grad-capm（毕业卡普姆）

Capital Asset Pricing Model (CAPM)

Overview

CAPM (Sharpe, 1964; Lintner, 1965) establishes a linear relationship between systematic risk and expected return. The model states that the expected return on any asset equals the risk-free rate plus a premium for bearing market risk, scaled by the asset's beta.

When to Use

• Estimating required rate of return for equity valuation
• Calculating cost of equity in WACC
• Comparing asset risk via beta
• Evaluating portfolio performance against the Security Market Line (SML)

When NOT to Use

• When the asset has significant exposure to size, value, or other factors beyond market risk
• For illiquid or non-traded assets where beta estimation is unreliable
• When market portfolio proxy is questionable (Roll's critique)

Assumptions

IRON LAW: CAPM only prices SYSTEMATIC risk — diversifiable (unsystematic)
risk earns NO premium. An asset's expected return depends solely on its
beta with the market portfolio.

Key assumptions:

1. Investors are mean-variance optimizers with homogeneous expectations
2. A risk-free asset exists for unlimited borrowing and lending
3. Markets are frictionless — no taxes, transaction costs, or short-selling constraints
4. All assets are infinitely divisible and publicly traded

Methodology

Step 1 — Identify Inputs

• Risk-free rate (Rf): government bond yield matching investment horizon
• Market return E(Rm): historical average or forward-looking estimate
• Beta: regression of asset returns against market returns

Step 2 — Compute Expected Return

E(Ri) = Rf + Bi x (E(Rm) - Rf). See references/derivation.md for the derivation from mean-variance optimization.

Step 3 — Plot on Security Market Line

Assets above the SML are undervalued (positive alpha); below are overvalued (negative alpha).

Step 4 — Interpret and Decide

• Beta > 1: amplifies market moves, higher risk-higher expected return
• Beta < 1: dampens market moves, lower risk-lower expected return
• Beta = 0: returns equal the risk-free rate

Output Format

⚠️ Decimal vs percent: When passing values to or from the bundled script, all rates (risk_free, market_return, beta_contribution, expected_return, alpha) are decimals — 0.05 means 5%, NOT 5.0. The narrative report below renders them as percentages for humans, but never mix the two in the same JSON object.

## CAPM Analysis: [Asset / Portfolio]

### Inputs
| Parameter | Value | Source |
|-----------|-------|--------|
| Risk-free rate (Rf) | x% | [source] |
| Market return E(Rm) | x% | [source] |
| Beta | x.xx | [estimation method] |

### Expected Return
- E(Ri) = Rf + B x (E(Rm) - Rf) = x%

### SML Assessment
- Alpha = Actual return - Expected return = x%
- Interpretation: [undervalued / overvalued / fairly priced]

### Limitations in This Context
- [Note any assumption violations]

Gotchas

• Beta is backward-looking; future beta may differ from historical estimates
• Choice of market proxy matters enormously (Roll's critique, 1977)
• CAPM assumes a single risk factor; empirical evidence supports multi-factor models
• Risk-free rate selection (T-bill vs T-bond) affects results significantly
• Beta estimation is sensitive to return frequency (daily vs monthly) and sample period
• CAPM fails to explain the low-beta anomaly (low-beta stocks outperform predictions)

Scripts

Run python scripts/capm.py --verify to execute built-in sanity tests.

References

• Sharpe, W. (1964). Capital asset prices. *Journal of Finance*, 19(3), 425-442.
• Lintner, J. (1965). The valuation of risk assets. *Review of Economics and Statistics*, 47(1), 13-37.
• Roll, R. (1977). A critique of the asset pricing theory's tests. *Journal of Financial Economics*, 4(2), 129-176.