Hedge Funds puzzles, solved step by step

The spread must be Rs 0.60: a bid Rs 0.30 below the mid and an offer Rs 0.30 above it. Call the half-spread s. Each uninformed order earns her s on average. Each informed order trades on the side where the value really is Rs 1 away, so she loses 1 minus s. Breaking even needs 0.7s = 0.3(1 - s), which gives s = 0.30.

Why do some of the people who trade with you cost you money?

A shop that buys second-hand phones pays the same price to every seller, but the sellers who know their phone is faulty are the keenest to sell. A quote is an offer made to everyone, and the people who know more take it only when it hurts you. The spread is the fee charged to the whole crowd to pay for the few who know more, which is called adverse selectionThe tendency for the counterparties who choose to trade with you to be the ones who know the trade is good for them..

How do you set up the break-even?

Charge a half-spread s on each side. An uninformed order is as likely to be a buy as a sell and carries no information, so on average it earns you s; an informed order buys at your offer only when the value is Rs 1 above the mid, so it costs you 1 minus s. Weight each by how often it arrives and set the total to zero: 0.7s - 0.3(1 - s) = 0, so s = 0.30 and the spread is Rs 0.60.

What moves the spread?

Two things: how much of the flow is informed and how much the informed know. In general the half-spread is s = aV, where a is the informed share and V is how far the true value sits from the mid, so doubling either doubles the spread. Here 0.3 x 1 = 0.3. That is why quotes widen ahead of a results announcement, when more of the flow may know something, and in small stocks, where one piece of news moves the value a long way. The model leaves out inventory risk and competing quoters, both of which move a real spread; say so.

Yes: buy the call, sell the put, short the stock and lend Rs 970.87, which collects about Rs 14.13 today with nothing owed at expiry. Put-call parity says the call less the put should equal the stock less the present value of the strike, 1,000 minus 1,000/1.03, about Rs 29.13. The market prices that gap at Rs 15, so the call is cheap against the put by about Rs 14.

What does a call minus a put actually pay?

Imagine agreeing today to buy a used car in six months at a fixed price. If car prices rise you gain the difference; if they fall you lose it. A long call with a short put at the same strike does exactly that: at expiry it pays the stock price minus the strike, whichever way the stock moved. So it must cost the same as buying the stock today with money borrowed against the strike, 1,000 minus 970.87, which is Rs 29.13. That identity is put-call parityFor European options on the same stock, strike and expiry, call minus put equals the stock price minus the present value of the strike, less the value of any dividends..

How do you build the trade?

Buy what is cheap and sell what is dear. The option version of the forward, long call and short put, costs Rs 15; the cash version, long stock bought with borrowed money, is worth Rs 29.13, so buy the option version and sell the cash version. Buy the call for 60, sell the put for 45, short the stock for 1,000 and lend 970.87 at 3%. The cash left over today is Rs 14.13, worth about Rs 14.55 at expiry.

What could stop the arbitrage working?

Say what you assumed. The lock holds only if the options are European, the stock pays no dividend before expiry, and you can short the stock without a borrowing fee that eats the Rs 14. A dividend lowers the forward and so the fair gap; an American put can be exercised against you early; a hard-to-borrow stock charges a fee. A gap this size in a liquid name usually means one of those is present, so your first question back is which one.

Six flips on average for two heads in a row, and four for a head then a tail. Track how far along the pattern you are. For HH, a tail at any point sends you back to the start, including right after a head. For HT, a head after a head keeps you one step away, so progress is never lost. Solving the two small chains gives 6 and 4.

Why do two equally likely patterns take different times?

Think of two ladders where a slip costs you differently. On one, slipping from the first rung drops you to the ground; on the other, you can only ever slip back to the first rung. Both patterns are equally likely in any given pair of flips, but after one head the wrong next flip costs you everything for HH and nothing for HT. A head then a tail breaks HH and restarts it; a head then a head is still a perfect start for HT.

How do you solve the chain?

Let E0 be the expected flips still needed from the start and E1 after one head. For HH, E0 = 1 + E1/2 + E0/2 and E1 = 1 + E0/2, because a tail from one head sends you back, and these solve to E1 = 4 and E0 = 6. For HT, the one-head state just waits for a tail, which takes 2 flips on average, and reaching the first head takes 2 more, giving 4. Say the states out loud before the algebra; the interviewer wants to hear them named.

What is the general pattern?

Patterns that can fail back to nothing take longer. The expected wait for n heads in a row is 2 to the power n + 1, minus 2: 2, 6 and 14 flips for one, two and three heads. This is a Markov chainA process whose next step depends only on the current state, not on how it got there, so it can be solved state by state. at heart, and the same state-by-state method handles anything that depends on a path: a streak, a barrier, a drawdown rule on a trading book.

No. The square root of the unbiased variance underestimates the standard deviation on average. The square root is concave, so by Jensen's inequality the average of the square roots is below the square root of the average. For normal data with two observations the estimate averages about 0.80 sigma; the bias shrinks as the sample grows, to about 6% at five observations and under 1% at thirty.

Why does taking a square root break unbiasedness?

Two square rooms have floor areas of 4 and 16 square metres, so their sides are 2 and 4 metres. Average the areas, 10, and take the root: 3.16 metres. Average the sides instead: 3 metres. Averaging and then taking a square root gives a bigger answer than taking square roots and then averaging, because the square root bends downwards. The sample variance is right on average, so the average of its square roots must fall short of the true standard deviation.

How big is the bias?

It depends on the sample size and on the distribution. For normal data the expected sample standard deviation is c4 times sigma, with c4 about 0.80 at n = 2, 0.94 at n = 5, 0.97 at n = 10 and 0.99 at n = 30. At n = 2 you can check it directly: the sample standard deviation is the gap between the two draws divided by the square root of 2, and the average gap between two normal draws is 2 sigma over the square root of pi, which leaves the square root of 2/pi, about 0.798.

Does it matter in practice?

Sometimes. With a year of daily returns the bias is a rounding error; with a handful of monthly returns for a new fund it is not. A manager with five monthly returns has a volatility estimate that averages about 6% too low under normality, which flatters a Sharpe ratioAverage excess return divided by the standard deviation of returns, a measure of return per unit of risk. before anyone has looked at fat tails. Dividing by c4 removes the bias for normal data, but the fix depends on the distribution, so name the assumption. And unbiased is not the same as most accurate.

The margin rises from 15% to about 19.8%, and operating profit rises 40%. Each dollar now buys 10% more rupees, which turns Rs 60 of dollar revenue into Rs 66, so revenue reaches Rs 106. Costs are all in rupees and stay at Rs 85, so profit climbs from Rs 15 to Rs 21, and 21 over 106 is 19.8%. A 6% rise in revenue becomes a 40% rise in profit.

Why does a weaker rupee help this company so much?

Think of a tutor in Pune who teaches some students abroad for fees in dollars and pays rent and a salary in rupees. When the rupee weakens, each dollar fee converts into more rupees while the rent does not move. A currency move lands on the revenue billed in the foreign currency and on nothing else, so when every cost is local the whole gain drops into profit. Here the gain is Rs 6, on a profit that started at only Rs 15.

How do you size the profit move quickly?

Chain three numbers. The dollar buys 10% more rupees, 60% of revenue is exposed, so revenue rises 6%; a thin 15% margin turns that Rs 6 into a 40% rise in profit. The ratio of the profit change to the revenue change, 40 over 6, almost 7 times, is the company's operating leverageHow much operating profit moves for a given move in revenue, high when most costs are fixed or unaffected by the change. to the currency. An analyst on a long-short book uses that multiplier to judge how much of a stock's move a currency view is really driving.

What would you check before trusting the new margin?

Three things that often shrink the gain. Hedges, pricing and imported inputs: a company that sold its dollars forward keeps the old rate until the hedges roll off, buyers abroad may push for lower dollar prices, and any imported materials get dearer too. Each one eats into the Rs 6. The 19.8% is the unhedged, first-round answer; naming those three tells the interviewer you would not stop at it.

About 11.6% in normal markets and 14.2% in a crisis. With equal weights and equal volatilities, portfolio variance is 15% squared times (1 + rho)/2. At a correlation of 0.2 that is 225 x 0.6 = 135, a volatility of 11.6%; at 0.8 it is 225 x 0.9 = 202.5, a volatility of 14.2%. About 77% of the diversification benefit disappears just when it is needed.

Where does correlation enter the arithmetic?

Picture an umbrella shop and an ice-cream stall owned by one family. On ordinary days one does well when the other is quiet, and the family income is smoother than either shop's. A city-wide power cut shuts both at once. Portfolio variance is each asset's own variance, weighted, plus a co-movement term that carries the correlation, so when correlation jumps the co-movement term grows and the smoothing shrinks. Here the own terms are 56.25 each and the co-movement term is 2 x 0.5 x 0.5 x rho x 225.

What are the two numbers?

Normal markets: 112.5 plus 2 x 0.25 x 0.2 x 225 = 22.5 gives 135, and the square root is 11.6%. Crisis: the co-movement term rises to 90, variance to 202.5, and volatility to 14.2%, a 22.5% jump in risk with no change in either asset's own volatility. The benefit of holding two assets instead of one falls from 3.4 points to 0.8. In a real crisis each asset's volatility usually rises too, so this is the milder case.

What would a risk manager do with this?

Stop sizing positions on correlations measured in calm markets. A risk limit set on a normal-market correlation understates crisis risk by about 22% here, before any rise in the assets' own volatility. Platforms that run many books run stressed-correlation scenarios for this reason, and ask each manager what the book looks like if everything moves together. The honest limitation: nobody knows the crisis correlation in advance; 0.8 is an assumption, and the answer should say so.

About 31% of the ending wealth. Rs 1 lakh growing at 10% a year becomes Rs 6.73 lakh in 20 years; at 8% after the fee it becomes Rs 4.66 lakh. The fee takes Rs 2.07 lakh, about 31% of what the investor would otherwise have had, although each year it looks like only a fifth of the return.

Why does a fifth of each year's return become almost a third of the wealth?

Think of a mango tree whose fruit you replant. If a neighbour takes one mango in every five each year, you lose those mangoes and every tree they would have grown. A fee taken every year removes money that would itself have compounded for the remaining years, so its cost grows faster than the fee rate suggests. The share lost is 1 minus (1.08/1.10) to the 20th: each year the investor keeps 98.2% of what the gross path would have held, and 0.982 to the 20th is about 0.69.

How do you estimate it without a calculator?

Use the rule of 72. At 10% money doubles about every 7.2 years, so 20 years is about 2.8 doublings, near 7 times; at 8% it doubles every 9 years, about 2.2 doublings, near 4.6 times. The exact numbers are 6.73 and 4.66, so the estimate lands within a few per cent, and 4.6 over 6.9 already tells you roughly a third is gone. Say the estimate first, then the exact figure.

What does an allocator take from this?

That fees should be judged against the return they leave, over the holding period, not as a percentage in a single year. A manager charging 2% needs to beat a cheaper alternative by about 2 points a year, every year, just to leave the investor level; over 20 years the difference is 31% of the pot. Hedge fund fees often add a share of profits on top, which widens the gap further. The illustration assumes a steady 10%; real returns vary, but the compounding of the fee does not depend on that.

About 31 managers, 1,000 halved five times. Each year roughly half the unbeaten managers beat the market by luck, so 500 survive year one, 250 year two, then 125, 62.5 and 31.25. A perfect five-year record is something luck hands to about 3 managers in every 100, so in a large crowd it cannot on its own separate skill from chance.

Why does a crowd produce streaks even without skill?

Fill a stadium with a thousand people and ask each to toss a coin five times. Someone will throw five heads, and about 31 will. A result that is rare for one person is almost certain somewhere in a large group, so the question is never whether a flawless record exists but how many you would expect by chance. Each manager's chance is 1 in 32; across 1,000 managers that is 31.25 expected perfect records.

If some managers really are skilled, how much does a perfect record tell you?

Suppose 5% of the thousand are skilled and beat the market 60% of the time. They produce about 3.9 perfect records, while the 950 unskilled produce about 29.7, so a manager with five perfect years is skilled only about 12% of the time. A 60% manager has only a 8% chance of five perfect years, so most skilled managers do not have flawless records either. The record is weak evidence in both directions.

What should you look at instead?

Longer records, more decisions per year and a reason. Skill shows up more reliably in many independent decisions than in a handful of annual outcomes, and in a process that explains where the edge comes from. Allocators also check how many managers were in the starting pool, because the funds still reporting are the ones that survived; the ones that were closed after bad years have dropped out of the data. That is survivorship biasThe distortion that comes from studying only the survivors of a process, whose results look better than those of the whole starting group., and it makes every surviving record look stronger than it is.

About 8.5x, not 10x. Rs 300 crore of the Rs 1,000 crore market value is cash, so the operating business is valued at Rs 700 crore. The cash earns 6% after tax, Rs 18 crore, so the operating business earns Rs 82 crore. Rs 700 crore over Rs 82 crore is about 8.5 times earnings. The headline 10x blends a cheaper business with cash priced at about 16.7 times its income.

Why is the headline P/E misleading here?

Imagine buying a shop for Rs 10 lakh that has Rs 3 lakh sitting in its till. You are really paying Rs 7 lakh for the shop itself. A company's price and earnings both include its cash, and cash is valued very differently from a business: here it earns 6% after tax, so it is worth about 16.7 times its income. Blending the two gives the headline 10x, which is neither the price of the cash nor the price of the business.

How do you work it?

Strip the cash out of both the numerator and the denominator. The price of the operating business is 1,000 minus 300, Rs 700 crore; its earnings are 100 minus 6% of 300, Rs 82 crore; and 700 over 82 is 8.54x. Taking the cash out of only the price gives 7.0x, which is too low because the Rs 100 crore still contains the cash's interest. The check: 700 plus 300 is 1,000 and 82 plus 18 is 100.

When would you not strip out all the cash?

When the cash is not really free. Cash needed to run the business, cash trapped abroad that would be taxed on the way home, or cash a management team is likely to spend badly may deserve less than full value. A long-short analyst uses the ex-cash multiple to compare this business with peers that hold no cash, and then asks how much of the Rs 300 crore shareholders will actually see. That judgement moves the answer between about 8.5x and the headline 10x.

It makes no difference: switching and sticking each win half the time. List the six equally likely cases of car position and the host's random pick. Two of them reveal the car, and you have seen that they did not happen. The four that remain split two and two. The knowing host gives two thirds only because he never risks the car, which pushes those two cases into the switch column.

Where does the usual two-thirds answer come from?

In the standard game the host knows where the car is and always opens a goat door. Your first pick is right one time in three, and nothing the knowing host does can change that, so the other two thirds sit on the remaining closed door. His choice carries information because it is forced: when the car is behind door 2, he must open door 3, and when it is behind door 3, he must open door 2.

What changes when the host picks at random?

Picture a friend who does not know the answer to a quiz question and strikes out one option on a whim. If that option happens to be wrong, you have learned less than if someone who knew had struck it. The random host is that friend. Write out six cases: the car behind door 1, 2 or 3, each with the host's coin choosing door 2 or door 3. In two of the six the random host opens the car door, and the goat you saw rules those two out, which removes switch wins rather than stick wins.

Count what is left. The car behind door 1 survives both host choices: two cases where sticking wins. The car behind door 2 survives only when the host opened door 3, and the car behind door 3 only when he opened door 2: two cases where switching wins. Two against two.

Why would an interviewer want the intuitive answer broken rather than recited?

Because the lesson travels to every desk. The same observation carries different information depending on the process that produced it. A strong track record shown by a manager who launched ten funds and closed the nine that did badly is a host choosing which door to open for you. Before you update on evidence, ask whether the source could have shown you something else, and whether it chose what to show.