3.4 Portfolio Mathematics and Safety-First

Portfolio Mathematics and Safety-First

Portfolio math starts with a simple insight: investors hold combinations of assets, so the risk and return of the whole portfolio matter more than the stand-alone properties of each asset. A stock with high volatility can still improve a portfolio if its returns offset other holdings at the right times.

Portfolio expected return is a weighted average: E(Rp) = w1E(R1) + w2E(R2) + ... + wnE(Rn). Weights normally sum to 1. A short position has a negative weight, and leverage can make total long weights exceed 1. For Level I, read weights carefully and keep signs intact.

Portfolio variance is more complex because asset returns move together. For two assets, Var(Rp) = w1^2 sigma1^2 + w2^2 sigma2^2 + 2w1w2 Cov12. Covariance measures whether two assets tend to move together. Positive covariance increases portfolio variance; negative covariance reduces it.

Correlation standardizes covariance: rho12 = Cov12 / (sigma1 sigma2). Correlation ranges from -1 to +1. A correlation of +1 means perfect co-movement, so diversification benefit is limited. A correlation below +1 allows diversification. A correlation of -1 can eliminate risk for a specific mix of two risky assets.

The key exam idea is that portfolio standard deviation is usually less than the weighted average of individual standard deviations when correlation is below +1. Expected return is linear in weights. Risk is not linear because covariance matters. This is why combining imperfectly correlated assets can improve the risk-return tradeoff.

Covariance can be estimated from scenarios. First calculate each asset's deviation from its expected return in each state. Multiply paired deviations, weight by state probability, and sum. A positive covariance means above-average return for one asset tends to occur with above-average return for the other.

Safety-first analysis focuses on the chance of failing to meet a minimum acceptable return, RL. Roy's safety-first ratio is SFRatio = [E(Rp) - RL] / sigma_p. If returns are normally distributed, the portfolio with the highest safety-first ratio has the lowest probability of falling below the threshold.

Shortfall probability can be estimated with a z-score: z = (RL - E(Rp)) / sigma_p. If expected return is 10%, standard deviation is 8%, and the threshold is 2%, then z = -1.0. Under normality, the shortfall probability is the probability of a standard normal variable below -1.0.

Safety-first is useful when avoiding a target breach is more important than maximizing average return. Examples include spending needs, liability payments, or capital preservation. It does not replace full portfolio analysis, because it usually assumes normality and summarizes risk through standard deviation.

For the exam, build the calculation in layers. Compute expected returns first. Then compute covariance or correlation as needed. Then compute portfolio variance and standard deviation. Finally apply the safety-first or shortfall formula. This order reduces formula mixing.