Portfolio Mathematics

How a portfolio's expected return and risk follow from its holdings: weights, covariance and correlation, the covariance matrix, diversification, covariance from a joint probability function, and Roy's safety-first rule for limiting the chance of falling below a minimum return.

1. 1. Portfolio expected return and varianceA portfolio's expected return is a plain weighted average of its holdings, but its risk is not, because risk also depends on how the holdings move together.Video · 6 minFree13 min
2. 2. Covariance and correlationCovariance tells you whether two returns move together; correlation rescales it to a unit-free number between −1 and +1 so you can judge how strongly.13 min
3. 3. The covariance matrix and many-asset portfoliosWith more than two assets, portfolio variance is the weighted sum of every entry in the covariance matrix, and the covariance entries quickly outnumber the variances.13 min
4. 4. Correlation and diversificationLower covariance between holdings lowers portfolio risk without lowering expected return; that free risk reduction is the diversification benefit.12 min
5. 5. Covariance from a joint probability functionWhen returns are forecast with scenarios, covariance is the probability-weighted sum of the cross-products of each asset's deviation from its expected return.14 min
6. 6. Shortfall risk and Roy's safety-first criterionIf the investor's main fear is ending below a minimum return, choose the portfolio with the highest safety-first ratio: the most standard deviations between the expected return and the threshold.13 min
7. 7. Shortfall probability, the Sharpe ratio and VaRUnder normality, the probability of missing the threshold is $N(-\text{SFRatio})$; the same idea links the safety-first ratio to the Sharpe ratio and underpins value at risk.12 min