Reading 84
MODULE 84.1: SYSTEMATIC RISK AND BETA
Describe the implications of combining a risk-free asset with a portfolio of risky assets.
For a portfolio that combines a risky portfolio (weight \(w_A\)) with a risk-free asset (weight \(w_B = 1 - w_A\)):
\[E(R_P) = w_A E(R_A) + w_B E(R_B)\]
\[\sigma_P^2 = w_A^2\sigma_A^2 + w_B^2\sigma_B^2 + 2 w_A w_B \rho_{AB}\sigma_A\sigma_B\]
Because the risk-free asset has \(\sigma_B = 0\) and zero correlation with the risky portfolio, this reduces to:
\[\sigma_P = w_A\sigma_A\]
Plotting risk vs. expected return for varying weights produces a straight line that begins at the risk-free rate \(R_f\) and passes through the risky portfolio's risk-return point.
A linear capital allocation line begins at \(R_f\) on the vertical axis (where \(\sigma_P = 0\)) and rises through the risky portfolio's point in \((\sigma_P, E(R_P))\) space. Points on the segment between \(R_f\) and the risky portfolio represent partial investment in the risky portfolio; points beyond represent borrowing at \(R_f\) to invest more than 100% in the risky portfolio.
當投資組合中同時持有一個風險組合（權重 \(w_A\)）與一個無風險資產（權重 \(w_B = 1 - w_A\)）時，預期報酬率仍為加權平均，但因為無風險資產的標準差為 0、與風險組合的相關係數也為 0，組合標準差化簡為：
\[\sigma_P = w_A\sigma_A\]
因此，在風險—報酬圖上，所有可能的組合會落在一條從 \(R_f\) 開始、通過風險組合點的直線上。線段內代表「部分投入風險組合」、線段外代表「以 \(R_f\) 借款再加碼投資風險組合」。
Explain the capital allocation line (CAL) and the capital market line (CML).
The line of possible risk—return combinations from blending a risk-free asset with a particular risky portfolio is the capital allocation line (CAL). Each investor's optimal CAL is the one that produces the most-preferred set of portfolios given their risk-return preferences. If different investors have different return, risk, and correlation expectations, each will have a different optimal risky portfolio and a different CAL.
A simplifying assumption of modern portfolio theory (and CAPM) is homogeneous expectations — all investors share the same estimates of risk, return, and correlations. Under this assumption every investor faces the same efficient frontier and selects the same risky portfolio: the one whose CAL is tangent to the efficient frontier. That tangency portfolio must be the market portfolio of all risky assets (because every investor holding any risky asset holds the same risky portfolio).
Under homogeneous expectations the optimal CAL for all investors is called the capital market line (CML):
\[E(R_P) = R_f + \left(\frac{E(R_M) - R_f}{\sigma_M}\right)\sigma_P\]
The intercept is \(R_f\); the slope \(\dfrac{E(R_M) - R_f}{\sigma_M}\) is the market risk premium per unit of market risk. The numerator \(E(R_M) - R_f\) is the market risk premium. An investor accepting no risk earns \(R_f\); for each unit of market risk \(\sigma_M\) accepted, the investor expects one extra unit of market risk premium.
If investors can both lend and borrow at \(R_f\), they can construct portfolios to the right of the market portfolio on the CML by borrowing at \(R_f\) (margin) and overinvesting in the market portfolio.
Investors who view markets as informationally efficient typically use a passive strategy — invest in an index proxy for the market portfolio plus a risk-free asset. Investors who believe their valuations differ from market prices use active management — overweight perceived undervalued securities and underweight overvalued ones relative to market weights.
資本配置線（CAL）：以某一風險組合與無風險資產搭配，所有可能組合在風險—報酬圖上形成的直線。每位投資者依自身偏好選擇最適 CAL；若每位投資者對報酬、風險、相關性有不同預期，每人會有不同的最適風險組合與不同的 CAL。
同質預期假設下（現代投資組合理論與 CAPM 的核心假設之一）：所有投資者面對相同的效率前緣，都選擇與效率前緣相切的同一個風險組合——即市場組合。此時所有投資者共用的最適 CAL 就稱為資本市場線（CML）：
\[E(R_P) = R_f + \frac{E(R_M)-R_f}{\sigma_M}\,\sigma_P\]
截距為 \(R_f\)，斜率 \(\dfrac{E(R_M)-R_f}{\sigma_M}\) 為「每單位市場風險的市場風險溢酬」。若投資者能以 \(R_f\) 借款，便可將點移到市場組合右側（保證金交易）。
相信市場有效率者採被動策略（指數投資 + 無風險資產）；相信自有評價優於市價者採主動管理（偏離市場權重以反映自身觀點）。
Explain systematic and nonsystematic risk, including why an investor should not expect to receive additional return for bearing nonsystematic risk.
Diversifying across imperfectly correlated assets reduces a portfolio's risk below the weighted average risk of its components. The portion of risk eliminated by diversification is unsystematic risk (also called unique, diversifiable, or firm-specific risk). The portion that remains is systematic risk (also called nondiversifiable or market risk). Because the market portfolio holds all risky assets, all diversifiable risk is gone and only systematic risk remains.
The concept applies to individual securities. Firms whose returns track the market closely (e.g., luxury goods producers like Ferrari or Harley-Davidson) have high systematic risk; firms whose returns are largely insensitive to broad economic conditions (e.g., utilities) have low systematic risk. Total risk decomposes as:
Total risk = Systematic risk + Unsystematic risk
Empirical studies suggest 12–30 stocks are typically enough to capture most diversification benefit; beyond that, portfolio standard deviation flattens at the level of systematic risk.
Portfolio standard deviation falls steeply as the first stocks are added, then asymptotes to the level of systematic (market) risk after roughly 30 holdings. The remaining risk is nondiversifiable.
分散投資於相關係數小於 1 的資產，可使組合風險低於各資產風險的加權平均。被分散掉的稱為非系統風險（個別公司風險、可分散風險）；剩下無法分散掉的稱為系統風險（市場風險、不可分散風險）。市場組合涵蓋所有風險資產，因此非系統風險為零，僅留下系統風險。
總風險 = 系統風險 + 非系統風險
研究指出大約 12–30 檔股票就能取得近 90% 的分散效益，再增加成份對降低標準差的邊際效用很小。
教授提醒：均衡的預期報酬只依賴系統風險（非總風險）。因為分散成本極低，市場不會對「可被免費消除的風險」給予額外報酬補償。例如生技小型股總風險很高但多為公司特有風險、系統風險低，理論上其要求報酬可能低於一檔總風險較低但市場敏感度高的成熟製造業股票。
Explain return generating models (including the market model) and their uses.
Return-generating models estimate expected security returns from sensitivities to a set of factors. Factors fall into three types:
- Macroeconomic factors — GDP growth, inflation, consumer confidence, interest rates.
- Fundamental factors — earnings, earnings growth, firm size, R&D spending, book-to-market.
- Statistical factors — derived purely from data; have no theoretical basis and may reflect data-mining artefacts.
General multi-factor model in excess returns:
\[E(R_i) - R_f = \beta_{i1}\,E(\text{Factor 1}) + \beta_{i2}\,E(\text{Factor 2}) + \cdots + \beta_{ik}\,E(\text{Factor k})\]
The \(\beta\)'s are factor sensitivities (factor loadings). The first factor is often \(E(R_M) - R_f\). The Fama–French model uses three factors: firm size, book-to-market ratio, and excess market return. Carhart adds a fourth factor for price momentum.
A single-factor (single-index) model in excess returns uses only the excess return on the market:
\[E(R_i) - R_f = \beta_i\,[E(R_M) - R_f]\]
The market model is the simplest variant and is used to estimate beta and abnormal returns:
\[R_i = \alpha_i + \beta_i R_M + e_i\]
where \(R_i\) is the asset return, \(R_M\) is the market return, \(\beta_i\) is the slope, \(\alpha_i\) is the intercept, and \(e_i\) is the abnormal return (deviation from expected). Setting \(\alpha_i = R_f(1 - \beta_i)\) makes the market model consistent with the excess-returns single-index form. Beta in the market model measures how sensitive asset \(i\)'s return is to the market return.
報酬生成模型用「對若干因子的敏感度」來估計預期報酬。因子分三類：
- 總體經濟因子：GDP、通膨、消費者信心、利率。
- 基本面因子：盈餘、盈餘成長、公司規模、研發支出、帳面/市值比。
- 統計因子：純以資料推導，缺乏理論基礎，可能是資料探勘的假關係。
多因子模型（超額報酬形式）：
\[E(R_i) - R_f = \beta_{i1}E(F_1) + \beta_{i2}E(F_2) + \dots + \beta_{ik}E(F_k)\]
常見模型：Fama–French 三因子（規模、帳面/市值、市場超額報酬）；Carhart 四因子再加動能。
最簡單的單因子（單指數）模型：\(E(R_i) - R_f = \beta_i [E(R_M) - R_f]\)。
市場模型是其簡化形式：\(R_i = \alpha_i + \beta_i R_M + e_i\)，用來估計 \(\beta_i\) 與異常報酬 \(e_i\)。當 \(\alpha_i = R_f(1-\beta_i)\) 時，市場模型即與單指數模型超額報酬形式一致。
Calculate and interpret beta.
Beta is a standardized measure of an asset's covariance with the market:
\[\beta_i = \frac{\text{Cov}(R_i, R_M)}{\sigma_M^2}\]
Using \(\rho_{i,M} = \dfrac{\text{Cov}_{i,M}}{\sigma_i\sigma_M}\), so that \(\text{Cov}_{i,M} = \rho_{i,M}\sigma_i\sigma_M\), beta can also be written:
\[\beta_i = \rho_{i,M}\,\frac{\sigma_i}{\sigma_M}\]
Market index standard deviation \(\sigma_M = 20\%\).
(1) If \(\sigma_A = 30\%\) and \(\rho_{A,M} = 0.8\):
\[\beta_A = 0.8 \times \frac{30\%}{20\%} = 1.2\]
(2) If \(\text{Cov}(R_A, R_M) = 0.048\):
\[\beta_A = \frac{0.048}{0.20^2} = \frac{0.048}{0.04} = 1.2\]
In practice, beta is estimated by regressing asset excess returns on market excess returns. The fitted line is the asset's security characteristic line (SCL); its slope is the estimated beta. A line steeper than 45° indicates \(\beta > 1\) — the asset is more sensitive to systematic factors than the overall market (whose beta is 1 by definition).
Beta 將資產與市場的共變數標準化：
\[\beta_i = \frac{\text{Cov}(R_i,R_M)}{\sigma_M^2} = \rho_{i,M}\,\frac{\sigma_i}{\sigma_M}\]
例題：市場標準差 20%。
- 若 \(\sigma_A = 30\%\)、\(\rho_{A,M} = 0.8\)：\(\beta_A = 0.8 \times 30/20 = 1.2\)。
- 若 \(\text{Cov}(R_A,R_M) = 0.048\)：\(\beta_A = 0.048 / 0.04 = 1.2\)。
教授提醒：兩種公式都要會。實務上以資產超額報酬對市場超額報酬作迴歸所得直線稱為證券特徵線（SCL），其斜率即估計的 \(\beta\)。市場本身 \(\beta = 1\)；斜率大於 1 代表該資產對系統風險敏感度高於市場。
- A. 6.0% / 6.8%
- B. 8.0% / 4.8%
- C. 10.0% / 6.6%
- A. Beta risk.
- B. Unsystematic risk.
- C. Total risk.
- A. lending portfolio.
- B. borrowing portfolio.
- C. inefficient portfolio.
- A. can increase or decrease.
- B. decreases at a decreasing rate.
- C. decreases at an increasing rate.
- A. unique plus diversifiable risk.
- B. market plus nondiversifiable risk.
- C. systematic plus unsystematic risk.
- A. statistical factors.
- B. macroeconomic factors.
- C. fundamental factors.
- A. 1.0
- B. 1.5
- C. 2.0
MODULE 84.2: THE CAPM AND THE SML
Explain the capital asset pricing model (CAPM), including its assumptions, and the security market line (SML). Calculate and interpret the expected return of an asset using the CAPM.
The only priced (relevant) risk for an individual asset is its covariance with the market, \(\text{Cov}_{i,M}\). Plotting expected return against \(\text{Cov}_{i,M}\) yields one form of the security market line (SML):
\[E(R_i) = R_f + \frac{E(R_M) - R_f}{\sigma_M^2}\,\text{Cov}_{i,M}\]
Replacing the standardized covariance with beta \(\beta_i = \text{Cov}_{i,M}/\sigma_M^2\) gives the capital asset pricing model (CAPM):
\[\boxed{\,E(R_i) = R_f + \beta_i\bigl[E(R_M) - R_f\bigr]\,}\]
In equilibrium, the expected return on a risky asset equals the risk-free rate plus a beta-adjusted market risk premium. Beta measures systematic (covariance) risk.
\(E(R_M) = 8\%\), \(R_f = 2\%\), \(\beta_A = 1.2\). Required return on Stock A:
\[E(R_A) = 2\% + 1.2(8\% - 2\%) = 9.2\%\]
Because \(\beta_A > 1\), \(E(R_A) > E(R_M)\).
CAPM assumptions:
- Risk aversion — investors require higher expected return for greater risk.
- Utility maximization — investors choose portfolios that maximize expected utility given their preferences.
- Frictionless markets — no taxes, no transaction costs, no other trading impediments.
- One-period horizon — all investors share the same one-period planning horizon.
- Homogeneous expectations — identical estimates of expected returns, standard deviations, and correlations.
- Divisible assets — investments are infinitely divisible.
- Competitive markets — investors are price takers; no single trade affects prices.
CML vs. SML. The CML's x-axis is total risk (\(\sigma\)) — only efficient portfolios plot on it. The SML's x-axis is beta — every properly priced asset and portfolio (efficient or not) plots on it. A low-beta stock is not necessarily low total-risk: a speculative biotech with high firm-specific uncertainty can have a low beta because most of its risk is unsystematic. All assets and portfolios with \(\beta = 1\) sit at the same point on the SML as the market portfolio, regardless of their total risk.
個別資產唯一被定價的風險，是其與市場的共變數 \(\text{Cov}_{i,M}\)。將共變數標準化（除以 \(\sigma_M^2\)）即得 \(\beta\)，由此導出 CAPM：
\[E(R_i) = R_f + \beta_i [E(R_M) - R_f]\]
例題：\(E(R_M) = 8\%\)、\(R_f = 2\%\)、\(\beta_A = 1.2\)，則 \(E(R_A) = 2\% + 1.2(6\%) = 9.2\%\)。
CAPM 七大假設：風險厭惡、效用最大化、無摩擦市場、單期投資期間、同質預期、資產無限可分割、競爭性市場（價格接受者）。
CML 與 SML 之差異：CML 橫軸為總風險（\(\sigma\)），只有效率組合會落在其上；SML 橫軸為 \(\beta\)，所有正確定價的證券（無論是否分散）都應落在 SML 上。低 \(\beta\) 不等於低總風險；研發中之生技股總風險很高、卻可能因為公司特有風險占大宗而具有低 \(\beta\)。所有 \(\beta = 1\) 的資產在 SML 上落點相同（無論總風險）。
Describe and demonstrate applications of the CAPM and the SML.
In equilibrium, a security's expected return equals its required return. Analysts compare a forecast return (from price targets, dividends, etc.) with the required return implied by CAPM/SML to identify mispriced securities.
Risk-free rate \(= 7\%\), market return \(= 15\%\). Forecast data:
| Stock | Price Today | E(Price) in 1 Yr | E(Div) in 1 Yr | Beta |
|---|---|---|---|---|
| A | $25 | $27 | $1.00 | 1.0 |
| B | $40 | $45 | $2.00 | 0.8 |
| C | $15 | $17 | $0.50 | 1.2 |
Forecast vs. required:
| Stock | Forecast Return | Required Return (CAPM) | Verdict |
|---|---|---|---|
| A | (27−25+1)/25 = 12.0% | 7 + 1.0(15−7) = 15.0% | Overvalued — short |
| B | (45−40+2)/40 = 17.5% | 7 + 0.8(8) = 13.4% | Undervalued — buy |
| C | (17−15+0.5)/15 = 16.6% | 7 + 1.2(8) = 16.6% | Properly valued |
- If the point falls above the SML → return is greater than required → security is undervalued (buy).
- If the point falls below the SML → return is less than required → security is overvalued (sell / short).
- On the SML → properly priced.
均衡下，預期報酬 = 必要報酬。分析師將自己估計的報酬與 CAPM/SML 推算的必要報酬作比較，以判斷是否錯誤定價。
例題：\(R_f = 7\%\)、\(R_M = 15\%\)：
- A 股：估計 12.0% < 必要 15.0% ⇒ 高估，建議放空。
- B 股：估計 17.5% > 必要 13.4% ⇒ 低估，建議買進。
- C 股：估計 16.6% = 必要 16.6% ⇒ 合理定價。
口訣：畫在 SML 上方 = 低估（買）；畫在 SML 下方 = 高估（賣／放空）；在 SML 上 = 合理定價。
Calculate and interpret the Sharpe ratio, Treynor ratio, M², and Jensen's alpha.
To compare actively managed portfolios with benchmarks fairly, returns must be adjusted for the risk taken. The four most common risk-adjusted measures:
1. Sharpe ratio — excess return per unit of total risk; equals the slope of a portfolio's CAL.
\[\text{Sharpe} = \frac{E(R_P) - R_f}{\sigma_P}\]
Higher is better. Useful for both concentrated and diversified portfolios. Only meaningful in comparison with another portfolio's Sharpe.
2. M² (M-squared) — same ranking as Sharpe but expressed as a percentage. Construct a hypothetical portfolio \(P^*\) by levering or de-levering \(P\) so that \(\sigma_{P^*} = \sigma_M\):
\[M^2 = R_f + \frac{\sigma_M}{\sigma_P}\,(R_P - R_f)\]
The amount by which \(M^2\) exceeds \(R_M\) is M² alpha. \(M^2\) can be derived from Sharpe: \(M^2 = \text{SR}\cdot\sigma_M + R_f\). If a portfolio's Sharpe ratio exceeds the slope of the CML, then \(M^2 > R_M\) and M² alpha is positive.
\(R_P = 10\%\), \(\sigma_P = 20\%\), \(R_f = 5\%\), \(R_M = 11\%\), \(\sigma_M = 30\%\).
Sharpe = \((10-5)/20 = 0.25\); \(M^2 = 0.25(0.30) + 0.05 = 12.5\%\). M² alpha \(= 12.5\% - 11\% = 1.5\%\).
3. Treynor measure — excess return per unit of systematic risk; slope of a line in \((\beta, R)\) space.
\[\text{Treynor} = \frac{R_P - R_f}{\beta_P}\]
4. Jensen's alpha — percentage return in excess of the SML's required return for the same beta.
\[\alpha_P = R_P - \bigl[R_f + \beta_P(R_M - R_f)\bigr]\]
Portfolios above the SML have positive Jensen's alpha and Treynor > the market's; portfolios above the CML have positive M² alpha and Sharpe > the CML's.
- Single-manager portfolio that may carry firm-specific risk → use Sharpe or M² (total-risk-based).
- Multi-manager fund whose overall portfolio is well diversified (only systematic risk left) → use Treynor or Jensen's α (beta-based).
- Caveat: \(E(R_M)\), the market risk premium, and beta itself are estimated with error — interpret these measures with care.
四大風險調整後績效指標：
- Sharpe Ratio：以「總風險」為分母 \(\dfrac{R_P - R_f}{\sigma_P}\)，等於該組合 CAL 的斜率，數值越高越好；可用於含非系統風險的集中組合。
- M²（M-squared）：將組合槓桿/去槓桿至 \(\sigma_{P^*} = \sigma_M\)，得到 \(M^2 = R_f + \dfrac{\sigma_M}{\sigma_P}(R_P - R_f)\)。\(M^2\) 與市場報酬之差稱M² alpha。排名與 Sharpe 相同，但以百分比表示。
- Treynor 衡量：以「系統風險（β）」為分母 \(\dfrac{R_P - R_f}{\beta_P}\)。
- Jensen's alpha：實際報酬 − SML 上同 β 處的必要報酬 \( = R_P - [R_f + \beta_P(R_M - R_f)]\)。
M² 例題：\(R_P = 10\%\)、\(\sigma_P = 20\%\)、\(R_f = 5\%\)、\(R_M = 11\%\)、\(\sigma_M = 30\%\)：Sharpe = 0.25；\(M^2 = 0.25 \times 0.3 + 0.05 = 12.5\%\)；M² alpha = 1.5%。
選用原則：單一經理人、組合可能含個別風險 ⇒ 用 Sharpe / M²（總風險）；多經理人且整體已分散 ⇒ 用 Treynor / Jensen's α（系統風險）。注意 \(E(R_M)\)、市場風險溢酬、β 都是估計值，存在估計誤差。
- A. Securities that plot above the SML are undervalued.
- B. Investors expect to be compensated for systematic risk.
- C. Securities that plot on the SML have no value to investors.
- A. 7.2%
- B. 12.0%
- C. 13.2%
- A. 11.9%
- B. 14.0%
- C. 16.8%
- A. overpriced — short it.
- B. underpriced — buy it.
- C. underpriced — short it.
- A. overpriced — do not buy.
- B. underpriced — buy it.
- C. properly priced — buy it.
- A. Sharpe ratio.
- B. Jensen's alpha.
- C. Treynor measure.
Combining a risk-free asset with a portfolio of risky assets gives a linear set of risk-return combinations. The line lets investors with different risk tolerances dial overall portfolio risk and return up or down.
The combinations of any risky portfolio with the risk-free asset form the capital allocation line (CAL). When the risky portfolio is the market portfolio (under homogeneous expectations), the CAL becomes the capital market line (CML).
Systematic (market) risk comes from factors that affect all risky assets and cannot be diversified away. Unsystematic (firm-specific) risk can. Because diversification is essentially free, investors are not rewarded for bearing unsystematic risk; equilibrium returns depend only on systematic risk.
A return-generating model estimates expected return from exposures to factors (macro, fundamental, statistical). The simplest form is the market model:
\[R_i = \alpha_i + \beta_i R_M + e_i\]
Beta:
\[\beta_i = \frac{\text{Cov}(R_i, R_M)}{\sigma_M^2} = \rho_{i,M}\,\frac{\sigma_i}{\sigma_M}\]
The market's average beta is 1; \(\beta = 0\) means returns are uncorrelated with the market.
CAPM assumptions: investors are risk-averse, utility-maximizing, rational; markets are frictionless; same one-period horizon; homogeneous expectations; investments infinitely divisible; investors are price takers. The SML is the graphical CAPM (expected return vs. beta) and applies to any security or portfolio.
\[E(R_i) = R_f + \beta_i\bigl[E(R_M) - R_f\bigr]\]
Compare a forecast return to the SML's required return. Forecast above SML → undervalued (buy); below → overvalued (sell/short); on the SML → fairly priced.
Sharpe = \((R_P - R_f)/\sigma_P\). M² = \(R_f + (\sigma_M/\sigma_P)(R_P - R_f)\); M² alpha = \(M^2 - R_M\). Treynor = \((R_P - R_f)/\beta_P\). Jensen's α = \(R_P - [R_f + \beta_P(R_M - R_f)]\). Use total-risk measures (Sharpe, M²) for portfolios that may bear unsystematic risk; use beta-based measures (Treynor, Jensen) when the overall portfolio is well diversified.
【LOS 84.a】無風險資產 + 風險組合：所有可能組合落在一條直線上，投資者可依風險偏好調整整體部位的風險與報酬。
【LOS 84.b】任意風險組合與無風險資產搭配形成 CAL；同質預期下，當該風險組合即為「市場組合」，CAL 即成為 CML。
【LOS 84.c】系統風險（市場風險）無法分散；非系統風險（個別公司風險）可分散。由於分散成本接近零，市場不對非系統風險給予補償，均衡報酬只依賴系統風險。
【LOS 84.d】報酬生成模型以資產對因子（總體、基本面、統計）的暴露估計預期報酬；最簡形式為市場模型 \(R_i = \alpha_i + \beta_i R_M + e_i\)。
【LOS 84.e】\(\beta_i = \dfrac{\text{Cov}(R_i, R_M)}{\sigma_M^2} = \rho_{i,M}\dfrac{\sigma_i}{\sigma_M}\)；市場 \(\beta = 1\)，\(\beta = 0\) 表示與市場零相關。
【LOS 84.f】CAPM 七大假設（風險厭惡、效用最大化、無摩擦、單期、同質預期、無限可分割、競爭性市場）。SML 即 CAPM 的圖像（預期報酬 vs. β），適用於所有證券與組合。
【LOS 84.g】\(E(R_i) = R_f + \beta_i [E(R_M) - R_f]\)。
【LOS 84.h】估計報酬 vs. SML 必要報酬：上方=低估（買）；下方=高估（賣/放空）；落在 SML 上=合理定價。
【LOS 84.i】Sharpe = \((R_P - R_f)/\sigma_P\)；M² = \(R_f + (\sigma_M/\sigma_P)(R_P - R_f)\)；Treynor = \((R_P - R_f)/\beta_P\)；Jensen's α = \(R_P - [R_f + \beta_P(R_M - R_f)]\)。組合可能含非系統風險用 Sharpe/M²；已分散組合用 Treynor/Jensen。