Sharpe, Treynor & Jensen Measures

(Complete Lecture Notes + Numerical Problems + Comparison Table + Diagram Description)

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A. LECTURE NOTES (READY TO TEACH / WRITE IN EXAMS)

1. Introduction

In portfolio management, investors are not concerned only with return, but also with the risk taken to earn that return. To evaluate portfolio performance on a risk-adjusted basis, three important measures are used:

• Sharpe Measure
• Treynor Measure
• Jensen’s Measure

These measures help compare different portfolios or mutual funds.

2. Sharpe Measure (Sharpe Ratio)

Definition

The Sharpe Measure evaluates portfolio performance by measuring excess return per unit of total risk.

Formula

Sharpe Ratio = (Rp − Rf) / σp

Explanation

• Considers total risk (standard deviation)
• Suitable for undiversified or partially diversified portfolios
• Higher value indicates better performance

Decision Rule

• Higher Sharpe Ratio → Better portfolio
• Negative Sharpe Ratio → Return below risk-free rate

3. Treynor Measure (Treynor Ratio)

Definition

The Treynor Measure evaluates portfolio performance by measuring excess return per unit of systematic risk.

Formula

Treynor Ratio = (Rp − Rf) / βp

Explanation

• Considers market risk (beta)
• Suitable for well-diversified portfolios
• Assumes unsystematic risk is negligible

Decision Rule

• Higher Treynor Ratio → Superior performance

4. Jensen’s Measure (Jensen’s Alpha)

Definition

Jensen’s Measure calculates the abnormal return of a portfolio over its expected return predicted by the CAPM.

Formula

α = Rp − [ Rf + βp (Rm − Rf) ]

Explanation

• Measures portfolio manager’s skill
• Based on CAPM
• Shows whether the manager has beaten the market

Decision Rule

• α > 0 → Superior performance
• α = 0 → Market-level performance
• α < 0 → Inferior performance

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B. FULLY SOLVED NUMERICAL PROBLEMS

Problem 1: Sharpe Ratio

Given:
Portfolio Return = 18%
Risk-free Rate = 6%
Standard Deviation = 12%

Sharpe Ratio = (18 − 6) / 12 = 12 / 12 = 1.0

Interpretation: The portfolio earns 1 unit of excess return per unit of total risk.

Problem 2: Treynor Ratio

Given:
Portfolio Return = 16%
Risk-free Rate = 5%
Beta = 1.1

Treynor Ratio = (16 − 5) / 1.1 = 11 / 1.1 = 10%

Interpretation: The portfolio provides 10% excess return per unit of systematic risk.

Problem 3: Jensen’s Alpha

Given:
Portfolio Return = 17%
Risk-free Rate = 6%
Market Return = 14%
Beta = 1.2

Expected Return = 6 + 1.2(14 − 6) = 15.6%
Jensen’s Alpha = 17 − 15.6 = 1.4%

Interpretation: Positive alpha indicates superior stock-selection ability.

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C. COMPARATIVE TABLE (EXAM-FRIENDLY)

Basis	Sharpe Measure	Treynor Measure	Jensen Measure
Risk Used	Total Risk (σ)	Systematic Risk (β)	Systematic Risk (β)
Model	Risk-Return	CAPM	CAPM
Best For	Undiversified portfolios	Diversified portfolios	Evaluating manager skill
Output	Ratio	Ratio	Alpha
Higher Value Means	Better performance	Better performance	Outperformance

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D. DIAGRAM / FLOWCHART (FOR SLIDES OR NOTES)

Portfolio Performance Evaluation | ------------------------------------- | | | Sharpe Measure Treynor Measure Jensen Measure | | | Total Risk (σ) Systematic Risk CAPM Expected Return | | | Excess Return / Excess Return / Actual − Expected Total Risk Market Risk (Alpha)

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E. SHORT EXAM ANSWER (5–6 LINES)

Sharpe, Treynor, and Jensen measures are risk-adjusted performance evaluation techniques. The Sharpe measure uses total risk to evaluate portfolio performance. The Treynor measure considers only systematic risk and is suitable for diversified portfolios. Jensen’s measure calculates abnormal return using the CAPM model and reflects the manager’s skill. A higher Sharpe and Treynor ratio and a positive Jensen’s alpha indicate superior performance.

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F. MEMORY TRICK (VERY IMPORTANT FOR EXAMS)

Sharpe → σ (Total Risk)
Treynor → β (Market Risk)
Jensen → α (Manager Skill)

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