Measures of Risk and Return

Quantitative risk and return measures are heavily tested on the CFP exam. Understanding when to use each measure--and how to calculate them--is essential for evaluating investment performance.

Standard Deviation

Standard deviation measures the dispersion of returns around the average (mean) return. It quantifies total risk--both systematic and unsystematic risk combined.

Key Characteristics

• Measures volatility or "how much returns flip-flop around the average"
• Higher standard deviation = more risky/volatile investment
• Appropriate for non-diversified portfolios or individual securities
• Expressed in the same units as the returns (percentages)

The Normal Distribution

Standard deviation relates to the normal distribution (bell curve). For a normally distributed set of returns:

Calculating Probability with Standard Deviation

Example: A fund has an expected return of 12% with a standard deviation of 6%. What is the probability of a return less than 0%?

Step 1: Determine how many standard deviations 0% is from the mean.

• Distance from mean: 12% - 0% = 12%
• Number of standard deviations: 12% / 6% = 2 standard deviations below the mean

Step 2: Use the normal distribution.

• 95% of returns fall within +/-2 standard deviations (between 0% and 24%)
• 5% falls outside this range (2.5% above 24%, 2.5% below 0%)
• Probability of return less than 0% = 2.5%

Comparing Investments Using Standard Deviation

Example: Which investment is more risky?

Asset B has higher returns but look at the consistency:

• Asset A ranges from 8% to 12% (4 percentage point spread)
• Asset B ranges from 19% to 21% (2 percentage point spread)

Asset A has higher standard deviation despite lower returns. Asset B is actually less volatile relative to its returns.

Variance

Variance is simply the standard deviation squared. While less intuitive, it is mathematically useful for portfolio calculations.

Key Points:

• Variance = Standard Deviation squared
• Standard Deviation = Square root of Variance
• Variance is always positive
• Used in portfolio risk calculations with covariance

Coefficient of Variation (CV)

The coefficient of variation standardizes risk per unit of return. It allows comparison of investments with different return levels.

Formula

CV = Standard Deviation / Mean Return

Interpretation

• Higher CV = More risk per unit of return (less desirable)
• Lower CV = Less risk per unit of return (more desirable)
• CV is useful when comparing investments with different expected returns

Example: Comparing Risk-Adjusted Performance

Asset A has the lower CV, meaning less risk per unit of return. Asset A offers better risk-adjusted performance.

Another Example

Asset B has the lower CV despite higher absolute volatility. The higher return more than compensates for the higher risk.

CFP Exam Tip: The asset with the LOWER coefficient of variation has the HIGHER risk-adjusted return. Think: lower risk per unit of return = better.

Beta

Beta measures systematic risk--the volatility of a security relative to the overall market. The market (typically S&P 500) has a beta of 1.0 by definition.

Interpreting Beta

When to Use Beta vs. Standard Deviation

For a well-diversified portfolio, unsystematic risk has been eliminated, so beta is the appropriate risk measure. For a concentrated portfolio or individual security, standard deviation captures total risk.

R-Squared (Coefficient of Determination)

R-squared (R2) measures what percentage of a portfolio's returns can be explained by the market's returns. It is calculated by squaring the correlation coefficient (r).

Formula

R-squared = (Correlation Coefficient) squared

Interpretation

Key Exam Points

• R-squared tells you if beta is meaningful for measuring risk
• High R-squared (>0.70): Beta is appropriate; use Treynor ratio
• Low R-squared (<0.70): Beta is unreliable; use Sharpe ratio (standard deviation)
• R-squared ranges from 0 to 1 (or 0% to 100%)

Example

Fund XYZ has a correlation of 0.90 to the S&P 500. What percent of Fund XYZ's return is due to the market?

R-squared = (0.90) squared = 0.81 = 81%

81% of Fund XYZ's return is explained by market movements. Beta is a reliable risk measure for this fund.

CFP Exam Tip: If the exam gives you R-squared but not correlation, you can tell if beta is appropriate. If R-squared > 0.70, use beta-based measures. If R-squared < 0.70, use standard deviation-based measures.

Correlation Coefficient

The correlation coefficient (r) measures how two investments move relative to each other. It ranges from -1 to +1.

Interpretation

Key Points

• Diversification benefits begin when correlation is less than +1
• The lower the correlation, the greater the diversification benefit
• Negative correlation provides the best diversification (rare in practice)
• Most stocks have positive correlations (0.3 to 0.8 typical)

Covariance

Covariance measures how two securities move together. It combines the standard deviations of two securities with their correlation.

Formula

COV(A,B) = Standard Deviation A x Standard Deviation B x Correlation A,B

Key Points

• Positive covariance: Assets move in the same direction
• Negative covariance: Assets move in opposite directions
• Covariance is used in portfolio variance calculations
• Unlike correlation, covariance is not standardized (harder to interpret)

Alpha (Jensen's Alpha)

Alpha measures the excess return of a portfolio above what the Capital Asset Pricing Model (CAPM) predicts. It indicates manager skill.

Formula

Alpha = Actual Return - Expected Return (from CAPM)

Or: Alpha = Rp - [Rf + Beta(Rm - Rf)]

Where:

• Rp = Portfolio's actual return
• Rf = Risk-free rate
• Beta = Portfolio's beta
• Rm = Market return

Interpretation

Example Calculation

A mutual fund returned 15% last year with a beta of 2.0. The risk-free rate was 3% and the market returned 8%. Did the manager outperform?

Step 1: Calculate expected return using CAPM. Expected Return = Rf + Beta(Rm - Rf) Expected Return = 3% + 2.0(8% - 3%) Expected Return = 3% + 2.0(5%) Expected Return = 3% + 10% = 13%

Step 2: Calculate alpha. Alpha = Actual Return - Expected Return Alpha = 15% - 13% = +2%

Interpretation: Positive alpha of 2% means the manager outperformed by 2% on a risk-adjusted basis.

CFP Exam Tip: Alpha is an ABSOLUTE measure. A positive alpha is good by itself--you do not need to compare to other funds. Sharpe and Treynor are RELATIVE measures that must be compared to rankings.

Summary: When to Use Each Measure