Portfolio Management
Portfolio management tests construction (diversification and the optimal risky portfolio), equilibrium pricing through CAPM, and performance measurement - including which ratio applies to which portfolio.
Construction
Two assets: σp² = w1²σ1² + w2²σ2² + 2w1w2ρσ1σ2. Adding a low-correlation asset reduces σp without reducing expected return. The efficient frontier is its upper half; combining it with the risk-free asset gives the CML, whose slope is the Sharpe ratio. Beta measures contribution to a diversified portfolio's risk: βp = Σ wiβi.
Performance measures
Sharpe = (Rp − Rf)/σp uses total risk and suits a standalone or undiversified portfolio. Treynor = (Rp − Rf)/βp uses systematic risk and suits a sub-portfolio within a larger fund. Jensen α = Rp − [Rf + βp(Rm − Rf)]. Information ratio = active return/active risk.
Worked example
Rp = 17%, Rf = 6%, σp = 18%, βp = 1.1, Rm = 13%. Sharpe = 11/18 = 0.61. Treynor = 11/1.1 = 10%. Jensen α = 17 − [6 + 1.1 × 7] = +3.3%. The positive alpha indicates the manager beat the CAPM benchmark; the Sharpe of 0.61 must be judged against the market's (13 − 6)/σm, not in isolation.
Traps
Using total risk in Treynor or beta in Sharpe; annualising returns by ×12 but risk by ×12 instead of ×√12; weights that do not sum to 1; treating a high raw return in a bull market as skill; forgetting that alpha depends on the chosen benchmark.