Sharpe vs Sortino vs Treynor Ratio: Complete Guide with Formulas, Examples, and Risk Comparison

Introduction

In investing, return alone does not provide a complete picture of performance. Two portfolios may generate the same return, yet one may involve significantly higher risk. This is why finance professionals rely on risk-adjusted return metrics to evaluate how efficiently an investment generates returns relative to the risk taken.
Three of the most widely used measures are the Sharpe Ratio, Sortino Ratio, and Treynor Ratio.

Why Risk-Adjusted Return Matters

Portfolio	Return	Risk (Std Dev)
A	12%	18%
B	12%	9%

Even though both give 12%, Portfolio B is better because it takes less risk.

Step 1: Types of Risk

Risk Type	Meaning	Used In
Total Risk	All ups and downs	Sharpe
Downside Risk	Only losses	Sortino
Market Risk	Beta (market movement)	Treynor

Step 2: Sharpe Ratio

Definition

Measures excess return per unit of total risk

$\text{Sharpe Ratio} = \frac{R_p – R_f}{\sigma_p}$Sharpe Ratio=σp​Rp​−Rf​​

Example

Return = 12%, Risk-free = 4%, Std Dev = 10% → Sharpe = 0.80

Full Solved Example (Step-by-Step)

Let’s take a more detailed real-world style example:

Year	Portfolio Return
1	10%
2	14%
3	6%
4	12%
5	8%

Step 1: Calculate Average Return
(10 + 14 + 6 + 12 + 8) / 5 = 10%

Step 2: Risk-free rate = 4%

Step 3: Calculate Standard Deviation (simplified)
Returns fluctuate around 10%, giving approx Std Dev ≈ 3%

Step 4: Apply formula
Sharpe = (10 − 4) / 3 = 2.00

Interpretation

This is a very strong Sharpe Ratio, meaning the portfolio generates high return relative to total volatility.

Step 3: Sortino Ratio

Definition

Measures excess return per unit of downside risk only

$\text{Sortino Ratio} = \frac{R_p – R_f}{\sigma_d}$Sortino Ratio=σd​Rp​−Rf​​

Full Solved Example (Step-by-Step)

Using same data:

Year	Return
1	10%
2	14%
3	6%
4	12%
5	8%

Target return = 8%

Step 1: Identify downside returns
Only values below 8% → 6%

Step 2: Calculate downside deviation
Difference from target = (6 − 8) = -2%
Square = 4
Average ≈ 4 → sqrt = 2% downside deviation

Step 3: Apply formula
Sortino = (10 − 4) / 2 = 3.00

Interpretation

Very high Sortino Ratio means minimal downside risk and strong performance.

Step 4: Treynor Ratio

Definition

Measures excess return per unit of market risk (beta)

$\text{Treynor Ratio} = \frac{R_p – R_f}{\beta_p}$Treynor Ratio=βp​Rp​−Rf​​

Full Solved Example (Step-by-Step)

Assume:

Portfolio	Return	Beta
X	10%	0.8

Risk-free = 4%

Step 1: Excess return
10 − 4 = 6%

Step 2: Divide by beta
6 / 0.8 = 7.5

Interpretation

For each unit of market risk, portfolio generates 7.5% return.

Step 5: Graphs with Real Points and Numbers

Step 6: Detailed Graph Reading with Numbers (Very Easy)

🔹 Graph 1: Sharpe Ratio (Total Risk vs Return)

Portfolio	Return	Risk
A	10%	5%
B	12%	10%
C	14%	20%

Risk-free = 4%

Sharpe:

• A = (10−4)/5 = 1.20
• B = 0.80
• C = 0.50

👉 Best = Portfolio A

Explanation:
A gives strong return with very low risk → highest efficiency

Visual logic:
Steepest line = best Sharpe

🔹 Graph 2: Sortino Ratio

Portfolio	Returns
A	10, 9, 11, 10, 9
B	15, -5, 18, -6, 20
C	12, 8, 13, 7, 11

Target = 8%

Downside:

• A = none
• B = large losses
• C = small losses

👉 Best = Portfolio A

Explanation:
A has almost zero downside → highest Sortino

Visual logic:
Less shaded area = better

🔹 Graph 3: Treynor Ratio

Portfolio	Return	Beta
A	10%	0.5
B	12%	1.0
C	14%	2.0

Treynor:

• A = 12
• B = 8
• C = 5

👉 Best = Portfolio A

Explanation:
A delivers good return with lowest market exposure

Visual logic:
Steepest slope = best

Step 7: Final Comparison Table

Portfolio	Sharpe	Sortino	Treynor	Best In
A	1.20	Highest	12	✅ All
B	0.80	Lowest	8	❌ Weak
C	0.50	Medium	5	❌ Risky

Key Takeaways

• Sharpe Ratio measures return per unit of total risk
• Sortino Ratio measures return per unit of downside risk
• Treynor Ratio measures return per unit of market risk
• Full solved examples show how each ratio is calculated step by step
• Graph 1 (Sharpe) identifies best portfolio using slope vs total risk
• Graph 2 (Sortino) identifies best portfolio using lowest downside area
• Graph 3 (Treynor) identifies best portfolio using return per beta
• Portfolio A performs best across all ratios due to efficient risk management
• High returns alone do not guarantee better performance if risk is excessive
• Combining all three ratios gives a complete and professional evaluation of investments