Descriptive Statistics

Quantitative measures used to summarize and describe data about investments and portfolios. This section covers: mean, median, mode, range, standard deviation, alpha, beta, Sharpe ratio, correlation, and R-squared.

Measures of Central Tendency

Before measuring how much returns vary, it helps to understand what they vary from. Central tendency measures establish the "center" of a distribution.

Why learn these basic statistics? Before you can measure investment risk, you need to know what "normal" looks like. These measures establish a baseline so you can then determine how much returns deviate from that baseline.

When to Use Each Measure

The NASAA study guide emphasizes understanding the differences between mean, median, and mode:

Key Insight: When outliers are present, the median is more representative than the mean. For example, if most portfolio returns cluster around 8-10% but one year had a 50% gain, the median better represents the typical year.

Example Calculation

Given annual returns of: 6%, 4%, 11%, 10%, 4%

• Mean: (6 + 4 + 11 + 10 + 4) ÷ 5 = 7%
• Median: Sorted: 4, 4, 6, 10, 11 → Middle value = 6%
• Mode: 4% appears twice → 4%

Note: If a distribution has no repeating values, there is no mode.

Range

Range = Spread between highest and lowest value

Example: Using the same returns (6%, 4%, 11%, 10%, 4%)

• Range: 11% − 4% = 7%

Range is simple but limited; it only considers two data points.

Standard Deviation

Now that you understand the mean, you can measure how much returns deviate from it. Standard deviation quantifies total risk (both systematic and unsystematic).

What Standard Deviation Tells You:

• Higher standard deviation = More volatile = Higher risk
• Lower standard deviation = More stable = Lower risk

The Normal Distribution (Bell Curve)

• 68% of returns fall within: ±1 standard deviation from mean
• 95% of returns fall within: ±2 standard deviations from mean
• 99% of returns fall within: ±3 standard deviations from mean

Example: A security with 8.7% expected return and 14.6% standard deviation:

• 68% chance: Returns fall between −5.9% and +23.3% (8.7 ± 14.6)
• 95% chance: Returns fall between −20.5% and +37.9% (8.7 ± 29.2)

Comparing Investments

Think of it this way: Standard deviation tells you how "spread out" an investment's returns are. An investment that goes up 5%, then up 6%, then up 4% has low standard deviation (consistent returns). An investment that goes up 30%, then down 20%, then up 15% has high standard deviation (wild swings). Both might have the same average return, but one is a roller coaster and the other is a smooth ride.

Exam Tip: Gotchas

• Standard deviation measures total risk (systematic + unsystematic). Beta measures only systematic risk. If asked which measures total risk, the answer is standard deviation.

Correlation

Correlation is a descriptive statistic that measures the strength and direction of the linear relationship between two securities' returns. The correlation coefficient (commonly denoted r) ranges from −1.0 to +1.0.

Correlation Coefficient Range

Think of it this way: A correlation coefficient tells you how predictable one security's movement is based on another's. At +1.0, if Security A rises 5%, you know Security B will also rise by a proportional amount. At 0, knowing Security A rose 5% tells you nothing about what Security B will do.

Example

Given these correlation coefficients, which pair has the strongest relationship (regardless of direction)?

• Assets A & B: correlation +0.90
• Assets C & D: correlation +0.47
• Assets E & F: correlation 0
• Assets G & H: correlation −0.88

Answer: A & B (+0.90) has the strongest positive relationship. G & H (−0.88) has a nearly-as-strong negative relationship.

Practical Application: Portfolio Construction

Securities with low or negative correlation move more independently from each other. This characteristic is relevant in portfolio construction because combining assets with low correlation can reduce overall portfolio volatility. (Portfolio construction strategies are covered in detail in the Portfolio Management Strategies unit.)

R-Squared (Coefficient of Determination)

• R-squared = correlation coefficient squared
• Represents the percentage of a portfolio's movement explained by the benchmark
• R-squared of 0.90 means 90% of the portfolio's returns are explained by the benchmark
• Higher R-squared makes beta more meaningful as a risk measure

Exam Tip: Gotchas

• Correlation of +1.0 provides NO diversification benefit. Maximum diversification benefit comes from combining assets with negative correlation. Zero correlation still provides some diversification benefit.

Beta

While standard deviation measures total risk, beta isolates systematic (market) risk.

Beta measures a security's volatility relative to the overall market (S&P 500 = beta of 1.0).

Beta Calculations:

• If beta = 1.2 and market rises 10% → Stock rises 12% (10% × 1.2)
• If beta = 0.85 and market falls 10% → Stock falls 8.5% (10% × 0.85)

Beta measures only systematic risk (not total risk).

Think of it this way: If the market is an ocean and stocks are boats, beta tells you how much a particular boat rocks when waves hit. A beta of 1.0 means the boat rocks exactly as much as the water moves. A beta of 1.5 means the boat rocks 50% more than the waves. A beta of 0.5 means the boat is more stable and only rocks half as much. Beta does not measure how good the boat is, just how it responds to the overall ocean (market).

Exam Tip: Gotchas

Beta measures systematic risk only. A stock with a beta of 1.0 still has unsystematic risk; it just moves with the market on average. Diversification reduces unsystematic risk but does NOT reduce beta.

Alpha

Alpha measures the excess return of an investment relative to its expected return based on its risk level.

Formula: Alpha = Actual Return - Expected Return (based on the Capital Asset Pricing Model, or CAPM)

• Alpha = R_portfolio - [R_f + Beta x (R_market - R_f)]
• R_f = risk-free rate; R_market = market return

Alpha Interpretation:

• Positive alpha - the investment outperformed its risk-adjusted expectation (manager added value)
• Negative alpha - the investment underperformed its risk-adjusted expectation
• Zero alpha - the investment performed exactly as expected for its risk level

Example: If a portfolio with beta of 1.1 earned 14% when the expected return was 12.8% (based on market conditions and risk level), the alpha would be +1.2%. This indicates the manager added 1.2% of value above what the portfolio's risk alone would have generated.

Exam Tip: Gotchas

Alpha is NOT simply how much an investment returned. It is how much EXTRA it returned above what was expected given its risk (beta). A fund that returned 12% with an expected return of 10% has an alpha of +2%, even if another fund returned 15% (which may have had higher risk).

Sharpe Ratio

Raw returns tell only part of the story. Two portfolios with identical returns may have vastly different risk profiles. The Sharpe ratio is a risk-adjusted performance measure that helps compare investments on equal footing.

Purpose: Measures excess return per unit of total risk (standard deviation)

Formula:

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

Where: Rp = portfolio return, Rf = risk-free rate (typically the 91-day/3-month Treasury bill rate), σp = standard deviation of portfolio (total risk)

Key Characteristics:

• Risk measure: Standard deviation (total risk = systematic + unsystematic)
• Higher Sharpe ratio = better risk-adjusted performance (more return per unit of total risk)
• Uses standard deviation in the denominator (total risk)
• The numerator is called the excess return (return above the risk-free rate)

Think of it this way: The Sharpe ratio asks, "For every unit of volatility you endured, how much extra return did you earn above the risk-free rate?"

Example 1: Single Portfolio

• Portfolio return: 12%
• Risk-free rate: 2%
• Standard deviation: 20%
• Sharpe = (12% - 2%) / 20% = 0.50

Example 2: Comparing Two Portfolios

Conclusion: Portfolio B has better risk-adjusted performance. It generated 0.75% of excess return for every 1% of volatility, compared to Portfolio A's 0.50%.

Exam Tip: Gotchas

• Sharpe uses standard deviation (total risk). If asked which ratio uses total risk, the answer is Sharpe. The Sharpe ratio is appropriate when the portfolio is the investor's entire holding, because unsystematic risk has not been diversified away.

Summary of Risk Measures