Hedge Funds puzzles, solved step by step

About 25 trading days, roughly five weeks. The market trades 0.2% of the shares a day and you take at most a fifth of that, so you can sell 0.04% of the company a day. A 1% stake divided by 0.04% a day is 25 days. In the desk's language the position is five days of volume, and exiting it without moving the price takes a month.

What is the one division that answers it?

Emptying a water tank through a tap that you are only allowed to open a fifth of the way: how long it takes is the tank size divided by the flow you actually use. Days to exit equals position size divided by your daily selling capacity, and capacity is market volume times your participation cap. Here that is 1% over (0.2% x 20%), or 1% over 0.04%: 25 days. Keeping everything in percent of shares means you never need the share count or the price.

Why does a portfolio manager care about this number?

Because the price can move a long way in 25 days. A position you cannot exit quickly carries more risk than its daily volatility suggests: if the stock falls 2% a day for a week, you have sold only a fifth of it. That is why many desks cap a position at a set number of days of volume, often a few days, and why liquidity sits beside volatility in position sizing. The measure has a name, days to liquidatePosition size divided by the volume you can realistically trade in a day; a common liquidity limit on hedge fund books., and interviewers like hearing it.

Say the limitations. Volume is not steady; it dries up exactly when you most want to sell, and in a sell-off everyone is trying to do the same thing. A 20% participation rate also moves the price against you, so the real exit costs more than the screen price. A stronger answer adds that you would stress the calculation with half the normal volume, which gives 50 days.

40%. Measure everything in units of the average loss. Each trade wins 1.5 units with probability p and loses 1 unit otherwise, so the expected result per trade is 1.5p minus (1 - p). Setting that to zero gives p = 1/2.5 = 40%. At a 45% hit rate the book makes 0.125 units a trade, so a trader who is wrong more often than right can still run a good business.

Why is a hit rate below 50% not a problem on its own?

A street vendor selling umbrellas can stand idle most days and still do well, because the rainy days pay for the dry ones. What decides whether a trading book makes money is the hit rate times the size of the wins against the miss rate times the size of the losses, not the hit rate alone. A win-loss ratio of 1.5 moves the breakeven from 50% down to 40%.

What does the general rule look like?

For a win-loss ratio R the breakeven hit rate is 1 over (1 + R). At R = 1 it is 50%, at R = 1.5 it is 40%, at R = 2 it is 33% and at R = 3 it is 25%. This is why portfolio managers are reviewed on both numbers together. A falling hit rate is fine if the winners are running further, and a rising hit rate is a warning sign if it comes from cutting winners early and letting losers run.

What would you add before calling 45% a good business?

Costs and sample size. Commission, slippage and financing come off every trade, winners and losers alike, so they raise the breakeven. If costs run at 0.05 units a trade, the book needs 1.5p - (1 - p) = 0.05, a hit rate of 42%. And a ratio of 1.5 measured over twenty trades is noisy; say you would want a longer record before trusting either number.