Hedge Funds puzzles, solved step by step
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005Without paper, what are 997 x 1,003 and 67 squared?Akuna Capitalchicago · 2024
Try it first
What is 997 x 1,003?
Show the worked solution
997 x 1,003 = 999,991 and 67 squared = 4,489. The first is a difference of squares: the numbers sit 3 either side of 1,000, so the product is 1,000,000 minus 9. The second uses the same identity the other way round: 67 squared is 64 x 70 plus 3 squared, which is 4,480 plus 9. Both take one line once you spot the round number nearby.
Why does multiplying around a round number work?
Take a square garden 10 metres a side and reshape it to 13 by 7. The fence is the same length, but the plot shrinks from 100 square metres to 91. It always shrinks by the square of how far you moved each side, here 3 squared, 9. Two numbers spaced equally around a midpoint multiply to the midpoint squared minus the gap squared. For 997 x 1,003 the midpoint is 1,000 and the gap is 3, so the answer is 1,000,000 minus 9.
Removing a b by b corner from an a by a square and standing the leftover strip on its end makes a rectangle a + b wide and a - b tall, which is why 997 x 1,003 is 1,000,000 - 9 = 999,991 and 67 squared is 64 x 70 + 9 = 4,489. The relationshipa the round midpoint, or the number being squared b the gap you choose to make a factor round What it says in wordsA product around a midpoint is the midpoint squared less the gap squared; run it backwards to square any number.How do you square a number like 67 in your head?
Push it to a round neighbour and repair the difference. Move 3 down to 64 and 3 up to 70, multiply those, then add back the 3 squared you took away: 64 x 70 = 4,480, plus 9 is 4,489. You choose the gap so one factor is round. The expansion route agrees: 67 is 70 minus 3, so 67 squared is 4,900 - 420 + 9, the same 4,489. Two methods landing on one number is your check.
Why would a fund ask arithmetic at all?
A trader checks prices, spreads and position sizes in their head all day, and a slip costs money before any spreadsheet catches it. Interviewers use speed on arithmetic like this as a proxy for how quickly you would catch a quote that does not add up. Say the identity as you use it, so that if you slip, the interviewer can see where and you can recover out loud.
Where candidates lose it
The trap is grinding 997 x 1,003 column by column under time pressure, dropping a carry and landing on 1,000,009 or 999,909. Two numbers either side of 1,000 is the whole question, and it is there to see whether you notice.
For 67 squared, candidates who remember 65 squared is 4,225 try to count up from there and lose track of the cross term. Pick the route that gives you one round multiplication, and say it out loud as you go.
What the interviewer asks next
- What is 48 x 52?
- What is 95 squared, and what is the fastest route?
- Estimate 1.03 to the tenth power in your head.
Asked at Akuna Capital, Hedge Fund, chicago, 2024 (Wall Street Oasis):
focusing on quick probability puzzles, mental math, and some data structure questions
030Your fund owns 1% of a company's shares. On a normal day 0.2% of the company's shares change hands, and your desk will not trade more than 20% of the day's volume. How many trading days do you need to exit the position?Multi-manager platformsLong-short equity funds
Try it first
Quick number:
Show the worked solution
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.
Selling 0.04% of the company each day, 20% of the daily 0.2% volume, the 1% stake falls in equal steps and is fully sold only after 25 trading days, with 0.80% still held after the first week. 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.
Where candidates lose it
The fast wrong answer is five days: 1% divided by 0.2%. It assumes you can be all the volume in the stock, which would crush the price. The participation cap is the whole point of the question.
The second loss is converting to shares and rupees before dividing. Everything is already a percentage of the same share count, so one division does it. Say the answer, then say why liquidity is a risk in its own right.
What the interviewer asks next
- Volume halves in a sell-off. How long now, and what would you do in the first week?
- The fund has a rule of no more than five days to liquidate. How big can the position be?
- How would you estimate the price impact of selling 20% of volume every day?
055Your average winning trade makes 1.5 times what your average losing trade loses. What hit rate do you need just to break even?Multi-manager platformsLong-short equity funds
Try it first
Winners are 1.5 times the size of losers. What hit rate breaks even?
Show the worked solution
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%.
The relationshipp the hit rate, the share of trades that win 1.5 the average win as a multiple of the average loss 1 - p the share of trades that lose What it says in wordsThe book breaks even when expected winnings per trade equal expected losses per trade.At a 30% hit rate expected wins of 0.45 units fall short of expected losses of 0.70; at 40% both are 0.60 and the book breaks even; at 45% wins of 0.675 beat losses of 0.55, a profit of 0.125 units a trade. 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.
Where candidates lose it
The instinct is to say 50% or more, because being right more often than wrong sounds like the definition of a good trader. The interviewer wants you to weigh each outcome by its size rather than count outcomes.
The other slip is inverting the ratio and answering 60%, which is the breakeven if losers were 1.5 times winners. Write the one equation, 1.5p = 1 - p, before you say a number.
What the interviewer asks next
- Your hit rate is 55% and your winners are 0.8 times your losers. Are you making money?
- Costs are 0.1 units a trade. What hit rate do you need now?
- Why might a manager's win-loss ratio fall as the fund grows?
