BFF5270 Chap.6 Portfolio Optimisation and Attribution
Portfolio Optimisation and Attribution
Portfolio optimisation maps combinations of risky assets using expected return, variance, correlation and constraints. This chapter distinguishes the global minimum-variance and tangency portfolios, applies two-fund interpolation and closes the investment process with allocation and selection attribution. Every output is treated as conditional on its inputs, not as a self-justifying recommendation.
Stress scenarios show when a precise base-case weight is too fragile to carry into a governed portfolio decision. Robustness matters more than reporting unstable weights with false precision.
What this chapter covers
- 01
Covariance inputs
- 02
Minimum variance
- 03
Tangency portfolio
- 04
Efficient frontier
- 05
Two-fund theorem
- 06
Target return
- 07
Allocation effect
- 08
Selection effect
Worked example · free
Interpolate a target return with two frontier funds
- 1Set 9.6 = 11.2w + 7.2 times 1 minus w.
- 1Solve 2.4 = 4.0w, giving w = 0.60 in T.
- 1Calculate equity: 0.60 times 70% plus 0.40 times 30% = 54%.
- 1Calculate property: 0.60 times 20% plus 0.40 times 25% = 22%.
- 1Calculate bonds: 0.60 times 10% plus 0.40 times 45% = 24%.
- 1Confirm the asset weights total 100%.
- 1Confirm the reconstructed expected return equals 9.6%.
Key terms
- Efficient Frontier
- The efficient frontier contains portfolios with the highest expected return for each feasible risk level.
- Minimum Variance
- The global minimum-variance portfolio is the risky combination with the lowest feasible variance.
- Two-Fund Theorem
- The two-fund theorem constructs frontier portfolios by mixing two known frontier portfolios.
- Allocation Effect
- The allocation effect captures performance associated with active asset-class weights.
- Selection Effect
- The selection effect captures performance associated with securities outperforming or underperforming their asset-class index.
Portfolio Optimisation and Attribution FAQ
What is the difference between GMV and tangency?
GMV is the risky portfolio with minimum variance. Tangency is the risky portfolio where the line from the risk-free rate has maximum slope, equivalent to the maximum Sharpe ratio for the inputs. They answer different optimisation questions and do not automatically represent the investor's final allocation.
Why can optimiser weights be unstable?
Optimisation is sensitive to expected returns, variances and correlations, especially when alternatives have similar estimates. Small input changes can produce large weight changes. Analysts should inspect sensitivity, concentration, turnover and mandate ranges rather than treating a precise output as certain or automatically implementable.
Why separate allocation from selection?
A manager can add value through asset-class weights while losing value through securities, or the reverse. Separating the effects identifies the decision source and makes governance useful. Reporting only total active return hides what could be repeated, corrected or constrained in the next period.
Exam move
Sketch the frontier and label GMV, tangency and the risk-free line without notes. Solve a two-fund target-return equation, apply the same mixing weight to every asset and perform three checks: fund weights, asset weights and target return. For attribution, begin with benchmark weights and returns, identify active weights, separate allocation from selection and reconcile the bridge.
Redraw the efficient frontier with standard deviation on the horizontal axis and expected return on the vertical axis. Put GMV at the leftmost feasible risky point, then draw the line from the risk-free rate that touches the frontier at tangency. Explain aloud why the highest-return point is not automatically tangency and why the mathematically efficient set does not decide suitability.
Build one sensitivity table that changes expected return, volatility or correlation and records how optimiser weights respond. Large weight movement from a small input change should lead to robustness controls, concentration limits or wider ranges, not false confidence in the original decimals. For two-fund problems, name which unknown weight the equation solves.
Substitute the result back into the return equation before applying it to asset weights. Keep precision until the final line, confirm each asset weight and confirm the full portfolio reaches the target. For attribution, compute benchmark and managed returns on a consistent basis, derive active weights and classify the contribution of asset allocation separately from security selection.
Use an unexplained residual as a diagnostic: inspect rounding, interaction treatment and definitions rather than forcing it into a convenient category. Finish with a governance sentence identifying which manager decision should be repeated, investigated or constrained. That final step connects mathematical construction to the reporting purpose of the unit.
Run a stress scenario after the base frontier: reduce risky expected returns, raise volatilities and increase correlations together. Compare which weights remain stable and which collapse into concentration. Report the decision range rather than one false-precision point.
Finally, trace a disappointing active return backward through attribution, implementation and the assumption register so forecast error is not confused with poor security selection. This backward audit completes the full funds-management process.
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