Desk Puzzle No. 4: Five Years of Genius

By EC Assets · Published 29 October 2026

Puzzle No. 4 is below; the solution to Desk Puzzle No. 3 follows it.

Puzzle No. 4: Five Years of Genius

You are screening managers. There are 1,024 of them, and to keep it clean, assume that in any given year each one either beats his benchmark or misses it, with no skill whatsoever - a coin flip, independent across years and managers.

The questions

1. After five years, how many of the 1,024 have beaten their benchmark in every single year? Not "how many would you expect if they were good" - how many pure coin flippers survive a five-year perfect record by luck alone?

2. Now allow for real skill. Suppose one manager in ten genuinely has skill, and a skilled manager beats his benchmark with probability 0.65 each year, while the other nine in ten are the coin flippers above. You find a manager with five perfect years. What is the probability that he is actually skilled? Guess before computing: is it above or below one in two?

3. You are unsatisfied by the answer and want to be 90 percent sure. How many consecutive winning years would you have to see?

Ground rules

Independence throughout, and no survivorship in the sample - you can see all 1,024, including the ones who did badly. Question 2 is one line of Bayes' theorem and needs no tables. Question 3 needs a logarithm, and the answer is an integer that will annoy you.

Fair hints

Alpha, Information Ratio and Sharpe Ratio define the quantities a track record is supposed to demonstrate. None of them tell you how long a record must be before it demonstrates anything - that is exactly what this puzzle asks, and what a five-year track record is almost never long enough to do.

The solution will be published here in next month's note, alongside Puzzle No. 5.

Solution: The Manager's Half

1. What you actually earn

The management fee comes off the gross return, and the performance fee takes a fifth of the rest:

net = (10% − 2%) × 0.8 = 6.4 percent per year

Over twenty years:

• Gross, with no fees at all: 100 × 1.10²⁰ = 672.75
• Net of 2 and 20: 100 × 1.064²⁰ = 345.81

Note in passing that 2 and 20 on a 10 percent gross return is arithmetically identical to a flat fee of 3.6 percentage points. The headline "two percent" describes a little over half of what you pay.

2. The split

Your profit is 245.81. The total profit the strategy generated was 572.75. The manager therefore took

(572.75 − 245.81) / 572.75 = 57.1 percent

The manager took more of the profit than the person whose capital was at risk, on a fund that never had a losing year, without ever putting a penny of his own money in front of the loss.

3. And it does not get better

This is the part worth keeping.

The share is a U. It is worst for weak funds, which is what everyone expects, and it is equally bad for brilliant ones, which almost nobody expects. It never falls below about 57 percent, and the best split available to an investor sits at an unremarkable ten percent gross.

The two arms of the U have different causes. On the left, the flat two percent is being charged against a small return and eats it proportionally. On the right, the twenty percent performance fee is being levied on a large return every year, and the fee that is taken out in year three is not there to compound for the following seventeen. The fee does not just cost you the fee - it costs you the growth of the fee, which is why the manager's share climbs with both the rate and the horizon.

There is no gross return at which the split becomes fair. That is not an accident of these particular numbers; it is what an annually crystallised performance fee does to a compounding series.

Further reading: Hedge Fund Fees, Fee Drag, Total Return vs Price Return.