Three assets return 6%, 9% and 12% with volatilities of 8%, 15% and 25%. Long only, at most 50% in any one, target volatility 12%. Write the optimisation problem and find approximate weights.

Maximise expected return subject to volatility of at most 12%, weights summing to one, each between 0 and 50%; the answer is about 32% A, 43% B and 25% C, returning 8.79%. Only the volatility target binds here. The 50% limit is slack at 12% and bites only below a 9.7% target, where A would pass half, or above 14.4%, where B would; above 17.5% the target cannot be reached at all.

Say it in words first: get as much expected return as possible without taking more than 12% of risk, using only the three assets, no shorting, and no more than half the money in any one. Then translate each phrase into symbols. An optimisation has three parts, and the interviewer marks each: the decision variables, the objective, and the constraints, with every practical rule from the mandate written as one line. It is the same as planning a week's meals on a budget: what you choose (the dishes), what you want most (taste), and the rules you cannot break (money, time, a doctor's limit on salt).

Build the covariance matrixThe table of how every pair of assets moves together. Each entry is the two volatilities multiplied by their correlation; the diagonal holds each variance. entry by entry: the A and B term is 0.08 x 0.15 x 0.2 = 0.0024, A and C is 0.006, and B and C is 0.01875. Then say what the problem is: a linear objective with a quadratic constraint, convex, so any local answer a solver finds is the global one. Mention the equivalent dual form too, minimising variance for a return target, because interviewers often ask which you would code.

Bracket the answer. Equal weights give a return of 9% at 12.7% volatility, above the ceiling, so you must tilt towards A. Half in A and half in B gives 7.5% at about 9.2%, under the ceiling with room to spare, so some C should come in. Trading off between those two points lands near 32% A, 43% B and 25% C, which a solver confirms: expected return 8.79% at exactly 12% volatility. Saying out loud that you are searching between a portfolio that breaks the risk limit and one that wastes it is what earns the credit.

Check each constraint against the answer. The volatility constraint is exactly met, so it binds. Every weight is strictly between 0 and 50%, so neither the long-only rule nor the position limit is active. The 50% limit matters only at the two ends: below a target of about 9.7%, where the safest portfolio would put more than half in A, and above about 14.4%, where B's weight reaches half the portfolio; from there C absorbs the extra risk, and once B and C are each at 50% the portfolio cannot exceed 17.5% volatility. So at 12% you could delete the position limit and get the same answer, which is worth saying because it tells the client what the mandate rule is really protecting against.

Name the weakness of the formulation itself. Mean-variance optimisers treat the expected returns as known, and small errors in them move the weights far more than errors in the risks do. A one point cut in C's expected return shifts weight back towards B noticeably. Practical fixes are to shrink the return estimates towards a common value, add a turnover or transaction cost term to the objective, and report how much the weights move when each input is nudged. The interviewer is listening for whether you trust the output of an optimiser more than its inputs deserve.