A manager runs a 3% tracking-error budget with equal active positions in five stocks, each with 30% idiosyncratic volatility and uncorrelated. How large can each active weight be?

Each active weight can be about 4.5%, giving 22.4% of total active exposure. Uncorrelated risks add in variance, so tracking error is w times 30% times the square root of 5, and setting that to 3% gives w of 4.47%. A linear split, 3% over five positions of 0.6% each, would allow only 2% weights and use just 1.34% of the budget. If the residuals were correlated at 0.2, the weight would fall to about 3.3%.

Imagine five friends each tossing a coin for Rs 100. Any one of them can lose Rs 100, but the five together almost never all lose at once, so the swing of the group's total is far smaller than five times Rs 100: it is Rs 100 times the square root of 5, about Rs 224. The same rule governs active positions whose residual returns are unrelated. The variance of the total is the sum of the variances, so the standard deviation grows with the square root of the number of positions, not with the number itself. That is the whole reason a diversified set of views fits inside a budget that a single view of the same size would blow through.

Candidates who divide 3% by five and then by 30% get 2% weights. That is not unsafe; it is wasteful. Five 2% positions produce only 1.34% of tracking error, so the manager runs at less than half the risk the mandate allows and, if the views are any good, earns less than half the active return the budget was meant to buy. The linear split is the right answer in exactly one case: when the five residuals move together, correlation 1, so that the five positions are really one bet five times over. Ask the interviewer which world you are in before you divide.

Real residuals usually share something, a sector or a style. With a pairwise correlation of 0.2 the variance picks up the cross terms: five own terms plus twenty cross terms each weighted 0.2, which is 5 + 4 = 9, so the multiplier is 3 instead of 2.24. At 0.2 correlation each weight falls to about 3.33%, because part of each position's risk is now the same risk counted five times. The table shows how the budget stretches with the count when the positions are independent: one view gets 10%, ten views get about 3.2% each, twenty get about 2.2%.

Close with the limits. The 30% idiosyncratic volatility is itself an estimate, usually from a risk model whose residuals are only approximately uncorrelated, and a position that looks independent in the model can share a hidden factor in a crisis. Desks therefore size a little inside the formula and check realised tracking error against the budget every month, because the budget is a ceiling on risk taken, not a target to be hit exactly.