Quick Answer
Descriptive statistics summarize investment data. Mean, median, and mode describe the center of a distribution; range and standard deviation describe its spread. Standard deviation captures total risk, beta isolates systematic risk, alpha measures manager skill, correlation describes how two securities move together, R-squared measures how much of that movement a benchmark explains, and the Sharpe ratio measures risk-adjusted return.
Each measure answers a different exam question: what's typical, how volatile is it, and did the manager add value. Knowing which measure answers which question is tested more often than the arithmetic itself.
What Do Mean, Median, and Mode Tell You?
Central tendency measures establish the "center" of a distribution so you have a baseline to measure deviation from.
| Measure | Definition | How to Calculate | Best Use |
|---|---|---|---|
| Mean (Average) | Sum of values divided by count | Add all returns, divide by number of periods | General performance summary |
| Median | Middle value when sorted | Arrange in order, find middle number | When outliers exist |
| Mode | Most frequently occurring value | Find the value that appears most often | Identifying common outcomes |
Which Measure Should You Use?
| Measure | Best Used When | Limitation |
|---|---|---|
| Mean | Data is normally distributed | Skewed by outliers (extreme values) |
| Median | Outliers exist (skewed distribution) | Ignores the actual values (only position matters) |
| Mode | Identifying most common outcome | May not exist; not useful for continuous data |
When outliers are present, the median is more representative than the mean. If most portfolio returns cluster around 8-10% but one year had a 50% gain, the median better represents the typical year.
Which way did the outliers pull it? The gap between mean and median tells you which way the outliers sit:
- Mean > Median: A few unusually high values pulled the mean up. This is a right-skewed (positive skew) distribution.
- Mean < Median: A few unusually low values pulled the mean down. This is a left-skewed (negative skew) distribution.
Exam Tip: Gotchas
A portfolio with a median return of 7% but a mean return of 4% is left-skewed: a few exceptionally poor returns dragged the average below the typical result.
Worked Example
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%
A data set can have no mode (no repeating values), one mode, or multiple modes (two or more values tie for most frequent).
What Does Range Measure?
Range = highest value minus lowest value.
| Measure | Definition | How to Calculate | Best Use |
|---|---|---|---|
| Range | Spread between highest and lowest | Highest return minus lowest return | Quick volatility estimate |
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.
What Does Standard Deviation Measure?
Standard deviation quantifies total risk: both systematic and unsystematic.
- Higher standard deviation = more volatile = higher risk
- Lower standard deviation = more stable = lower risk
How Does the Normal Distribution (Bell Curve) Apply?
- 68% of returns fall within ±1 standard deviation from the mean
- 95% of returns fall within ±2 standard deviations from the mean
- 99% of returns fall within ±3 standard deviations from the mean
Example: A security with an 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)
How Do You Compare Investments by Volatility?
| Company | Returns Over 4 Years | Mean | Deviation from Mean | Volatility |
|---|---|---|---|---|
| A | 12%, 4%, 8%, 6% | 7.5% | -3.5% to +4.5% | Very Low |
| B | 7%, 8%, 9%, 6% | 7.5% | -1.5% to +1.5% | Lowest |
| C | 10%, 12%, -2%, 10% | 7.5% | -9.5% to +4.5% | Moderate |
| D | 15%, 20%, -8%, 3% | 7.5% | -15.5% to +12.5% | Highest |
An investment that moves up 5%, then 6%, then 4% has low standard deviation: consistent returns. One that moves up 30%, down 20%, then up 15% has high standard deviation: wide swings. Both can share the same average return; one is a smooth ride, the other a roller coaster.
Exam Tip: Gotchas
Standard deviation measures total risk (systematic + unsystematic). Beta measures only systematic risk. If asked which measure captures total risk, the answer is standard deviation.
What Does Correlation Tell You?
Correlation measures the strength and direction of the linear relationship between two securities' returns. The correlation coefficient (r) ranges from -1.0 to +1.0.
How Do You Read the Correlation Coefficient?
| Coefficient | Meaning | Interpretation |
|---|---|---|
| +1.0 | Perfect positive correlation | Two securities move together in perfect lockstep |
| 0 | No correlation | No linear relationship (independence is not implied) |
| -1.0 | Perfect negative correlation | Two securities move in exactly opposite directions |
| Between | Partial correlation | Strength increases as coefficient approaches ±1.0 |
At +1.0, if Security A rises 5%, Security B rises by a proportional amount. At 0, there is no linear relationship between the two, though they can still be related in a non-linear way.
Which Pair Has the Strongest Relationship?
Given these correlation coefficients, which pair moves together most predictably, 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.
Why Does Correlation Matter for Portfolio Construction?
Securities with low or negative correlation move more independently of each other. Combining assets with low correlation can reduce overall portfolio volatility. (Portfolio construction strategies are covered in the Portfolio Management Strategies unit.)
What Does R-Squared Add?
- 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
- A 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.
What Does Beta Measure?
While standard deviation measures total risk, beta isolates systematic (market) risk only. Beta measures a security's volatility relative to the overall market (the S&P 500 has a beta of 1.0).
| Beta | Interpretation | Example |
|---|---|---|
| 0 | No market correlation | 91-day T-bill |
| < 1 | Less volatile than market | Utilities, defensive stocks |
| = 1 | Moves with market | Index funds |
| > 1 | More volatile than market | Tech stocks, small caps |
| 2.0 | Twice as volatile as market | Highly leveraged funds |
| Negative | Moves opposite to market | Rare; some gold stocks |
Calculations:
- If beta = 1.2 and the market rises 10%, the stock rises 12% (10% x 1.2)
- If beta = 0.85 and the market falls 10%, the stock falls 8.5% (10% x 0.85)
- A beta of 2.0 means the security is twice as volatile as the market, common among highly leveraged funds
If the market is an ocean and stocks are boats, beta tells you how much a boat rocks when waves hit. A beta of 1.0 rocks exactly as much as the water moves; 1.5 rocks 50% more than the waves; 0.5 rocks only half as much. Beta does not measure how good the boat is, only how it responds to the market.
Exam Tip: Gotchas
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.
What Does Alpha Measure?
Alpha measures the excess return of an investment relative to its expected return based on its risk level (CAPM).
Formula: Alpha = Actual Return - Expected Return
-
Alpha = R_portfolio - [R_f + Beta x (R_market - R_f)]
-
R_f = risk-free rate; R_market = market return
-
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: A portfolio with a beta of 1.1 earned 14% when the expected return was 12.8%. Alpha = +1.2%, meaning 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% at higher risk.
What Does the Sharpe Ratio Measure?
Raw returns tell only part of the story; two portfolios with identical returns can carry very different risk. The Sharpe ratio is a risk-adjusted performance measure that lets you compare investments on equal footing.
Purpose: Measures excess return per unit of total risk (standard deviation).
Formula:
Where Rp = portfolio return, Rf = risk-free rate (typically the 91-day/3-month Treasury bill rate), σp = standard deviation of the portfolio (total risk). The numerator is called the excess return.
How Do You Compare Two Portfolios?
| Portfolio | Return | Risk-Free Rate | Std Dev | Sharpe Ratio |
|---|---|---|---|---|
| A | 12% | 2% | 20% | (12% - 2%) / 20% = 0.50 |
| B | 14% | 2% | 16% | (14% - 2%) / 16% = 0.75 |
A higher Sharpe ratio always means better risk-adjusted performance, so Portfolio B wins the comparison above.
Exam Tip: Gotchas
Sharpe uses standard deviation (total risk). If asked which ratio uses total risk, the answer is Sharpe. It is the appropriate measure when the portfolio is the investor's entire holding, because unsystematic risk has not been diversified away.
Summary of Risk Measures
| Measure | What It Measures | Risk Type | Key Formula Component |
|---|---|---|---|
| Standard Deviation | Dispersion of returns around the mean | Total risk | Variance of returns |
| Beta | Sensitivity to market movements | Systematic risk | Covariance with market |
| Alpha | Excess return vs. expected (CAPM) | Manager skill | Actual - Expected return |
| Sharpe Ratio | Return per unit of total risk | Total risk | Uses standard deviation |
| R-squared | % of returns explained by benchmark | Correlation | Correlation coefficient squared |
What Should You Check on Exam Day?
- Standard deviation captures total risk; beta captures systematic risk only. Know which question is being asked before you pick an answer.
- Mean is pulled toward outliers; median is not. A mean below the median signals a left-skewed distribution of unusually poor returns.
- Correlation of +1.0 gives zero diversification benefit; negative correlation gives the most.
- Alpha is excess return over the risk-adjusted expectation, not raw return.
- Sharpe ratio's denominator is standard deviation (total risk), so it fits an investor's entire portfolio.