MATH2070 Chap.9 Portfolio Risk and Efficient Frontiers
Portfolio Risk and Efficient Frontiers
Portfolio analysis turns asset weights, expected returns, variances and covariances into a constrained optimisation problem. Expected portfolio return is linear in the weights, while variance is a quadratic form that captures both stand-alone volatility and co-movement. Diversification depends on covariance, not merely on holding many assets.
The feasible set records all attainable risk-return pairs, and its upper-left boundary is the efficient frontier. Minimum-variance portfolios provide the key benchmark for comparing feasible allocations. The budget equation does not itself prohibit short positions, so sign and position bounds must be stated independently.
Correlation affects the shape of the opportunity set by changing covariance, with lower co-movement typically bending attainable combinations toward lower risk. Efficiency is always relative to the permitted feasible set: a portfolio is dominated only when another allowed choice offers at least as much return with no more risk. Investor preference then selects among efficient alternatives rather than redefining the frontier.
What this chapter covers
- 01
Portfolio weights and budget constraints
- 02
Expected portfolio return
- 03
Variance and covariance
- 04
Diversification effects
- 05
Minimum-variance portfolios
- 06
Feasible sets
- 07
Efficient frontiers
Worked example · free
Compare two feasible portfolios
- 1Verify that each pair of weights satisfies the budget constraint.
- 1Calculate each portfolio's expected return from the weighted asset returns.
- 2Calculate both variances including the covariance cross-term.
- 1Compare return and risk and state whether dominance holds.
Key terms
- Portfolio weight
- The fraction of total wealth allocated to an asset.
- Covariance
- A measure of how two asset returns move together around their means.
- Feasible set
- All risk-return combinations attainable under the portfolio constraints.
- Efficient frontier
- The feasible boundary offering maximum expected return for a given risk level.
- Long-only constraint
- A restriction requiring every portfolio weight to be non-negative. Together with the budget equation, it confines allocations to combinations without short sales and can remove portions of an unrestricted efficient frontier.
- Dominated portfolio
- A feasible allocation for which another permitted portfolio offers at least as much expected return with no more risk and one strict improvement. A dominated point cannot belong to the relevant efficient frontier. The comparison must use portfolios allowed by the same weight and short-sale restrictions.
Portfolio Risk and Efficient Frontiers FAQ
Why is portfolio variance not a weighted average of asset variances?
Returns move together, so covariance terms also affect total risk. For two assets, the variance includes a cross-term involving both weights and their covariance, which can reduce or increase risk.
What does diversification accomplish?
Diversification combines assets whose returns are not perfectly positively correlated. The resulting covariance structure can make portfolio risk lower than a simple average of stand-alone risks, although diversification cannot automatically remove every source of risk.
How do I recognise an efficient portfolio?
A portfolio is efficient when no feasible alternative offers a higher expected return at the same or lower risk. Graphically, efficient portfolios occupy the relevant upper-left boundary of the feasible risk-return set.
Exam move
Start with the constraint set rather than the variance formula. Record the weight vector, budget equation and any long-only, short-sale or position bounds. Check whether the weights supplied or calculated actually sum to one. Translate expected returns and standard deviations into a consistent decimal or percentage-point convention before constructing covariances.
A negative weight is meaningful when short selling is allowed and an error when the stated bounds prohibit it. Write portfolio return and risk in different forms. Expected return is linear in weights, while variance is a quadratic form. For two assets, expand both squared terms and the doubled covariance cross-term before inserting numbers. If correlation is given, convert it to covariance using both standard deviations.
Carry variance without premature rounding, take one square root at the end and compare the resulting standard deviation with the component risks for a reasonableness check. Connect calculations to the opportunity-set diagram. Vary a two-asset weight across several values, plot expected return against standard deviation and identify the minimum-variance point.
Mark the lower branch as inefficient only after finding a feasible alternative with the same risk and higher return. Explore correlations of one, zero and minus one to see how covariance changes curvature and diversification. Redraw the set under long-only restrictions so that efficiency is always judged relative to the permitted feasible set.
For optimisation questions, state whether return is fixed while variance is minimised or risk is fixed while return is maximised. Use the budget and target-return constraints alongside the quadratic objective, then interpret multipliers or candidate weights within their bounds. Keep an error ledger for weights, return units, covariance conversion, the factor of two and frontier claims.
A final answer should report allocation, expected return, standard deviation and whether the point is efficient under the actual restrictions, not merely present a vector of decimals.
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