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  1. 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

    Try it first

    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.

    How many of 20 positions go to zero in a year, each at a 10% chance12.2%00%27.0%1-5%28.5%2-10%19.0%3-15%9.0%4-20%3.2%5-25%0.9%6-30%0.2%7-35%<0.1%8-40%LostBook15% or worseThree or more zeros32.3%of yearsExpected zeros = 20 x 10% = 2, so three is only one more than the 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≥3)=1−∑k=02(20k)(0.1)k(0.9)20−k≈1−0.677=0.323P(X \ge 3) = 1 - \sum_{k=0}^{2} \binom{20}{k} (0.1)^k (0.9)^{20-k} \approx 1 - 0.677 = 0.323
    Xthe number of positions that go to zero
    \binom{20}{k}the number of ways to choose which k names fail
    0.1 and 0.9the 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?
  2. 026You have five trade ideas. Each needs some units of risk budget and carries an expected profit: A needs 4 units for Rs 9 crore, B 3 units for Rs 7 crore, C 5 units for Rs 10 crore, D 2 units for Rs 4 crore and E 6 units for Rs 11 crore. Your budget is 10 units and you cannot take part of an idea. Which ideas do you take?Betting and sizingCoreBridgewater AssociatesNew York · 2025

    Try it first

    Before you enumerate: which set do you expect to win?

    Show the worked solution

    Take B, C and D: 10 units for Rs 21 crore. Ranking by profit per unit of risk picks B (2.33), then A (2.25), then D (2.00), which uses 9 units for Rs 20 crore and leaves one unit idle. Swapping A for C costs a little ratio but fills the budget and adds Rs 1 crore. With whole ideas, you check the few sets that fill the budget rather than trust the ratio alone.

    Why is profit per unit of risk the right place to start?

    Packing a suitcase for a flight with a weight limit, you favour the things that give the most use per kilo. When the budget is the scarce thing, the useful measure is profit per unit of budget, not profit alone. E earns the most in rupees but only 1.83 per unit, the worst of the five. B earns 2.33 per unit, A 2.25, and C and D 2.00 each. If ideas could be split, you would simply fill from the top of that list and the answer would be exact.

    Why does the ratio ranking fail here?

    Ideas come whole. Take B and A and you have used 7 units; D fits for 9 units, and nothing else fits in the last one. A unit left idle earns nothing, so a set with a slightly lower average ratio that uses the full budget can beat the greedy pick. Replace A (4 units) with C (5 units) and the budget is full: B, C and D earn 7 + 10 + 4 = Rs 21 crore against Rs 20 crore. This is the knapsack problemChoosing whole items, each with a size and a value, to maximise total value without exceeding a capacity., and the fix is to check the handful of sets that nearly fill the budget.

    Every full set that fits 10 risk units, ranked by expected P&LIdea A4u, Rs 9 cr, 2.25/uIdea B3u, Rs 7 cr, 2.33/uIdea C5u, Rs 10 cr, 2.00/uIdea D2u, Rs 4 cr, 2.00/uIdea E6u, Rs 11 cr, 1.83/uRisk units usedExpected P&LB + C + DBCDRs 21 crbest: all 10 units usedA + B + DABDRs 20 crper-unit ranking picks thisA + EAERs 20 crA + CACRs 19 crB + EBERs 18 crA + B + CABCRs 26 crover budgetbudget: 10 units05
    B, C and D use all 10 risk units for Rs 21 crore, the best set that fits; ranking by profit per unit picks B, A and D, which uses 9 units for Rs 20 crore, and A, B and C would earn Rs 26 crore but needs 12 units.

    With five ideas there are only 31 possible sets, so enumerate the full ones out loud: B, C, D for 21; A, B, D for 20; A, E for 20; A, C for 19. On a real book with hundreds of positions a desk uses an optimiser, but the logic is the same, and interviewers want to hear that you know the ratio rule is a starting point and not a proof.

    Say the limitation too. Expected profit is an estimate, and the gap here is Rs 1 crore on Rs 21 crore. If C's estimate is shakier than A's, a portfolio manager could reasonably prefer the greedy set. The arithmetic tells you the best set on the stated numbers; confidence in each number decides whether that edge is real.

    Where candidates lose it

    The common loss is ranking by profit per unit, taking the top three, and stopping. That is right for divisible positions and wrong here, because the tenth unit of budget sits unused and the interviewer built the numbers to punish exactly that.

    The opposite loss is ranking by rupee profit and grabbing E first. E is the least efficient idea on the list. Say the ratio rule, then check the sets that fill the budget, then say how sure you are of each estimate.

    What the interviewer asks next

    • The budget rises to 11 units. What changes?
    • Ideas B and C are highly correlated, so together they use 9 units instead of 8. Does your answer move?
    • You can take half of any idea for half its units and half its profit. What do you take now?

    Asked at Bridgewater Associates, Generalist, New York, 2025 (Wall Street Oasis): Given a list of items and their utilitites, give an algorithm to maximise utility

  3. 051You start with Rs 1,000 and can bet any amount on a coin that wins 60% of the time at even money, as many times as you like. What fraction of your money do you bet each time, and what happens if you bet twice that?Betting and sizingCoreMillennium ManagementAtlanta · 2025

    Try it first

    Which fraction of your money gives the fastest long-run growth?

    Show the worked solution

    Bet 20% of your current money each time; bet 40% and the long-run growth is gone. For an even-money bet the Kelly fraction is the win probability minus the loss probability, 0.6 minus 0.4. At 20% the typical path grows about 2.0% a bet. At double that, the average log growth is about -0.2% a bet, so the same edge now earns nothing over time.

    Why not bet everything when the odds are in your favour?

    Think of a shopkeeper who puts all of today's takings into tomorrow's stock. Six days in ten the stock sells and she doubles her money; on the other four she is back to zero, and one zero ends the shop. Wealth compounds, so a single wipe-out is permanent, and the stake that maximises the average payoff of one round is not the stake that maximises what you end with after many. Betting 100% has the highest expected value per round and a certainty of eventual ruin.

    How do you find the best fraction?

    Maximise the growth rate, which is the average of the log of what each bet does to your money. Bet a fraction f and a win multiplies your wealth by 1 + f, a loss by 1 - f. The growth rate 0.6 ln(1 + f) + 0.4 ln(1 - f) peaks where f equals the edge, 0.6 - 0.4 = 20%, at about 2.01% a bet. Over 100 bets that turns Rs 1,000 into about Rs 7,490 on the typical path. This rule is the Kelly criterionA sizing rule that picks the bet size maximising the long-run growth rate of wealth, the expected log return per bet..

    The relationship
    g(f)=pln⁡(1+f)+qln⁡(1−f),f∗=p−q=0.6−0.4=0.2g(f) = p\ln(1+f) + q\ln(1-f), \qquad f^{*} = p - q = 0.6 - 0.4 = 0.2
    pthe chance of winning, 0.6
    qthe chance of losing, 0.4
    fthe fraction of current money bet
    g(f)the expected log growth per bet
    What it says in wordsGrowth per bet is the probability-weighted average of the log gain and the log loss, and it is highest when you bet the edge.
    Growth per bet against the fraction you bet: the peak sits at the edge-4%-2%+2%0%0%10%20%30%40%50%Fraction of current money bet each timeKelly: bet 20%, growth +2.0% a betDouble Kelly, 40%: -0.2% a beta real edge, and no growth leftshaded: negative long-run growth
    Expected log growth per bet rises to about 2.0% at a 20% stake, falls back to slightly below zero, -0.2%, at a 40% stake and turns sharply negative beyond it, so over-betting a real edge can remove all long-run growth.

    What happens at twice the Kelly bet?

    Near the peak the growth curve is close to a parabola centred on 20%. Double the Kelly fraction and you sit as far down the far side as not betting at all sits on the near side: growth of about zero, -0.24% a bet. Over 100 bets the typical path ends near Rs 783, below where you started, although every single bet had a positive expected value. The extra swings cost more in compounding than the extra stake earns.

    Say the limitation too. Kelly assumes you know the 60% exactly. On a desk the edge is an estimate, and overestimating it pushes you to the right of the peak, where the curve falls fastest. That is why many traders bet half Kelly: 10% here keeps about 75% of the growth with half the swings.

    Where candidates lose it

    The fast wrong answer is to bet big because the odds favour you, often 60% because that is the chance of winning. A candidate who only computes expected value per round will always pick the largest stake, and the interviewer is waiting to see whether you notice that money compounds.

    The second loss is reaching 20% and being unable to say what happens beyond it. Have the shape ready: growth peaks at the edge, is roughly zero at twice the edge, and is negative after that.

    What the interviewer asks next

    • The coin now pays 2 to 1 when you win and still wins 60% of the time. What is the Kelly fraction?
    • You are only 80% sure the coin is 60/40 rather than fair. How does that change your bet?
    • What would you pay to play 100 rounds of this game starting with Rs 1,000?

    Asked at Millennium Management, Software Engineering Intern Interview, Atlanta, 2025 (Wall Street Oasis): Start with 1000 and bet each round. Kelly criterion would be very useful for this step.