Key Takeaways
- Mean (average) adds all values and divides by the count; median is the middle value when data is ordered
- Standard deviation measures total volatility/risk — how far returns deviate from the average
- Beta measures systematic (market) risk: β=1.0 matches market; β>1.0 is more volatile; β<1.0 is less volatile
- Alpha measures excess return: positive alpha = outperformance; negative alpha = underperformance
- Sharpe ratio = (Return - Risk-Free Rate) / Standard Deviation — higher is better for risk-adjusted returns
Descriptive Statistics and Risk Measures
Investment advisers use statistical measures to analyze investment returns, compare risk profiles, and evaluate portfolio performance. Understanding these measures is essential for the Series 66 exam.
Measures of Central Tendency
These statistics describe the "center" of a data set.
Mean (Average)
The mean is calculated by adding all values and dividing by the number of values.
Example: Returns of 10%, 15%, 5%, and -7% Mean = (10 + 15 + 5 + (-7)) / 4 = 23 / 4 = 5.75%
Median
The median is the middle value when data is arranged in order. It's less affected by extreme values (outliers) than the mean.
Example: Same returns ordered: -7%, 5%, 10%, 15%
- With an even number of values, average the two middle numbers
- Median = (5% + 10%) / 2 = 7.5%
Mode
The mode is the most frequently occurring value. If no value repeats, there is no mode. Mode is rarely used in investment analysis.
Range
The range is the difference between the highest and lowest values.
Example: Range = 15% - (-7%) = 22%
Measures of Dispersion (Risk Measures)
Standard Deviation
Standard deviation measures how far individual returns deviate from the average (mean) return. It quantifies total volatility.
| Standard Deviation | Risk Level |
|---|---|
| Low (e.g., 5%) | Lower volatility, more predictable |
| High (e.g., 25%) | Higher volatility, less predictable |
Key Point: Standard deviation measures total risk — both systematic (market) risk and unsystematic (company-specific) risk.
Variance
Variance is standard deviation squared. While mathematically important, the Series 66 focuses more on standard deviation.
Risk-Adjusted Performance Measures
Beta (β)
Beta measures systematic risk — how sensitive a security is to market movements compared to a benchmark (usually the S&P 500).
| Beta | Interpretation | Example |
|---|---|---|
| β = 1.0 | Moves exactly with the market | Index funds |
| β = 1.5 | 50% more volatile than market | Growth tech stocks |
| β = 0.7 | 30% less volatile than market | Utility stocks |
| β = 0 | Unrelated to market movements | Some hedge strategies |
| β < 0 | Moves opposite to market | Inverse ETFs, gold (sometimes) |
Example: If the market rises 10% and a stock has β = 1.5:
- Expected stock return = 10% × 1.5 = 15%
Beta and Diversification
Beta measures only systematic risk (market risk), which cannot be diversified away. Unsystematic risk (company-specific) can be reduced through diversification.
Alpha (α)
Alpha measures the excess return of an investment compared to what its beta would predict. It indicates whether a fund manager adds or destroys value.
Formula: Alpha = Actual Return - Expected Return (based on beta)
| Alpha | Interpretation |
|---|---|
| Positive | Outperformed expectations (manager added value) |
| Zero | Performed as expected |
| Negative | Underperformed expectations (manager destroyed value) |
Example: A fund with β = 1.2 should return 12% if the market returns 10% (10% × 1.2). If the fund actually returned 14%, alpha = 14% - 12% = +2% (outperformance).
Alpha in Passive vs. Active Funds
| Fund Type | Expected Alpha |
|---|---|
| Passively managed (index funds) | Near zero (designed to match benchmark) |
| Actively managed | Positive alpha is the goal |
Sharpe Ratio
The Sharpe ratio measures risk-adjusted return — how much excess return an investor receives per unit of risk taken.
Formula: Sharpe Ratio = (Portfolio Return - Risk-Free Rate) / Standard Deviation
The risk-free rate is typically the 91-day (3-month) Treasury bill rate.
| Sharpe Ratio | Interpretation |
|---|---|
| Higher | Better risk-adjusted performance (more "bang for the buck") |
| Lower | Worse risk-adjusted performance |
Example:
- Portfolio return: 12%
- Risk-free rate: 3%
- Standard deviation: 15%
- Sharpe ratio = (12% - 3%) / 15% = 0.60
Comparing Investments Using Sharpe Ratio
| Investment | Return | Std Dev | Sharpe Ratio | Verdict |
|---|---|---|---|---|
| Fund A | 15% | 20% | 0.60 | More efficient |
| Fund B | 18% | 30% | 0.50 | Less efficient |
Fund A is more efficient despite lower returns because it achieved better risk-adjusted performance.
Summary: Beta vs. Standard Deviation
| Measure | Risk Type | Diversifiable? |
|---|---|---|
| Beta | Systematic (market) risk | No |
| Standard Deviation | Total risk (systematic + unsystematic) | Partially |
Exam Tip: Beta measures MARKET risk only (systematic). Standard deviation measures TOTAL risk. The Sharpe ratio measures risk-adjusted returns — higher is better. You likely won't calculate Sharpe ratio on the exam, but understand the concept and components.
A stock with a beta of 1.5 would be expected to:
Which measure indicates that a portfolio manager has outperformed expectations based on the level of risk taken?
Standard deviation measures which type of risk?
The Sharpe ratio uses which of the following in its denominator?