Hedge Funds puzzles, solved step by step

1,225 correlations and 1,275 parameters in all. A covariance matrix is symmetric, so only the cells above the diagonal carry new information: one per pair of stocks, 50 x 49 / 2 = 1,225. The diagonal holds the 50 variances. The count grows with the square of the number of names, which is why large books estimate risk through a handful of factors instead.

Why do you count pairs rather than cells?

In a class of 50, how many handshakes happen if everyone shakes everyone else's hand once? Each person shakes 49 hands, but every handshake has been counted twice, once from each side, so it is 50 x 49 / 2 = 1,225. A correlation is a handshake: it belongs to a pair, and the pair A and B is the same pair as B and A. The diagonal is each stock paired with itself. Its correlation is 1 and needs no estimating, but the diagonal of the covariance matrix holds each stock's variance, which does.

Why does the count become a problem for a big book?

Because it grows with the square of the names. Ten times as many stocks needs about a hundred times as many correlations: 500 names need 124,750 of them plus 500 variances, 125,250 parameters. A year of daily returns on 500 names is 250 x 500 = 125,000 numbers, fewer than the parameters being estimated. With fewer days than stocks the sample matrix is singular: some combinations of positions appear to carry zero risk, and an optimiser will pile into exactly those.

What does a factor model buy you?

A factor model says each stock's return is driven by a few shared drivers, such as the market, its sector and its size, plus noise of its own. With 5 factors, 50 stocks need 250 loadings, 50 specific variances and 15 factor covariances: 315 numbers instead of 1,275. At 500 names it is 3,015 instead of 125,250. The limitation is worth saying: any risk the factors do not name is assumed independent across stocks, and in a crowded unwind that assumption is the first to break.

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.

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.

Light rope one at both ends and rope two at one end at the same moment; when rope one burns out, light rope two's other end, and it burns out at 45 minutes. Two flames always meet after burning 60 minutes of rope between them, so rope one takes 30 minutes however uneven it is. Rope two then has 30 minutes left, which two flames finish in 15.

Why does lighting both ends halve the time on an uneven rope?

Picture two people eating a long, uneven sandwich from opposite ends, each chewing through whatever is at their end. However the filling is spread, they meet once the whole sandwich has been eaten between them, and together they finish in half the time one would take. Two flames consume the rope's total burn time twice as fast, so a 60 minute rope lit at both ends is gone in 30 minutes, wherever the flames happen to meet. Length tells you nothing here; burn time is the only quantity you can trust.

Why is this really a question about information?

The rope hides where its time is stored, much as an order book hides how much size is waiting behind a price. The solution uses only what is known, the total burn time, and never what is not, how it is spread along the rope. Anyone who cuts a rope in half is assuming evenness that the first sentence ruled out. Say that out loud before you give the method: naming what you may not assume is half of a good answer.

Expect the follow-up. The same trick measures 15 minutes as an interval, the gap between rope one going out and rope two going out. Each rope lit from its second end at a known moment halves whatever burn time it has left, and chaining those halvings is how you reach times such as 52.5 minutes with a third rope. Walk through the chain in order, one lighting at a time.

One round trip makes at most 16; unlimited round trips make 26. For one trade, walk the prices once, carrying the lowest price so far and the best sale against it: buy at 96, sell at 112. For many trades, add every day-on-day rise and skip every fall: 8 + 11 + 7 = 26. Without short selling the falls are simply sat out, never profited from.

How do you find the best single trade without checking every pair?

Imagine walking down a street of shops that all sell the same phone, planning to buy once and sell once further along. You do not need to compare every pair of shops: carry the cheapest price seen so far in your head, and at each shop ask what selling here would make against it. One pass, keeping the running minimum and the best gap found so far, gives the best single trade. Here the running minimum drops to 96 on day 2 and the best gap appears on day 7: 112 - 96 = 16.

Why is the many-trade answer just the sum of the rises?

Any rise from a low to a later high is the sum of the daily steps inside it, and some of those steps may be falls. With no short selling and no costs, the most you can make is the total of every positive day-on-day move, 26 here, because trading only the up steps collects everything a longer trade would and skips its falls. In practice that is three round trips: buy 96, sell 104; buy 99, sell 110; buy 105, sell 112.

What does the interviewer add next?

Costs. Once each round trip costs something, the sum of rises overstates the profit, because small moves stop being worth trading. With a cost of 6 per round trip, the three separate trades net 2 + 5 + 1 = 8, the best two-trade split nets 9, and the single trade from 96 to 112 nets 10, so the single trade now wins. The general version is a short dynamic programme that tracks the best profit on each day while holding and while flat.

About 64.2%. A useless signal clears a 5% bar by luck one time in twenty. The chance that all 20 stay insignificant is 0.95 to the twentieth, about 35.8%, so the chance that at least one looks like a discovery is 64.2%. On average the search turns up one false signal, as 20 x 5% suggests, but at least one appears in roughly two searches out of three.

Why does testing more ideas manufacture a winner?

Ask twenty friends to flip a coin five times each. Any one of them flips five heads only once in 32 tries, yet the chance that at least one of the twenty does is 47.0%, and that friend will look gifted. Each test is a lottery ticket for a false discovery, and buying twenty tickets makes a win likely even when nothing works. A signal chosen because it looked best among twenty has not passed a 5% test; it has passed a 64.2% one.

How do you correct for it?

Tighten the bar to match the number of tries. The Bonferroni correction tests each signal at 5% divided by 20, which is 0.25%, and that brings the chance of any false discovery back to 4.9%. The cost is power: a real but modest signal now struggles to get through. The other defence is data the search never touched: choose the best signal on one period, then test it once on another.

What does a quant fund take from this?

Research teams run thousands of tests, and the ones that get presented are the survivors. Count every test, including the ones you ran and forgot, because the significance of the survivor depends on how many were tried. That is why systematic funds keep research logs and hold data back, and why a backtest with a t-statistic of 2 means much less after a large search than after one planned test. Say the limitation: the 64% assumes independent tests; correlated signals give a lower figure, but rarely a comfortable one.

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.

161/36, about Rs 4.47. The higher face equals k in 2k minus 1 of the 36 equally likely rolls: 1, 3, 5, 7, 9 and 11 rolls for k from 1 to 6. Multiply each value by its count, add to 161, and divide by 36. The lower face averages 91/36, about 2.53, and the two add to 7, the average total of two dice, which is a quick check.

How many of the 36 rolls give each maximum?

Think of two runners and a prize for the faster time: the winning time is better than a typical single runner's because you always keep the better of two. The higher face is at most k in k x k of the 36 rolls, so it equals exactly k in k squared minus (k minus 1) squared, which is 2k minus 1 rolls. That gives 1 roll with a maximum of 1, 3 with a maximum of 2, and on up to 11 with a maximum of 6. On the grid those cells form L shapes that grow as you move towards the corner.

How do you check 4.47 in ten seconds?

Use the pair. The higher face plus the lower face always equals the total of the two dice, so their averages must add to 7. The lower face is at least k in (7 minus k) squared rolls, which gives an average of 91/36, about 2.53. 4.47 plus 2.53 is 7.00. A second method that lands exactly is what makes an interviewer stop checking your arithmetic and move on to the follow-up.

The follow-up is usually a game. If you could pay to roll one die or to roll two and keep the higher, the second is worth about Rs 0.97 more. That gap, the value of a free second look, is the same idea as an option: the right to choose after seeing the outcome is worth paying for.

About 8% a year, two points below the 10% average. Compound growth is roughly the average return minus half the variance: 10% minus 0.5 x 0.20 squared, which is 10% minus 2%. Check with two years of +30% and -10%: the average is 10% and the volatility 20%, but 1.30 x 0.90 is 1.17, which compounds at 8.17% a year. The drag grows with the square of volatility.

Why does the average return overstate what you end up with?

A shopkeeper whose sales rise 50% one month and fall 50% the next has not broken even: 100 becomes 150 and then 75. A loss is applied to a bigger base after a gain, and a gain to a smaller base after a loss, so swings always drag compound growth below the simple average. The bigger the swings, the bigger the drag. The simple average of yearly returns is called the {term('arithmetic mean', 'The plain average of the yearly returns, adding them up and dividing by the number of years.')}; the rate your money actually grows at is the geometric mean, and it is always the lower of the two when returns vary.

How big is the drag, and where does half the variance come from?

For returns that are not too large, the geometric mean is close to the arithmetic mean minus half the variance. With 20% volatility the variance is 0.04, half of that is 0.02, so a 10% average compounds at about 8%. The two-year example makes it concrete: +30% and -10% average 10% with a standard deviation of 20%, and 1.30 x 0.90 = 1.17, a compound rate of 8.17% a year. The rule of thumb says 8.00%, close enough to trust in an interview.

Add why an allocator asks this. Two funds with the same average return and different volatility do not leave investors with the same money. Cutting volatility from 20% to 10% raises compound growth by 1.5 points with no change in the average, which is part of why lower-volatility strategies can be worth more than their averages suggest. The limitation: the half-variance rule is an approximation that weakens for very volatile or fat-tailed returns.

1 over root 2, about 0.71. The covariance of X with X + Y is Var(X) plus Cov(X, Y), which is sigma squared plus zero. The standard deviation of X + Y is root 2 times sigma because the variances add. So the correlation is sigma squared over (sigma x root 2 sigma), which is 1/root 2. X explains half the variance of the sum, and the correlation is the square root of that half.

What is the fastest way to set it up?

A two-member team's score is the sum of both players' scores. If the players are equally good and play independently, knowing one player's score tells you something about the team total, but only half the story. Split the covariance: Cov(X, X + Y) = Cov(X, X) + Cov(X, Y) = sigma squared + 0. The variance of the sum is sigma squared + sigma squared = 2 sigma squared, because independent variances add. Correlation is covariance over the product of standard deviations: sigma squared over (sigma x root 2 sigma) = 1/root 2.

Why is the answer not 0.5?

Because 0.5 is the R squaredThe share of one variable variance explained by another; for a simple regression it is the correlation squared., not the correlation. X explains exactly half of the variance of X + Y, and correlation is the square root of the share of variance explained, so it is root 0.5, about 0.707. The geometric picture makes it stick: treat independent variables as arrows at right angles, and correlation as the cosine of the angle between arrows. X + Y sits at 45 degrees to X, and cos 45 is 0.707.

Give the general version to show you own it. If Y has variance k times X's, the correlation is 1/root(1 + k): the more noise you add, the lower it falls. That is the logic behind a noisy signal: a forecast that is half signal and half independent noise, by variance, correlates about 0.71 with the signal, not 0.5.

Net income falls Rs 7.5, cash rises Rs 2.5 and the balance sheet shrinks by Rs 7.5 on both sides. Pre-tax profit falls 10, tax falls 2.5, so net income falls 7.5. The cash flow statement starts at -7.5 and adds back the non-cash 10: cash up 2.5. On the balance sheet, cash is up 2.5 and fixed assets are down 10, so assets fall 7.5, matched by retained earnings down 7.5.

Why does cash go up when an expense goes up?

Imagine your employer lets you deduct the wear on your car from taxable income. No money leaves your pocket for the wear itself, but your tax bill falls. Depreciation is an expense that costs no cash but reduces tax, so the only cash effect is the tax saved: 25% of Rs 10, Rs 2.5. That is the {term('depreciation tax shield', 'The tax saved because depreciation is deductible even though it uses no cash; equal to depreciation times the tax rate.')}, and it is the one number the question is testing.

What order do you walk it in so nothing gets lost?

Income statement first, then cash flow, then balance sheet, one line each. Net income is the bridge: it closes the income statement, opens the cash flow statement, and lands in retained earnings on the balance sheet. Income statement: depreciation +10, pre-tax -10, tax -2.5, net income -7.5. Cash flow: -7.5 plus 10 added back, cash +2.5. Balance sheet: cash +2.5, fixed assets -10, so assets -7.5; retained earnings -7.5, so the two sides move together.

Add one sentence on why a hedge fund analyst cares. Two companies with identical operations can report different earnings because of depreciation choices, while their cash generation differs only by the tax effect. That is one reason investors look at cash flow alongside earnings before trusting a P/E.