Case 078Performance evaluation and manager selectionWarm up

An active large cap fund charges 1.8% and is expected to deliver 2.0% of gross alpha with 5% tracking error; an index fund charges 0.2%. What is the chance the active fund beats the index fund in one year, and over ten?

The situation

A client is choosing between Mayurakshi Large Cap Fund, an actively managed fund with a total expense ratio of 1.8%, and a large cap index fund charging 0.2%. Your manager research team is unusually positive on Mayurakshi: it expects the manager to beat the index by 2.0% a year before fees.

The fund's returns wander around the index's with a tracking errorThe standard deviation of the gap between a fund return and its benchmark return, a measure of how far the fund strays from the index in a typical year. of 5% a year. Assume yearly outcomes are independent and roughly normal.

Your task

What is the probability Mayurakshi beats the index fund in any one year, and over a ten-year holding period? What does that tell the client?

Quick check

Your team is right about the 2% gross alpha. Roughly how often does Mayurakshi beat the index fund in a single year?

Worked solution

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

30-second answerThe answer to give first

About 53% in any one year and about 60% over ten years. The fund's 2.0% gross edge shrinks to 0.4% after its 1.8% fee against the index fund's 0.2%. With 5% tracking error, one year is close to a coin toss; ten years narrows the spread of the average edge to 1.58% but still leaves a 40% chance of trailing. Even a genuinely skilled manager rarely looks skilled after these fees.

Step 1What is the fund's real edge over the index fund?

Compare like with like: the client's alternative is not the index, it is an index fund that also charges. Gross alpha of 2.0% less the 1.8% fee plus the 0.2% the index fund would have cost leaves an expected edge of 0.4% a year. It is like a taxi driver who knows a shortcut worth ten minutes but charges nine minutes' worth extra for it: you are still ahead, barely, and only on average.

Step 2How likely is it to show up in one year?

The edge of 0.4% sits inside a yearly spread of 5%. Divide one by the other and you get 0.08, the number of standard deviations the centre sits above zero. That gives about a 53% chance of beating the index fund in a single year, close to a coin toss even though the manager is skilled. Before fees the same manager would win about 66% of years; the fee does most of the damage.

Mayurakshi's expected edge over the index fund is only 0.4% a year, so 53% of the one-year curve lies above zero; averaging over ten years narrows the curve and lifts the share above zero to about 60%.

Step 3Does holding for ten years fix it?

It helps, slowly. Averaging ten independent years shrinks the spread of the average edge by the square root of ten, from 5% to 1.58%. The centre stays at 0.4%, so the chance of being ahead over ten years rises only to about 60%. To be 90% sure of coming out ahead, the client would need about 257 years. The luck in active returns washes out at the square root of time, and the fee is charged every single year.

The relationship

P(\text{win over } n \text{ years}) = \Phi\left(\frac{\alpha - f + f_I}{\sigma_{TE}/\sqrt{n}}\right) = \Phi\left(\frac{0.4}{5/\sqrt{10}}\right) \approx 0.60

alpha	expected return above the index before fees, 2.0%
f, f_I	the active fund's fee, 1.8%, and the index fund's fee, 0.2%
sigma_TE	tracking error, 5% a year
n	years held; the spread of the average falls with the square root of n

What it says in wordsThe chance of coming out ahead is the net edge divided by how noisy the average edge is over the holding period.

Step 4What would change the answer?

The fee, far more than the skill. Cut Mayurakshi's fee to 0.8% and the expected edge becomes 1.4%: the one-year chance rises to about 61% and the ten-year chance to about 81%. Tell the client the honest version: the fund may be well run, but at this price the odds of being better off are only modestly above even, and the 2% gross alpha is itself an estimate that could be wrong.

Where candidates lose it

Candidates compare the 2% alpha with zero and say the fund wins most years. They forget two things at once: the fee comes off the alpha, and the alternative also has a fee. The net edge is 0.4%, not 2%.

The second loss is assuming time fixes everything. Averaging reduces noise only with the square root of years, so a small edge needs decades to show itself reliably, while the fee is certain every year.

What the interviewer asks next

What gross alpha would give a 75% chance of beating the index fund over ten years?
How would you test whether the research team's 2% estimate is believable?
Why might a fund with lower tracking error be easier to hold even with the same edge?

Company names and figures are illustrative.