Portfolio MathematicsLocked: included in All Access

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 minLocked: included in All Access13 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.Locked: included in All Access13 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.Locked: included in All Access13 min
4. 4. Correlation and diversificationLower covariance between holdings lowers portfolio risk without lowering expected return; that free risk reduction is the diversification benefit.Locked: included in All Access12 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.Locked: included in All Access14 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.Locked: included in All Access13 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.Locked: included in All Access12 min