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  1. 060A fund has a tracking error budget of 4% a year and runs 20 active positions, each carrying the same amount of active risk, with the positions uncorrelated. How much active risk can each position carry?Risk, volatility and drawdownHardRisk and complianceIndian AMCs

    Try it first

    How much active risk can each position carry?

    Show the worked solution

    About 0.89% each. Uncorrelated risks add in squares. Twenty positions of active risk s give a total variance of 20 s squared, which must equal 4 squared, 16. So s squared is 0.8 and s is 0.89%. The straight split of 4% over 20, 0.20%, would use under a quarter of the budget, because it assumes every position fails at the same time.

    Why can each position carry more than a twentieth of the budget?

    Think of twenty friends each guessing the weight of a cake. Each guess is off by about 100 grams, but in random directions, so the errors partly cancel and the total of their errors is nowhere near 2 kilograms. Independent errors grow with the square root of how many there are, not in proportion, because some push up while others push down. Tracking errorThe standard deviation of the gap between a fund return and its benchmark return, usually quoted per year. works the same way. Twenty uncorrelated bets of equal size s give a total of s x sqrt(20), about 4.47 s.

    Uncorrelated risks add in squares, not in straight lines0.89%0.89%0.89%0.89%0.89%0.89%0.89%0.89%0.89%0.89%0.89%0.89%0.89%0.89%0.89%0.89%0.89%0.89%0.89%0.89%20 positions, 0.89% active risk eachAdd the squares20 x s² = 4² = 16s² = 16 / 20 = 0.80s = 0.89%Tracking error4%the budgetStraight-line split: 4% / 20 = 0.20% eachUses only sqrt(20 x 0.20²) = 0.89% of the budget: about 22% of the risk you were allowed to take
    Twenty uncorrelated positions of 0.89% active risk each have squares that add to 16, whose square root is the 4% budget, while the straight-line split of 0.20% each would use only about 22% of the risk allowed.

    How do you solve it, and what happens if the bets are not independent?

    Set the total equal to the budget and solve. 4 = s x sqrt(20), so s = 4 / 4.47 = 0.89%: each position may carry almost four and a half times the naive answer. The catch is the word uncorrelated. If every pair of bets has a correlation of 0.2, the variance picks up 20 x 19 cross terms, each worth 0.2 s squared. The total variance becomes 96 s squared and each position can carry only 0.41%. Correlated bets behave more like one large bet.

    The relationship
    σTE2=Ns2  ⇒  s=σTEN=4%20≈0.89%\sigma_{TE}^2 = N s^2 \;\Rightarrow\; s = \frac{\sigma_{TE}}{\sqrt{N}} = \frac{4\%}{\sqrt{20}} \approx 0.89\%
    sigma TEthe tracking error budget, 4% a year
    Nthe number of independent active positions, 20
    sthe active risk of each position
    What it says in wordsWith independent bets the budget is shared out in variance, so each bet's risk is the budget divided by the square root of the count.

    This is why risk teams care about the true number of independent bets more than the number of line items. Twenty stocks that are all quietly a bet on falling interest rates are close to one position, and they would breach the budget at a fraction of the size the arithmetic above allows. The limit: tracking error is a one-number summary of normal-times behaviour, and correlations between bets tend to rise in a sell-off, exactly when the budget matters.

    Where candidates lose it

    Almost everyone says 0.20%. It assumes risks add like rupees, which happens only when every position moves together. The interviewer wants to hear the word variance before any number.

    The second trap is answering 0.89% and stopping. Say that it rests on zero correlation, and show how a modest correlation of 0.2 shrinks the allowance to 0.41%.

    What the interviewer asks next

    • The fund adds 20 more uncorrelated positions. What can each one carry now?
    • One position is twice the size of the others in risk terms. How does that change the allowance for the rest?
    • Why do correlations between active bets tend to rise in a market sell-off?