Citadel / Citadel Securities interview questions

Quantitative Trader · Interview · Expected value · reported by a candidate

A fair die is rolled and you receive the cube of the result.

1. What is the expected value?
   Answer and worked solution

   Answer: 73.5

   Key idea. Average the six payouts directly.

   1. Cubes: 1, 8, 27, 64, 125, 216. Sum = 441.
   2. E = 441/6 = 73.5.

   Check. Shortcut: 1³ + … + n³ = (1 + … + n)² = 21² = 441.

2. Follow-up (ours): what is (E[X])³, and why is it smaller?
   Answer and worked solution

   Answer: 42.875

   Key idea. Jensen's inequality: for a convex function f, E[f(X)] ≥ f(E[X]).

   1. (E[X])³ = 3.5³ = 3.5 × 12.25 = 42.875.
   2. x³ is convex for x > 0, so E[X³] (73.5) exceeds it.

   Check. The same idea is why options gain value with volatility: their payoff is convex.

Interviewers may push further: What is the variance of the payout?

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Quant Trader Intern · Interview · Expected value · reported by a candidate

You roll a fair die until every face has appeared at least once.

Also reported at Da Vinci.

1. What is the expected number of rolls?
   Answer and worked solution

   Answer: 14.7

   Key idea. Coupon collector: one stage per new face.

   1. With k faces seen, a roll is new with probability (6 − k)/6: mean wait 6/(6 − k).
   2. Sum: 6(1 + ½ + ⅓ + ¼ + ⅕ + ⅙) = 6 × 2.45.
   3. = 14.7.

   Check. For n faces it grows like n ln n.

Interviewers may push further: Roughly how does this grow for an n-sided die? (n ln n.)

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Quant Trader Intern · Interview · Statistics · reported by a candidate

Which is more likely: 40 to 60 heads in 100 fair flips, or 400 to 600 heads in 1000?

1. What is the probability of 40–60 heads in 100 flips (to two decimal places)?
   Answer and worked solution

   Answer: 0.9648

   Key idea. Measure each range in standard deviations.

   1. 100 flips: σ = 5, so 40–60 is ±2σ. P ≈ 0.95; with the endpoints included, exactly 0.965.
   2. 1000 flips: σ ≈ 15.8, so 400–600 is about ±6.3σ: essentially certain.
   3. So the 1000-flip range is far more likely.

   Check. The same ±10% window covers more standard deviations as n grows (σ grows like √n).

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Quantitative Trader · Interview · Expected value · reported by a candidate

The numbers 1 to 5 are placed around a circle in random order.

1. What is the expected number of numbers smaller than both their neighbours?
   Answer and worked solution

   Answer: 5/3 ≈ 1.6667

   Key idea. Linearity of expectation over the 5 positions.

   1. A number and its two neighbours are three distinct values in random order.
   2. P(it's the smallest) = 1/3.
   3. E = 5 × 1/3 = 5/3.

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Semi-Systematic Trader · Interview · Market making · reported by a candidate

Make a market on the square root of 30.

1. First, your best estimate of √30 to two decimal places.
   Answer and worked solution

   Answer: 5.4772

   Key idea. Start from a nearby square and take one Newton step.

   1. 5.5² = 30.25, slightly high.
   2. Newton: x − (x² − 30)/(2x) = 5.5 − 0.25/11 ≈ 5.477.
   3. √30 ≈ 5.48. Quote tightly around it, e.g. 5.47 @ 5.49.

   Check. 5.477² ≈ 29.998.

   Trap. Quoting very wide on something you can compute: it signals you don't trust your own arithmetic.

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Trader · Interview · Behavioural · reported by a candidate

Behavioural questions candidates report:

• Why do you want to be a trader?

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