1.3 Descriptive Statistics and Risk Measures
Investment advisers evaluate client portfolios using statistical metrics to quantify historical dispersion, benchmark sensitivity, active manager value-add, and asset co-movements.
Key NASAA Exam Takeaways
- Standard Deviation measures total volatility (both systematic and unsystematic risk).
- Beta measures systematic (non-diversifiable) market risk relative to a benchmark index (S&P 500 = 1.0).
- Alpha measures risk-adjusted excess return generated by an active portfolio manager relative to CAPM expectation.
- Correlation coefficient (r) ranges from -1.0 to +1.0; a correlation of -1.0 offers maximum diversification benefit.
Normal Distribution Rule (Empirical Rule)
In a bell-shaped distribution: 68% of observations fall within +/- 1 standard deviation, 95% fall within +/- 2 standard deviations, and 99.7% fall within +/- 3 standard deviations.
Alpha Formula
Alpha = Realized Return - Expected Return [Risk-Free Rate + Beta * (Market Return - Risk-Free Rate)]. A positive alpha reflects manager skill.
| Metric | Formula / Range | Measures | Application |
|---|---|---|---|
| Beta | Covariance(Asset, Market) / Var(Market) | Systematic / Market Risk | Compare volatility against S&P 500 (Beta = 1.0) |
| Standard Deviation | Square root of Variance | Total Risk (Systematic + Unsystematic) | Sharpe Ratio denominator |
| Sharpe Ratio | (Return - Risk-Free Rate) / Std Dev | Excess return per unit of total risk | Best for non-diversified portfolios |
| Treynor Ratio | (Return - Risk-Free Rate) / Beta | Excess return per unit of systematic risk | Best for well-diversified portfolios |
| Alpha | Actual Return - CAPM Expected Return | Manager performance value-add | > 0 indicates market outperformance |
Interactive Knowledge Checkpoint • Section 1.3
An equity portfolio has a beta of 1.2. If the broader market increases by 10%, what is the expected return of the portfolio according to beta alone?