy grail of active management — and, CAPM purists argue, usually a sign of a missing risk factor rather than magic.

The security market line (SML)

Plot expected return against beta and CAPM is a straight line from R_f (at beta = 0) through the market portfolio (at beta = 1). Assets above the line are under-priced (more return than their risk warrants); assets below are over-priced.

Practical notes

  • Beta is estimated, not given — it is the regression slope of the asset on the market and drifts over time. See Finance - Beta Calculator for rolling estimates, and Quantitative Methods - Regression Analysis for the mechanics.
  • CAPM is a single-factor model. Real returns need more factors (size, value, momentum) — continue to Quantitative Methods - Factor Models.
  • Use a consistent period: annual R_f and R_m with annual beta, or scale everything to the same frequency.
  • CAPM underpins the discount rate in Discounted Cash Flow (DCF) and the appraisal ratio in Finance - Information Ratio.

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