The Interview Book · Careers

5Phone and Technical Screens

Twenty minutes into a video screen the candidate announces an expected value of 3.5 for a game whose payoff is never above three. The interviewer says nothing. The candidate who then says “that cannot be right: the payoff is capped at three” and finds the error in a minute has had a better screen than one who never made it, and the notes will say so. A screen is short, it is decided on a handful of observations, and two of the most valuable ones are how the candidate thinks when speaking and what the candidate does after a mistake.

5.1 What a screen is for

Definition 5.1 (Technical screen)

A technical screen is a short technical interview, by telephone or video, that decides whether a candidate goes on to the final round. It has a pass-or-fail outcome and a short rubric, usually two to four questions of increasing difficulty.

A screen asks less than a final round and forgives less. Its interviewer has thirty to forty-five minutes and one decision to make, so the rubric records a few things: did the candidate reach answers, did the candidate explain them, did the candidate check them, and would a colleague enjoy an afternoon of this. It does not record how much the candidate knows beyond the questions asked. Video removes most of the non-verbal channel, so the spoken account of the reasoning is almost all the interviewer has.

5.2 Thinking aloud

Definition 5.2 (Think-aloud protocol)

The think-aloud protocol is the practice of saying one’s reasoning while solving a problem, as it happens, without explaining it to the listener after the fact. The term comes from the psychology of problem solving, where verbal reports of this kind are recorded as data about the solver’s thinking (Ericsson and Simon).

The interviewer cannot score what the interviewer cannot hear. A silent minute followed by a correct number scores less than a minute in which the candidate says what the problem is, what method applies, and what the first step gives; and a wrong final number after a correct spoken method is often scored as a slip, not a failure. Thinking aloud is a skill with a structure.

Method 5.3 (Thinking aloud in a screen)

Restate the question in one sentence, with the assumptions you will make (“fair die, independent rolls”).
Name the method before using it (“first-step analysis on two states”, “linearity with indicators”).
Compute in steps small enough to say, writing the intermediate numbers where the interviewer can see them.
Check (the next section) and say the check.
Conclude with the number and one sentence of interpretation, then stop talking.
Silence is allowed when announced: “give me twenty seconds to set this up”.

5.3 Sanity checks

Definition 5.4 (Sanity check)

A sanity check is a fast test of an answer against a property it must have: a bound, a unit, a sign, a symmetry, a limiting case, a small case that can be computed by hand, or an order of magnitude from a second route. It cannot prove an answer right; it can prove an answer wrong in seconds.

Method 5.5 (The sanity checks, in the order to try them)

Range: a probability in $[0,1]$ ; an expectation between the smallest and largest outcomes; a variance non-negative; a correlation in $[-1,1]$ .
Units and sign: money, time and counts in the right units; a price that rises with the thing that should raise it.
Limits: set a parameter to 0, 1 or infinity and see whether the formula gives the obvious answer.
Small case: $n = 1$ or $n = 2$ by hand.
Symmetry: swapping two roles that should not matter must not change the answer.
Second route: an order of magnitude by a different argument.

Example 5.6 (A range check)

“What is the chance of at least one six in six rolls?” The quick answer $6 \times \tfrac16 = 1$ fails the range check at once: the event is not certain, since all six rolls can miss. The union bound overcounts the overlaps; the complement gives $1 - (5/6)^6 \approx 0.665$ .

Example 5.7 (A limit check)

A candidate proposes $\sigma\sqrt{w_1^2 + w_2^2}$ for the volatility of a two-asset portfolio with weights $w_1, w_2$ and equal volatilities $\sigma$ . Setting the correlation to 1 must return $\sigma(w_1 + w_2)$ , since the assets are then the same; the formula does not depend on the correlation at all, so it is the answer for correlation 0 only.

5.4 Recovering from an error

Errors in a screen are normal; what is scored is what happens next. Say that the check failed, say which step you suspect, redo that step aloud, and state the corrected answer with the check that now passes. Do not restart from the beginning in silence, and do not argue that the wrong answer was nearly right.

Asking for a hint is also allowed. It costs a little on the rubric and saves the screen: a candidate stuck for ten minutes gives the interviewer nothing to write down. The good form is specific: “I have set up the two states and I am not sure whether the absorbing state counts as a step; is that the right way to think about it?” A vague “can you help me?” costs more and teaches less.

Coding screens add one more check: run the code on a small case, by hand if there is no interpreter. The function in Listing 5.1 was submitted in a screen; the small case finds its error (Interview question 5.3).

def moving_averages_buggy(prices, k):
    """Average of each window of k consecutive prices (as submitted in the screen)."""
    out = []
    for i in range(len(prices) - k):
        out.append(sum(prices[i : i + k]) / k)
    return out

Listing 5.1. A moving average as submitted in a screen. Run it by hand on four prices with

k = 2

. code/interviews/05-phone-and-technical-screens/python/iv_screen_windows.py

The last five minutes of a screen usually belong to the candidate. One or two questions that show the candidate has thought about the role (what a first year works on, how work is reviewed, what distinguishes people who do well) are worth more than questions whose answers are on the firm’s website.

5.5 Worked answers

Example 5.8 (A screen question answered aloud)

“A fill ratio $X$ and a price improvement factor $Y$ are independent and uniform on $[0,1]$ . What is the chance that their product exceeds one half?”

Aloud. “I’ll draw the unit square, $X$ across and $Y$ up. The product exceeds a half above the hyperbola $y = 1/(2x)$ , which only enters the square for $x$ above a half. So the area is the integral from a half to one of $1 - 1/(2x)$ , which is $\tfrac12 - \tfrac12\ln 2$ , about $0.5 - 0.347 = 0.153$ . Checks: for a general level $c$ the same steps give $1 - c + c\ln c$ ; at $c = 0$ that is 1 and at $c = 1$ it is 0, both right; and the answer must be below $1/4$ , because a product above a half needs both numbers above a half. So 0.15 is plausible.” Three things score beyond the number: the picture chosen out loud, the integral set up before it is computed, and two limit checks in ten seconds.

Example 5.9 (Recovering from a wrong first answer)

“How many rolls of a die, on average, until you see a six immediately followed by a roll that is not a six?”

Aloud. “Two consecutive specified rolls, so 36?” The candidate stops: “That’s the answer for two sixes in a row, where a failure sends me back to the start. Here it doesn’t: if a six is followed by another six, I’ve still just rolled a six. Let me redo it. Waiting for the first six takes 6 rolls on average. From ‘just rolled a six’, each roll ends the wait with probability 5/6 and otherwise leaves me in the same state, so 6/5 more rolls. Total $6 + 6/5 = 7.2$ . Check: it has to be at least 6, since I need a six first, and much less than 36, since the second roll is easy. 7.2 passes both.” The interviewer writes down the recovery, not the false start: the failed check was named, the faulty step identified, and the new answer checked.

Example 5.10 (A range check on units)

“A stock’s daily returns have a standard deviation of 1.5%. What is its annualised volatility?” A candidate who multiplies by 252 says 378% and should hear the alarm at once: a large-cap stock with a volatility of nearly four hundred per cent a year is outside every range they know. Variances add over independent days, so the standard deviation grows like the square root of time: $1.5\% \times \sqrt{252} \approx 1.5\% \times 15.9 \approx 23.8\%$ , an ordinary equity volatility.

5.6 Question bank

Interview question 5.1 ★ trader • market maker

A candidate says that the expected value of the larger of two fair dice is 3.5. Without computing, why is that wrong? Then compute it.

Solution

Solution of Interview question 5.1.

The larger of two dice is at least the first die and strictly larger whenever the second is higher, so its expectation exceeds the first die’s 3.5: a range check. $\P(\max \le k) = (k/6)^2$ , so $\P(\max = k) = (2k - 1)/36$ and $\E[\max] = \sum_k k(2k-1)/36 = 161/36 \approx 4.47$ .

What the interviewer is looking for: a one-sentence argument that rejects the answer, then the distribution of the maximum.

Interview question 5.2 ★ researcher, risk • systematic fund

Ten independent alerts each fire on a given day with probability 0.15. A colleague says the chance that at least one fires is 1.5. Say what is wrong in one sentence and give the right number.

Solution

Solution of Interview question 5.2.

A probability cannot exceed one: adding the ten probabilities counts the days on which several alerts fire more than once. The complement gives $1 - 0.85^{10} \approx 0.803$ .

What the interviewer is looking for: the range check and the complement.

Interview question 5.3 ★ developer • proprietary firm

Run Listing 5.1 by hand on the prices 10, 11, 12, 13 with $k = 2$ . What does it return, what should it return, and how would you fix it so that it also runs in $O(n)$ time?

Solution

Solution of Interview question 5.3.

With four prices and $k = 2$ the loop runs for $i = 0, 1$ and returns $[10.5, 11.5]$ ; the windows are $[10, 11], [11, 12], [12, 13]$ , so the answer should be $[10.5, 11.5, 12.5]$ . There are $n - k + 1$ windows and the loop stops one early (with $n = k$ it returns nothing). Fix the bound to range(len(prices) - k + 1), and for $O(n)$ keep a running sum: add the entering price, subtract the leaving one. Guard $k \le 0$ and $k > n$ . The chapter’s code tests the fix against a direct computation on 200 random cases.

What the interviewer is looking for: running a small case by hand, the count of windows, and the running-sum improvement.

Interview question 5.4 ★★ trader, risk • bank

Two assets each have a volatility of 20% and their correlation is 0.5. A candidate estimates the volatility of an equally weighted portfolio at 25%. Which check rejects this at once? What is the right figure?

Solution

Solution of Interview question 5.4.

A portfolio’s volatility cannot exceed the weighted average of its components’ volatilities, reached only at correlation 1; that average is 20%, so 25% is impossible. The variance is $0.25 \times 0.04 + 0.25 \times 0.04 + 2 \times 0.25 \times 0.5 \times 0.04 = 0.03$ , so the volatility is $\sqrt{0.03} \approx 17.3\%$ .

What the interviewer is looking for: the bound from the triangle inequality and the two-asset variance formula.

Interview question 5.5 ★★ researcher, developer • any

You conjecture in a screen that a $2 \times n$ board can be tiled by $2 \times 1$ dominoes in $2^{n-1}$ ways. Test it on small cases before the interviewer does. What is the right count?

Solution

Solution of Interview question 5.5.

Small cases: $n = 1$ gives 1, $n = 2$ gives 2, $n = 3$ gives 3, where the conjecture says 4. It is refuted at $n = 3$ . The last column is either one vertical domino, leaving a $2 \times (n-1)$ board, or two horizontal dominoes, leaving $2 \times (n-2)$ : $T_n = T_{n-1} + T_{n-2}$ with $T_1 = 1$ , $T_2 = 2$ , the Fibonacci numbers 1, 2, 3, 5, 8, 13.

What the interviewer is looking for: testing a conjecture on small cases before stating it, and the recursion on the last column.

Interview question 5.6 ★★ bank, researcher • bank

Twelve minutes into a derivation you are stuck, and there are fifteen minutes left. What exactly do you say, and what should you avoid?

Solution

Solution of Interview question 5.6.

Say where you are and what blocks you, specifically: “I have the SDE for the discounted price and I need its expectation; I am not sure whether the Itô term vanishes here. Would it help to take the expectation before applying the formula?” Then use the hint and finish aloud. Avoid silence, avoid restarting from scratch without saying why, and avoid bluffing a step you cannot justify. If the time runs out, state the answer you would expect and how you would check it: an interviewer can score a correct plan.

What the interviewer is looking for: a specific request for help, and continued structured progress.

Interview question 5.7 ★★★ trader • market maker

You roll a die and may roll once more, keeping the second result if you do. A candidate argues: “I re-roll half the time; when I keep the first roll it averages 5; so the game is worth 4.5.” Find the error, give the value, and give the value when up to three rolls are allowed.

Solution

Solution of Interview question 5.7.

The error is in the re-roll branch: when you re-roll you get a fresh die worth 3.5 on average, not 4. Re-roll when the first roll is 1, 2 or 3 (below 3.5): the value is $\tfrac12 \times 5 + \tfrac12 \times 3.5 = 4.25$ , which is $17/4$ . With three rolls, the last two are worth 4.25, so keep a first roll of 5 or 6 and re-roll otherwise: $\tfrac13 \times 5.5 + \tfrac23 \times 4.25 = 14/3 \approx 4.67$ .

What the interviewer is looking for: backward induction on the continuation value, and spotting the double-counted average.

Interview question 5.8 ★★★ researcher, developer • systematic fund

You state in a screen that the expected number of rolls of a fair die until every face has appeared is about 14.7. The interviewer asks how you would check that in thirty seconds without a computer. What do you say?

Solution

Solution of Interview question 5.8.

Three checks. Bounds: at least 6 rolls, and the last new face alone takes 6 rolls on average, so the answer is well above 6. Structure: after $k$ faces are seen, a new one appears with probability $(6 - k)/6$ , so the wait is geometric with mean $6/(6-k)$ ; summing gives $6(1 + \tfrac12 + \dots + \tfrac16) = 14.7$ . Small case: for a coin ( $n = 2$ ) the formula gives $2(1 + \tfrac12) = 3$ , which is right (one toss, then a wait of mean 2 for the other side). The chapter’s simulation over ten seeds agrees.

What the interviewer is looking for: bounds, the geometric decomposition and a small-case check, all stated in seconds.

Sources and further reading

K. A. Ericsson and H. A. Simon, Protocol Analysis: Verbal Reports as Data, MIT Press, 1984; revised edition 1993.
G. Pólya, How to Solve It, Princeton University Press, 1945, on looking back at a solution.

Terms defined in this chapter

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