grad-capm
用 CAPM 和 beta 估計資產預期報酬與風險溢酬
中文翻譯
--- name: "grad-capm" description: "運用資本資產定價模型（CAPM）估計預期報酬並評估風險與報酬的取捨。當使用者需要計算資產的預期報酬、把 beta 解讀為系統性風險暴險、評估一項投資是否補償了風險，或問到「我該預期多少報酬」、「風險溢酬是多少」或「beta 如何影響定價」時使用。" metadata: category: "WP-26 財務理論" tags: ["CAPM", "beta", "systematic-risk", "expected-return", "Sharpe", "risk-premium"] --- # 資本資產定價模型（CAPM） ## 概述 CAPM（Sharpe, 1964；Lintner, 1965）建立了系統性風險與預期報酬之間的線性關係。這個模型指出，任何資產的預期報酬等於無風險利率，加上承擔市場風險的溢酬（再乘上該資產的 beta）。 ## 何時使用 - 為股權估值估計必要報酬率 - 計算 WACC 中的股權成本 - 透過 beta 比較資產風險 - 對照證券市場線（SML）評估投資組合績效 ## 何時不要用 - 資產對規模、價值或市場風險以外的其他因子有顯著暴險時 - 對缺乏流動性或未公開交易的資產，beta 的估計不可靠 - 市場投資組合的代理變數有疑慮時（Roll 的批評） ## 假設 ``` 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. ``` 關鍵假設： 1. 投資人是具有同質預期的均值-變異數最佳化者 2. 存在一個無風險資產，可以無限制地借入與貸出 3. 市場是無摩擦的：沒有稅、交易成本或放空限制 4. 所有資產都是無限可分割且公開交易的 ## 方法論 ### 步驟 1：找出輸入值 - 無風險利率（Rf）：與投資期間相符的政府公債殖利率 - 市場報酬 E(Rm)：歷史平均或前瞻性的估計 - Beta：資產報酬對市場報酬做迴歸 ### 步驟 2：計算預期報酬 E(Ri) = Rf + Bi x (E(Rm) - Rf)。從均值-變異數最佳化推導的過程，見 `references/derivation.md`。 ### 步驟 3：畫在證券市場線上 在 SML 之上的資產被低估（正的 alpha），在其下則被高估（負的 alpha）。 ### 步驟 4：解讀並決策 - Beta > 1：放大市場的波動，風險較高、預期報酬較高 - Beta < 1：減弱市場的波動，風險較低、預期報酬較低 - Beta = 0：報酬等於無風險利率 ## 輸出格式 > ⚠️ **小數 vs 百分比**：傳值給隨附的 script 或從它取值時，所有利率 > （`risk_free`、`market_return`、`beta_contribution`、`expected_return`、`alpha`）都是 > **小數**，`0.05` 代表 5%，不是 `5.0`。下面的敘述型報告為了方便人閱讀，會以 > 百分比呈現，但同一個 JSON 物件裡絕對不要把兩種混用。 ```markdown ## 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] ``` ## 常見陷阱 - Beta 是回頭看的，未來的 beta 可能跟歷史估計不同 - 市場代理變數的選擇影響極大（Roll 的批評，1977） - CAPM 假設只有單一風險因子，實證證據支持多因子模型 - 無風險利率的選擇（短期國庫券 vs 長期公債）對結果有顯著影響 - Beta 的估計對報酬頻率（日 vs 月）和樣本期間很敏感 - CAPM 無法解釋低 beta 異常（低 beta 股票的表現優於預測） ## Scripts | Script | 說明 | 用法 | |--------|-------------|-------| | `scripts/capm.py` | 計算 CAPM 預期報酬與 alpha | `python scripts/capm.py --help` | 執行 `python scripts/capm.py --verify` 可跑內建的合理性測試。 ## 參考資料 - 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.
英文原文
--- name: "grad-capm" description: "Apply the Capital Asset Pricing Model (CAPM) to estimate expected returns and assess risk-return tradeoffs. Use this skill when the user needs to calculate expected return on an asset, interpret beta as systematic risk exposure, evaluate whether an investment compensates for risk, or when they ask 'what return should I expect', 'what is the risk premium', or 'how does beta affect pricing'." metadata: category: "WP-26 財務理論" tags: ["CAPM", "beta", "systematic-risk", "expected-return", "Sharpe", "risk-premium"] --- # 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. ```markdown ## 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 | Script | Description | Usage | |--------|-------------|-------| | `scripts/capm.py` | Compute CAPM expected return and alpha | `python scripts/capm.py --help` | 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.