In portfolio theory, risk is the uncertainty of future returns, measured by how widely outcomes are spread around the expected value.
E(R) = sum of probability x return.
| Scenario | Probability | Return | Probability x return | Deviation from mean | Probability x deviation squared |
|---|---|---|---|---|---|
| Boom | 0.30 | 30% | 9.0% | +20 | 120 |
| Normal | 0.50 | 10% | 5.0% | 0 | 0 |
| Bust | 0.20 | -20% | -4.0% | -30 | 180 |
| Total | 1.00 | 10.0% | 300 |
Expected return is 10%. Variance is 300 (in percent squared) and standard deviation is the square root, 17.3%.
Sample variance = sum of squared deviations from the mean / (n minus 1). Standard deviation of returns is called volatility. To annualize, multiply monthly volatility by the square root of 12 and daily volatility by the square root of 252: a 3% monthly volatility is about 10.4% a year.
If returns were normally distributed, about 68% of outcomes would fall within one standard deviation of the mean and about 95% within two. A stock with a 10% expected return and 20% volatility would land between minus 30% and +50% in about 95 years out of 100. Real returns have fatter tails than the normal curve: extreme losses happen more often than the model predicts.
Semi-deviation counts only returns below the mean or a target. Maximum drawdown is the largest peak-to-trough fall. Value at risk (VaR) is the loss not expected to be exceeded with a given confidence over a given period. These capture what investors fear: losses, not upside surprises.
Covariance measures whether two assets move together. Correlation rescales it to between minus 1 and +1:
+1 means the assets move in lockstep; 0 means no linear relationship; minus 1 means they move exactly opposite. If covariance is 0.006 and the volatilities are 20% and 10%, correlation = 0.006 / (0.20 x 0.10) = 0.30. Correlations are not stable; they tend to rise in market crises, exactly when diversification is needed most.
Investors dislike risk: between two investments with the same expected return, they choose the less volatile one. They accept more risk only for more expected return. The rate at which they trade the two differs from person to person.
Five monthly returns: 2%, -1%, 3%, 0% and 1%. The mean is 1%. Deviations are 1, -2, 2, -1 and 0; their squares sum to 10. Sample variance = 10 / 4 = 2.5, so monthly volatility is 1.58%, and annualized volatility is 1.58% x square root of 12 = 5.5%. In practice use at least three to five years of monthly data; five points are only for illustration.
A portfolio's value goes 100, 140, 91, 120. The largest peak-to-trough fall is from 140 to 91, a drawdown of 35%. To get back to 140 from 91 requires a gain of 140 / 91 minus 1 = 53.8%. Drawdown captures what investors live through in a way that volatility does not.
With several assets, risk is summarized in a covariance matrix: variances on the diagonal and covariances off it. For n assets there are n variances and n x (n minus 1) / 2 distinct covariances: 3 for three assets, 45 for ten, nearly 125,000 for 500 stocks. Estimating so many numbers reliably is hard, which is why practitioners use factor models (Module 6) to simplify the matrix.
Willingness to bear losses is psychological; capacity is financial (horizon, income stability, other assets, obligations). A young professional with a secure income may have high capacity but low willingness; a retiree who loves risk may have little capacity. An investment policy uses the lower of the two (Module 9).
Clients understand 'in a bad year, one in twenty, you could lose about $X' better than '15% standard deviation'. For a $200,000 portfolio with 10% volatility and a 6% expected return, a one-in-twenty bad year under a normal approximation is about 6% minus 1.645 x 10% = -10.5%, a loss of about $21,000; real tails are worse.
Using five years of monthly returns for two stocks and one bond fund, compute each one's annualized volatility and the three pairwise correlations. Find the maximum drawdown of each.
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