Descriptive Statistics

Now that you understand how to value cash flows over time, you need tools to measure and compare investment returns and risk. Descriptive statistics give you that toolkit.

Measures of Central Tendency

Central tendency tells you "what's typical" for a data set of returns.

• Mean (Arithmetic Average): Sum of all values divided by the number of values

  • Most commonly used measure of average return
  • Weakness: sensitive to extreme outliers (one very large or small value can distort the mean significantly)

• Median: The middle value when data is arranged in order

  • Resistant to outliers: extreme values do not move the median
  • Better measure of "typical" when data is skewed (e.g., income distributions, hedge fund returns)

• Mode: The most frequently occurring value

  • A data set can have multiple modes (bimodal, multimodal) or no mode at all
  • Less commonly tested than mean and median

• Range: Difference between the highest and lowest values

  • Simplest measure of dispersion
  • Limitation: only considers two data points, ignoring everything in between

Exam Tip: Gotchas

• When a question mentions "skewed data" or "outliers," the correct measure of central tendency is the median, not the mean. The mean gets pulled toward outliers; the median does not.

Standard Deviation - Measuring Total Risk

Standard deviation measures how far individual returns typically deviate from the mean return.

• Higher standard deviation = greater volatility = greater risk
• Lower standard deviation = more consistent returns = lower risk
• Standard deviation measures total risk, which includes both:
  • Systematic risk (market risk; cannot be diversified away)
  • Unsystematic risk (company-specific risk; can be diversified away)

Think of it this way: If two funds both averaged 8% annually, but Fund A's returns ranged from 6% to 10% while Fund B's ranged from -5% to 21%, Fund B has a much higher standard deviation. Same average return, very different ride.

Exam Tip: Gotchas

• Standard deviation measures total risk. Beta measures only systematic (market) risk. If a question asks about "total risk" or "volatility," the answer involves standard deviation. If it asks about "market risk" or "systematic risk," the answer involves beta.

Risk-Adjusted Performance Metrics

Raw returns don't tell the full story. A 12% return with wild swings is very different from a 12% return with steady growth. Risk-adjusted metrics account for the risk taken to earn those returns.

Memory Aid:

• Standard deviation = Sum of all risk (total)
• Beta = market risk only (Broad market sensitivity)
• Alpha = Added value by the manager
• Sharpe ratio = Standard deviation in the denominator (total risk)

Beta - Systematic Risk

Beta measures how sensitive an investment is to overall market movements (systematic risk only).

• A stock with beta of 1.5 is expected to move 1.5% for every 1% move in the market
• Beta only captures systematic risk: the risk that remains after full diversification

Alpha - Manager Skill

Alpha measures the excess return of an investment relative to what was expected given its level of risk (as measured by beta).

• Positive alpha: The investment outperformed its risk-adjusted benchmark (the manager added value)
• Negative alpha: The investment underperformed its risk-adjusted benchmark
• Zero alpha: The investment performed exactly as expected for its risk level
• Alpha is the primary measure of active management skill
• Jensen's Alpha formula: Alpha = Actual Return - [Risk-Free Rate + Beta x (Market Return - Risk-Free Rate)]

Think of it this way: If the market returned 10% and a fund's beta predicted it should earn 8%, but the fund actually earned 11%, the manager generated 3% alpha. That extra return came from skill (or luck), not just riding the market.

Exam Tip: Gotchas

• Positive alpha means the manager beat expectations after adjusting for risk, not simply that the fund went up. A fund can have a positive return and still have negative alpha if it underperformed what its risk level predicted.

Sharpe Ratio - Return Per Unit of Total Risk

The Sharpe ratio measures how much excess return you earn for each unit of total risk.

• Formula: Sharpe Ratio = (Portfolio Return - Risk-Free Rate) / Standard Deviation
• Higher Sharpe ratio = better risk-adjusted performance
• Uses standard deviation (total risk), not beta
• Useful for comparing portfolios or funds with different risk levels

Exam Tip: Gotchas

• The Sharpe ratio uses standard deviation in the denominator (total risk). It does NOT use beta. If you see a question about "risk-adjusted return per unit of total risk," that is the Sharpe ratio. If you see "risk-adjusted return per unit of systematic risk," that would be the Treynor ratio (less commonly tested on the Series 66).

Correlation - The Key to Diversification

Correlation is a statistical measure of how two investments move relative to each other, ranging from -1.0 to +1.0.

Why correlation matters:

• Combining assets with low or negative correlation reduces overall portfolio risk
• This is the mathematical basis for diversification
• Perfectly negative correlation is not required to benefit; anything below +1.0 provides some risk reduction
• Diversification eliminates unsystematic risk but cannot eliminate systematic risk
• Research suggests holding approximately 30 or more securities effectively diversifies away most unsystematic risk

Exam Tip: Gotchas

• Diversification does NOT reduce all risk. It eliminates unsystematic (company-specific) risk only. Systematic (market) risk remains no matter how many securities you hold. A perfectly diversified portfolio still has market risk.
• Correlation of +1.0 means zero diversification benefit, not maximum. Two assets that move in perfect lockstep provide no risk reduction when combined.