Case 030Investment and portfolio riskCore

An enhanced index fund with a 2% tracking error budget wants three stock overweights and a sector tilt. Compute the tracking error and check it against the budget.

MSCIMonterrey · 2013

The situation

Tulsivan Enhanced Index Fund tracks a large-cap index and is allowed a tracking error of 2% a year: the standard deviation of its return minus the index return. The manager proposes four active bets.

Overweight three stocks by 2 percentage points each; each stock has residual volatility of 30% a year, the part of its movement not explained by the index. Tilt 5 points into a sector whose return relative to the index has volatility of 6% a year. The overweights are funded by small underweights spread across the rest of the index. Treat the four bets as independent of one another.

Your task

What is the fund's tracking error, how much of the budget does it use, and what assumption is the answer most sensitive to?

Quick check

Roughly what tracking error do the four bets produce?

Worked solution

Try it on paper, then open one step at a time.

30-second answerThe answer to give first

Tracking error is about 1.08%, comfortably inside the 2% budget. Each stock bet contributes 2% times 30%, or 0.6%, and the sector bet 5% times 6%, or 0.3%. Because the bets are independent they add in squares, not in a line, so the total is the square root of 1.17, and the bets use about 29% of the variance budget. If the bets moved together the answer would be 2.1%, over budget.

Step 1What does each bet contribute on its own?

A bet's contribution is its size times the volatility of what it bets on. A 2 point overweight in a stock with 30% residual volatility adds 0.6% of tracking error; a 5 point sector tilt at 6% adds 0.3%. The stock bets are smaller in weight but three times as risky per bet, because single names move far more against the index than a whole sector does. The residual volatilityThe part of a stock return that the index does not explain, measured as a standard deviation. It is what an overweight actually bets on. is the right input, not the stock's total volatility, because the index part of the move is shared with the benchmark and cancels.

Step 2Why do independent bets add in squares?

Toss four coins and count heads minus tails: some cancel, and the total rarely reaches four. Independent bets behave the same way, some winning while others lose, so the spread of the total grows with the square root of the sum of squared spreads. Square each contribution, add the squares, and take the square root. Three stock bets give 0.36 each and the sector 0.09, a variance of 1.17 in squared percentage points.

The relationship
TE=∑i(wi σi)2=3×(0.02×0.30)2+(0.05×0.06)2=0.000117≈1.08%\text{TE} = \sqrt{\sum_i (w_i\,\sigma_i)^2} = \sqrt{3\times(0.02\times0.30)^2 + (0.05\times0.06)^2} = \sqrt{0.000117} \approx 1.08\%
w_ithe active weight of bet i, fund weight minus index weight
\sigma_ithe volatility of what bet i bets on, relative to the index
TEtracking error, the yearly standard deviation of fund return minus index return
What it says in wordsWith independent bets, tracking error is the square root of the sum of each bet's squared contribution.
Tracking error adds in squares: four bets use under a third of the budgetStock A overweight2 pts x 30% = 0.6%0.36Stock B overweight2 pts x 30% = 0.6%0.36Stock C overweight2 pts x 30% = 0.6%0.36Sector tilt5 pts x 6% = 0.3%0.09All four, independent1.17unused budgetbudget 4.00 = 2%²Tracking error = square root of 1.17 = 1.08%
Each stock overweight contributes 0.36 and the sector tilt 0.09 squared percentage points, a total of 1.17 against a budget of 4.00, which is a tracking error of 1.08% against the 2% limit.
Step 3How much room is left, and how should the manager use it?

Budgets are set in tracking error but spent in variance. The four bets use 1.17 of 4.00, about 29% of the budget, so every bet could be scaled up about 1.85 times before the fund reaches 2%. The squares also say something about how to spend it: many small independent bets are cheap, because doubling the number of bets raises tracking error by only about 1.4 times, while doubling the size of one bet doubles its contribution. That is why enhanced index funds tend to hold many modest tilts rather than a few large ones.

Step 4Which assumption could break the answer?

Independence. If the three stocks all come from the tilted sector, or all load on the same style such as small size or momentum, they move together and the straight sum becomes the better guide. At perfect co-movement the same bets give 2.1% tracking error, over the budget. A risk manager checks the fund's factor exposures and the correlation between the bets before accepting 1.08%, and says that the funding underweights add a little risk of their own, which this estimate leaves out.

The answer rests on the bets being independentIndependent bets (the case)1.08%Bets that move together2.10%2% budget
The same four bets give a tracking error of 1.08% if they are independent and 2.1% if they move together perfectly, so the verdict on the 2% budget depends on the correlation between them.

Where candidates lose it

Most candidates add the four contributions in a line and get 2.1%, then reject the proposal as over budget. That treats the bets as perfectly correlated, the opposite of what the question told you.

The second miss is using total stock volatility instead of residual volatility. A stock with 30% residual volatility may have 40% total volatility, but the index part of that move is also in the benchmark and does not create tracking error.

What the interviewer asks next

  • The three stocks are all in the tilted sector with a correlation of 0.5 among the bets. Recompute the tracking error.
  • How would you split a 2% budget between stock selection and sector bets?
  • Why might ex-post tracking error come in well above the 1.08% estimate?

Asked at MSCI, Financial Tools, Monterrey, 2013 (Wall Street Oasis): What's the tracking error formula? Why shouldn't we hire you?

Company names and figures are illustrative.