Hedge Funds puzzles, solved step by step

Puzzles: 100
Traced to a firm: 38
Topics: 14
Hard: 30

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001A book holds 20 independent positions of 5% each, and each has a 10% chance of going to zero over the year. What is the probability that you lose 15% or more of the book?Betting and sizingCoreMulti-manager platformsProp and quant trading firms
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Before calculating: roughly how likely is a loss of 15% or more?
Show the worked solution
About 32%. A 15% loss means three or more of the 20 positions go to zero. The count of zeros is binomial with 20 tries at 10%, so the chance of zero, one or two is 12.2% + 27.0% + 28.5% = 67.7%, and the chance of three or more is 32.3%. The book expects two blow-ups a year, so three is not a tail event.
Why is a rare event per name a common event per book?
Think of a wedding with twenty guests, each with a one in ten chance of arriving late. Any single guest is almost certainly on time, but a host who plans for nobody being late is planning badly: on average two will be. When you hold many independent risks, the question is not whether one fails but how many do, and the expected count here is 20 x 10% = 2. A loss of 15% needs three failures, which is one more than an average year.
With 20 positions each carrying a 10% chance of going to zero, the most likely outcomes are one or two zeros; three or more zeros, a loss of 15% or more, happen in 32.3% of years.
How do you count the ways to lose three or more?
Count the outcomes you can live with and subtract. It is faster to add up zero, one and two blow-ups and take them from one than to add up three through twenty. Zero needs all twenty to survive: 0.9 to the twentieth, 12.2%. One needs a single failure, with twenty choices of which name: 20 x 0.1 x 0.9 to the nineteenth, 27.0%. Two has 190 possible pairs: 190 x 0.01 x 0.9 to the eighteenth, 28.5%.
The relationship
$P(X \ge 3) = 1 - \sum_{k=0}^{2} \binom{20}{k} (0.1)^k (0.9)^{20-k} \approx 1 - 0.677 = 0.323$
X the number of positions that go to zero
\binom{20}{k} the number of ways to choose which k names fail
0.1 and 0.9 the chance one name fails, and survives
What it says in wordsThe chance of three or more failures is one minus the chance of zero, one or two.
What does a risk manager take from this?
Sizing each position so a single wipe-out is survivable does not make the book survivable. Five per cent a name feels small, yet a 15% drawdown is roughly a one in three year event on these odds, and a platform with a 10% drawdown limit would see it breached in 60.8% of years, because two zeros, the average outcome, already cost 10%. Say the limitation as well: the positions are assumed independent. In a sell-off failures cluster, and correlation fattens exactly the tail you have just computed.
Where candidates lose it
The fast wrong answer multiplies: 10% cubed is 0.1%, so three blow-ups look like a freak. That ignores the 1,140 different ways to choose which three names fail, and it ignores four, five and more.
The second loss is stopping at exactly three. The question says 15% or more, so either sum three through twenty or, far faster, take the complement of zero, one and two. Say which route you are taking before you start.
What the interviewer asks next
What is the chance of losing 10% or more?
If the 20 names are positively correlated, does the chance of a 15% loss rise or fall, and why?
How would you resize the book so a 15% loss happens less than one year in ten?
065A stock rises 50% or falls 40% each year with equal probability. Its expected return is positive, but what happens to a buy-and-hold investor over time? And what fraction of wealth should sit in the stock if the rest is held in cash and the mix is rebalanced every year?Betting and sizingHardMulti-manager platformsProp and quant trading firms
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Over many years, what happens to the typical buy-and-hold investor?
Show the worked solution
The typical buy-and-hold investor loses about 5.1% a year, yet a 25% stake rebalanced yearly grows about 0.6% a year. The average year returns +5%, but a good year and a bad year multiply wealth by 1.5 x 0.6 = 0.9. With 25% in the stock the two years multiply wealth by 1.125 x 0.9 = 1.0125. That 25% is the Kelly fraction, the stake that maximises the average log return.
How can a positive average return shrink your wealth?
Imagine a shop whose sales rise 50% in a good year and fall 40% in a bad one. After one of each it is at 90% of where it began, whatever the order. Wealth compounds by multiplying, so over many years what matters is the typical growth factor, the square root of 1.5 x 0.6, about 0.949, not the average return of +5%. The average is real, but it is carried by rare paths with long lucky streaks. After 20 years the typical investor holds about 0.35 of the starting money while the average across all paths is 2.65 times it.
The relationship
$g(f) = \tfrac12\ln(1 + 0.5f) + \tfrac12\ln(1 - 0.4f), \qquad g'(f) = 0 \;\Rightarrow\; f^{*} = 0.25$
f the fraction of wealth held in the stock, the rest in cash
g(f) the expected log growth per year of the rebalanced mix
f* the stake that maximises it
What it says in wordsPick the stake that makes the average log return per year as large as possible; here that is a quarter of your wealth.
Over 20 alternating years the all-stock investor falls to about 0.35 of the starting wealth while the average across all paths climbs to 2.65, and a 25% stake rebalanced every year grows to about 1.13, because growth depends on the log return, not the average return.
Why does holding less of the stock help?
Rebalancing to a fixed mix sells after gains and buys after losses, and a smaller stake shrinks the swings. With a fraction f in the stock, a good year multiplies wealth by 1 + 0.5f and a bad year by 1 - 0.4f; typical growth peaks where 0.5/(1 + 0.5f) equals 0.4/(1 - 0.4f), which gives f = 25%. At 25% a pair of years gives 1.125 x 0.9 = 1.0125, about 0.6% a year. The quick check is return over variance: 0.05 divided by 0.45 squared is 0.247.
What is the limit of this answer?
The 25% rests on knowing both outcomes and their odds exactly, on cash earning nothing and on free rebalancing. Change any of those and the fraction moves, and because a real edge is only an estimate, desks size well below the full Kelly number. The lesson to lead with in the room is the gap itself: a positive average return is not a positive growth rate, and position size decides which one you earn. That gap is called volatility dragThe shortfall of the compound growth rate below the average return, roughly half the variance of returns..
Where candidates lose it
Most candidates answer that the investor earns 5% a year, because that is the average. The interviewer built the numbers so the average and the typical outcome point in opposite directions, and wants to see you notice.
The second loss is concluding the stock is simply bad and putting nothing in it. Zero earns nothing; the point is that a small, rebalanced stake turns the same gamble into positive growth.
What the interviewer asks next
Cash now earns 3% a year. How does the best stake change?
What changes if you rebalance every two years instead of every year?
Two such stocks move independently. What happens if you hold half in each and rebalance yearly?
076Your stop-loss on a new long position sits 25% below your entry price, and the desk rule caps the loss on any one idea at 1% of the book. What is the largest position you can take?Betting and sizingWarm upMulti-manager platformsProp and quant trading firms
Try it first
Before you work it: how big can the position be, as a share of the book?
Show the worked solution
4% of the book. If the stop is hit you lose 25% of the position, and that loss must not exceed 1% of the book. So the position times 25% equals 1%, and the position is 1% divided by 0.25, which is 4%. On a Rs 1,000 crore book that is a Rs 40 crore position and a Rs 10 crore loss at the stop.
Why is the answer not simply 1%?
Think of lending a friend money for a trip when you know the worst case is that a quarter of it never comes back. If you can stand to lose Rs 1,000, you can lend Rs 4,000, because only a quarter of the loan is at risk. The loss cap limits what you can lose, and the stop decides what fraction of the position you can lose, so the size is the cap divided by the stop distance. A trader who puts on 1% because the cap is 1% has confused the bet with the damage.
A 4% position with a 25% stop, a 2% position with a 50% stop and a 10% position with a 10% stop all lose exactly 1% of the book at the stop; on a Rs 1,000 crore book the 4% position is Rs 40 crore and loses Rs 10 crore.
The relationship
$\text{size} = \frac{\text{loss cap}}{\text{stop distance}} = \frac{1\%}{25\%} = 4\%\ \text{of the book}$
loss cap the most the desk lets one idea lose, as a share of the book, here 1%
stop distance how far below entry the stop sits, as a share of the entry price, here 25%
What it says in wordsThe position is as large as it can be while still losing no more than the cap when the stop is hit.
What happens to the size as the stop moves?
Tighten the stop and the position can grow; widen it and the position must shrink. With a 10% stop the same 1% cap allows a 10% position, and with a 50% stop only 2%. Every pair on the curve loses exactly 1% of the book when the stop is hit. This is why a trader whose thesis needs room to breathe, say through an earnings print, carries a smaller position than one working to a tight technical level.
What does the stop not protect you against?
The arithmetic assumes you get out at the stop price. A stock that gaps through the stop overnight, on results or a regulatory order, fills below it, and the loss exceeds the cap. If the stock opens 40% down, the 4% position loses 1.6% of the book, not 1%. Risk managers therefore size to the stop and then check the gap risk separately, often capping single-name positions whatever the stop says. Give that limitation straight after the number.
Where candidates lose it
The quick wrong answer is 1%: candidates hear the loss cap and repeat it as the position size. That position would lose only 0.25% of the book at the stop, so the trader is using a quarter of the risk the desk allowed.
The second loss is stopping at 4% without saying that a stop is not a guarantee. One sentence on gap risk shows you know the rule sizes the planned loss, not the worst loss.
What the interviewer asks next
The stock gaps 40% below your entry overnight. What did you lose as a share of the book?
You want the same 1% cap across ten ideas with different stops. How do you set each size?
How would you size the position from the stock's volatility instead of a fixed stop?

X	the number of positions that go to zero
\binom{20}{k}	the number of ways to choose which k names fail
0.1 and 0.9	the chance one name fails, and survives

f	the fraction of wealth held in the stock, the rest in cash
g(f)	the expected log growth per year of the rebalanced mix
f*	the stake that maximises it

loss cap	the most the desk lets one idea lose, as a share of the book, here 1%
stop distance	how far below entry the stop sits, as a share of the entry price, here 25%

Try it first

Why is a rare event per name a common event per book?

How do you count the ways to lose three or more?

What does a risk manager take from this?

Where candidates lose it

What the interviewer asks next

Try it first

How can a positive average return shrink your wealth?

Why does holding less of the stock help?

What is the limit of this answer?

Where candidates lose it

What the interviewer asks next

Try it first

Why is the answer not simply 1%?

What happens to the size as the stop moves?

What does the stop not protect you against?

Where candidates lose it

What the interviewer asks next