Sharpe Ratio | Treynor Ratio | M² (Modigliani–Modigliani) | Jensen’s Alpha **📅 Date:** ➤ ⌈ [[2025-09-08-Mon〚Sharpe ratio, Treynor ratio, M², and Jensen’s alpha ▪Greek Letters 〛]]⌋ **💭 What:** [[💰 L065- 06+M², Sharpe Ratio, Treynor Ratio, Jensen’s Alpha — Quick Guide|Explained]] ➤ **Jensen’s Alpha**: If α > 0, the manager outperforms by generating returns above the CAPM benchmark. ➤ **CAPM**: Provides the expected return for a security given its Beta. - If Jensen’s Alpha > 0 → manager delivers excess return beyond CAPM’s required return. - If a security’s actual return > CAPM return → the stock is undervalued, and it signals a **buy opportunity**. ➤**M²** → benchmarked to the **market portfolio**. It scales the portfolio to the market’s standard deviation. _Meaning_: the return the portfolio would have earned if it carried the **same risk as the market portfolio**. ➤ **Sharpe Ratio** → higher is better. **Sharpe vs Treynor**: 1. **Treynor Ratio** assumes idiosyncratic risk is already diversified away, so it focuses only on **systematic risk (β)**. 2. **Sharpe Ratio** considers **total risk (σ)**, useful when the portfolio is the investor’s entire wealth. - In practice, Sharpe and Treynor serve different contexts (σ vs β). ➤**M²** makes Sharpe easier to interpret as a % return. ➤**Jensen’s Alpha** shows true excess return beyond CAPM — the manager’s added value. **👀 Snap:** ![[IMG_5916.jpeg]] ⇩ 🅻🅸🅽🅺🆂 ⇩ **🏷️ Tags**: #💰/Economy #💰/Formula #💰/公式 **🗂 Menu**: ⌈[[✢ M O C ➣ 09 ⌈S E P - 2 0 2 5⌉ ✢|2025 - S E P - MOC]]⌋ ⌈[[✢ L O G ➢ 09 ⌈A U G - 2 0 2 5⌉ ✢|2025 - S E P - LOG]] ⌋ #👾/Private ➤ ⌈[[💰 L065- 01 Common Greek Letters]]⌋ ➤ ⌈[[💰 L065- 02 - The Sharpe ratio, Treynor Ratio, M² & Jensen’s Alpha 「公式」]]⌋ ➤ ⌈[[💰 L065- 03 + Passive vs Active Management & Treynor Ratio]]⌋ ➤ ⌈[[💰 L065- 04 - Applications in Portfolio Construction(SML, CAPM, CAL,CML)]]⌋ ➤ ⌈[[💰 L065- 05+Idiosyncratic Risk (特质风险) -Unsystematic Risk]]⌋ ➤ ⌈[[💰 L065- 06+M², Sharpe Ratio, Treynor Ratio, Jensen’s Alpha — Quick Guide]]⌋ ➤ ⌈[[💰 058 - 02 - Sharpe Ratio]]⌋ ➤ ⌈[[💰 059-07 -Capital Allocation Line(Sharpe Ratio), Indifference Curves & Utility function]]⌋ ➤ ⌈[[💰 059-01- Millennium Hedge Fund – σ Sharpe Ratio & Sigma Strategy (风险管理）]]⌋ --- >[!question] If the **Treynor ratio is very high**, does that mean active management is effective? >Passive VS Active >- In active management, we cannot look at **Beta alone**, because Beta is delivered by the market itself. >- A high Treynor ratio means the portfolio generates **more return per unit of systematic risk** than the market — but true active value must also be shown through **alpha (excess return beyond CAPM)**. >>在active management 不止看beta 因为beta是市场deliver的， treynor很大可以理解return is bigger than the Market， You generate more return than per unit of systematic risk >[!quote] 用正确的东西影响自己判断的方式，很多时候做投资判断的时候是在摸着石头过河 ---- ## 📝 1. Sharpe Ratio - **Definition**: Measures **excess return per unit of total risk** (standard deviation). - **Formula**: ![[Screenshot 2025-09-08 at 21.15.05.png]] - **Interpretation**: - Higher Sharpe = better risk-adjusted performance. - Useful when portfolio is the entire investment (all risks matter). - **Case Study**: - Portfolio return = 10%, risk-free = 2%, σ = 15% - Sharpe = (10% − 2%) / 15% = 0.53 - Benchmark Sharpe = 0.40 → portfolio outperforms on risk-adjusted basis. --- ## 📝 2. Treynor Ratio - **Definition**: Measures **excess return per unit of systematic risk (Beta)**. - **Formula**: ![[Screenshot 2025-09-08 at 21.15.26.png]] - **Interpretation**: - Focuses only on **market risk**. - Useful when portfolio is already **well-diversified** (idiosyncratic risk ≈ 0). - **Case Study**: - Portfolio return = 12%, risk-free = 3%, β = 1.2 - Treynor = (12% − 3%) / 1.2 = 7.5% - Market Treynor = 6% → portfolio delivers higher return per unit of systematic risk. --- ## 📝 3. M² (Modigliani–Modigliani Measure) - **Definition**: Converts the Sharpe ratio into a measure that is **directly comparable to market returns**. - **Formula**: ![[Screenshot 2025-09-08 at 21.24.07.png]] - **Interpretation**: - Represents what the portfolio would return if it had the **same risk as the market**. - Easier for investors to understand → expressed as % returns. - **Case Study**: - Sharpe ratio = 0.50, σm = 12%, Rf = 2% - M² = 2% + (0.5 × 12%) = 8% - If market return = 7%, portfolio shows superior performance. --- ## 📝 4. Jensen’s Alpha (α) - **Definition**: Measures portfolio’s **excess return beyond CAPM prediction**. - **Formula**: ![[Screenshot 2025-09-08 at 21.24.14.png]] - **Interpretation**: - α > 0 → portfolio outperforms given its risk. - α < 0 → underperforms given its risk. - Direct measure of “manager skill” in active management. - **Case Study**: - Portfolio return = 11%, Rf = 3%, β = 1.1, market return = 9% - CAPM required = 3% + 1.1 × (9% − 3%) = 9.6% - Jensen’s α = 11% − 9.6% = +1.4% - Positive alpha → manager added value beyond risk exposure. ### CAPM Return Formula ![[Screenshot 2025-09-15 at 13.32.13.png]] --- ## 🔑 Key Differences | Measure | Risk Denominator | Best Use Case | |--------------------|---------------------------|---------------| | **Sharpe Ratio** | σ (total risk) | When portfolio is entire investment | | **Treynor Ratio** | β (systematic risk) | When portfolio is well-diversified | | **M²** | σm (market’s risk level) | Converts Sharpe into % returns | | **Jensen’s Alpha** | CAPM benchmark (expected) | Evaluating manager skill | --- ## 📝 Summary - **Sharpe Ratio** → excess return per total risk. - **Treynor Ratio** → excess return per systematic risk. - **M²** → risk-adjusted return in % terms (easy comparison to market). - **Jensen’s Alpha** → measures active manager’s value-add vs CAPM. All four are **risk-adjusted performance metrics**, but each fits different contexts (undiversified portfolio, diversified portfolio, communication with clients, manager evaluation).