Sharpe and Treynor ratios

The Sharpe Ratio and Treynor Ratio are two essential risk-adjusted performance measures used in portfolio and investment analysis. Both help investors evaluate whether the returns earned are sufficient given the level of risk taken.

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📈 1. Sharpe Ratio

📌 Formula:

$$\boxed{Sharpe\ Ratio = \frac{R_p - R_f}{\sigma_p}}$$

Where:

• $R_p$: Portfolio return

• $R_f$: Risk-free rate

• $\sigma_p$: Standard deviation of portfolio returns (total risk)

🔍 Interpretation:

• Measures excess return per unit of total risk.

• Higher Sharpe Ratio = Better risk-adjusted return.

• Helps compare diversified portfolios or mutual funds.

Sharpe Ratio	Interpretation
< 1	Suboptimal performance
1 – 1.99	Acceptable
2 – 2.99	Good
3+	Excellent

✅ Best Use Case:

When evaluating total risk (both systematic and unsystematic), such as mutual funds or ETFs.

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📈 2. Treynor Ratio

📌 Formula:

$$\boxed{Treynor\ Ratio = \frac{R_p - R_f}{\beta_p}}$$

Where:

• $R_p$: Portfolio return

• $R_f$: Risk-free rate

• $\beta_p$: Beta of the portfolio (systematic risk)

🔍 Interpretation:

• Measures excess return per unit of systematic risk.

• Ignores unsystematic (diversifiable) risk.

• Higher Treynor Ratio = Better compensation for market risk.

Treynor Ratio	Interpretation
Higher Ratio	More return per beta unit
Lower Ratio	Inefficient risk-taking

✅ Best Use Case:

When comparing well-diversified portfolios or benchmark-aware strategies where only systematic risk matters.

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🧠 Key Differences: Sharpe vs Treynor

Feature	Sharpe Ratio	Treynor Ratio
Risk Type	Total risk (std deviation)	Systematic risk (beta)
Use When	Portfolio is not fully diversified	Portfolio is well-diversified
Focus	Risk-adjusted return	Market risk-adjusted return
Suitable For	Any portfolio or mutual fund	CAPM-based portfolios, equity investments

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💡 Example

Assume:

• $R_p = 12\%$, $R_f = 4\%$

• $\sigma_p = 10\%$, $\beta_p = 1.2$

Sharpe Ratio:

$$\frac{12\% - 4\%}{10\%} = \boxed{0.8}$$

Treynor Ratio:

$$\frac{12\% - 4\%}{1.2} = \boxed{6.67\%}$$

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🛠️ Tools to Calculate:

• Excel: Using historical return series & risk-free rate

• Online platforms: Value Research, Morningstar, Portfolio Visualizer

• Python (Pandas, NumPy): For backtesting and custom metrics

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✅ Summary

Ratio	Measures	Use When
Sharpe	Return per unit of total risk	Portfolio may not be diversified
Treynor	Return per unit of market risk	Portfolio is well-diversified