Statistics: Risk, Return and Inference
What the exam tests
This domain tests the quantitative toolkit behind portfolio and market analysis: central tendency and dispersion, covariance and correlation, beta and regression, distributional assumptions, and how to annualise the numbers correctly.
Core rules
- Mean, median and mode describe location; variance and standard deviation describe dispersion around the mean. For returns, the geometric mean measures realised compound growth and is always at or below the arithmetic mean.
- Covariance measures joint movement in original units; correlation standardises it to the range -1 to +1. Diversification benefit depends on correlation, not on individual volatility.
- Portfolio variance for two assets: w1 squared x sd1 squared + w2 squared x sd2 squared + 2 x w1 x w2 x sd1 x sd2 x correlation.
- Beta measures sensitivity to the market: beta = covariance(asset, market) / variance(market). It captures systematic risk only; idiosyncratic risk is diversified away and is not rewarded.
- Annualise a daily standard deviation by multiplying by the square root of the number of trading days, typically 252.
Worked example: two-asset portfolio
Asset A: expected return 12%, standard deviation 20%, weight 60%. Asset B: expected return 6%, standard deviation 10%, weight 40%. Correlation 0.3.
- Expected return = 0.6 x 12 + 0.4 x 6 = 9.6%
- Variance = 0.36 x 400 + 0.16 x 100 + 2 x 0.6 x 0.4 x 20 x 10 x 0.3
- = 144 + 16 + 28.8 = 188.8
- Standard deviation = square root of 188.8 = 13.74%
The portfolio return sits between the two assets, but risk is below the 16% weighted average of volatilities because correlation is below one.
Traps that cost marks
- Dividing by n instead of n-1 when a sample standard deviation is asked for.
- Annualising volatility by multiplying by 252 instead of by the square root of 252.
- Quoting beta as if it measured total risk.
- Treating correlation as causation, or assuming a correlation of zero means independence in a non-normal distribution.