Risk & Performance Measures

Investment advisers use various metrics to evaluate portfolio risk and performance. Understanding these measures is essential for client reporting, manager evaluation, and investment selection.

Risk Measures

Standard Deviation

What it measures: Total volatility (dispersion of returns around the mean)

Interpretation Example:

• Fund A: 10% return, 8% standard deviation
• Fund B: 10% return, 20% standard deviation
• Fund A is less risky with the same return

Beta (β)

What it measures: Sensitivity to market movements (systematic risk)

R-Squared (R²)

What it measures: How much of a portfolio's movements are explained by the benchmark

Important: R² determines whether beta is meaningful. If R² is low, beta is unreliable.

Performance Measures

Alpha (Jensen's Alpha)

What it measures: Excess return above what CAPM predicts—manager skill

$\alpha = R_p - [R_f + \beta (R_m - R_f)]$

Where $R_p$ = Actual Portfolio Return

Example:

• Actual return: 14%
• Risk-free rate: 3%
• Market return: 10%
• Beta: 1.2

Expected return = $3% + 1.2 \times (10% - 3%) = 11.4%$

$\alpha = 14% - 11.4% =$ +2.6% (outperformed)

Sharpe Ratio

What it measures: Excess return per unit of total risk (standard deviation)

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

Where $\sigma_p$ = Portfolio Standard Deviation

Example:

• Portfolio return: 12%
• Risk-free rate: 2%
• Standard deviation: 20%

$\text{Sharpe Ratio} = \frac{12\% - 2\%}{20\%} = \frac{10\%}{20\%} = \mathbf{0.50}$

Treynor Ratio

What it measures: Excess return per unit of systematic risk (beta)

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

Where $\beta_p$ = Portfolio Beta

Example:

• Portfolio return: 12%
• Risk-free rate: 2%
• Beta: 1.25

$\text{Treynor Ratio} = \frac{12\% - 2\%}{1.25} = \frac{10\%}{1.25} = \mathbf{8.0}$

Comparing Sharpe and Treynor

Decision Rule

• If the portfolio is the investor's ONLY holding → Use Sharpe (total risk exposure)
• If the portfolio is ONE OF MANY holdings → Use Treynor (only systematic risk matters due to diversification)

Information Ratio

What it measures: Active return per unit of tracking error (consistency of outperformance)

$\text{Information Ratio} = \frac{R_p - R_b}{\text{Tracking Error}}$

Where:

• $R_b$ = Benchmark Return
• Tracking Error = standard deviation of the difference between portfolio and benchmark returns

Summary Comparison

In Practice

When evaluating managers:

• Look for positive alpha (value added)
• Compare Sharpe ratios for standalone investments
• Use Treynor ratio when adding to an existing diversified portfolio
• Check R-squared before relying on beta
• Higher information ratio indicates more consistent outperformance

On the Exam

Series 65 frequently tests:

• Calculating Sharpe ratio and Treynor ratio
• Knowing which risk measure each ratio uses
• Understanding when to use Sharpe vs. Treynor
• Interpreting positive vs. negative alpha

Key Takeaways

1. Standard deviation measures total risk; beta measures systematic risk
2. R-squared indicates how much the benchmark explains portfolio movements
3. Positive alpha = outperformance; negative alpha = underperformance
4. Sharpe uses standard deviation (total risk)—for undiversified portfolios
5. Treynor uses beta (systematic risk)—for diversified portfolios
6. Higher ratios = better risk-adjusted performance