Debt Capital Markets puzzles, solved step by step

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All topicsLeverage, coverage and cash flow9Mental maths and numeracy8Estimation and market sizing7Logic and brainteasers8Cost of capital and valuation riddles7Bond pricing and yield7Compounding, PIK and fees6Issuance and refinancing arithmetic8Credit spreads and default probability8Duration and convexity8Capital structure and recovery8Probability and expected value10Yield curve and forward rates6
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  1. 004What is the angle between the hour hand and the minute hand of a clock at 3:15?Logic and brainteasersWarm upSyndicate desks

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    Answer inside ten seconds.

    Show the worked solution

    7.5 degrees. The minute hand at 15 minutes points exactly at the 3, which is 90 degrees from 12. The hour hand moves half a degree every minute, so at 3:15 it sits at 3 times 30 plus 15 times 0.5, which is 97.5 degrees. The gap is 97.5 minus 90, or 7.5 degrees.

    Why is zero the wrong answer?

    Zero comes from picturing a clock as two independent pointers that jump. The hour hand moves continuously: it covers 30 degrees between one number and the next over 60 minutes, so it moves half a degree every minute. At quarter past, it has done a quarter of its journey from 3 to 4. Think of a train between two stations: fifteen minutes into an hour long run, you are not still standing on the first platform.

    At 3:15 the hour hand has already left the 3121256789101134gap 7.5 degreesminute hand (dark), hour hand (green, shorter)Minute hand15 min x 6 degrees90Hour hand3 x 30 + 15 x 0.597.5Gap between them97.5 minus 907.5The hour hand moves 30 degrees an hour,so half a degree every minute.
    At 3:15 the minute hand points exactly at the 3, 90 degrees from 12, while the hour hand has moved a quarter of the way to the 4, to 97.5 degrees, leaving a 7.5 degree gap between them.

    What is the general method, so any time works?

    Measure both hands from 12 in degrees. The minute hand sits at 6 degrees times the minutes; the hour hand sits at 30 degrees times the hours plus half a degree times the minutes. Subtract, and if the result is over 180, take 360 minus it to get the smaller angle. Saying the method before the number is what the interviewer is listening for, because the next question will be a harder time.

    The relationship
    θ=∣ 30h+0.5m−6m ∣=∣ 30h−5.5m ∣=∣ 90−82.5 ∣=7.5∘\theta = \left|\,30h + 0.5m - 6m\,\right| = \left|\,30h - 5.5m\,\right| = \left|\,90 - 82.5\,\right| = 7.5^\circ
    hthe hour, here 3
    mthe minutes, here 15
    5.5how many degrees a minute the minute hand gains on the hour hand
    What it says in wordsThe minute hand gains 5.5 degrees a minute on the hour hand, starting 30 degrees behind for every hour on the clock.

    Why does a capital markets desk ask this?

    It is a speed and care test, not a clock test. The question checks whether you notice that two things are moving when the obvious reading has only one moving. The same slip in a desk setting is quoting a yield as if the price had not moved since the morning, or pricing accrued interest as if the settlement date were today. After the answer, one sentence on the 5.5 degrees a minute rule shows you can generalise.

    Where candidates lose it

    Zero is the whole trap. It comes from answering the picture in your head rather than the mechanism, and it is said fast because the question sounds too easy to need thought.

    The second loss is the follow up. Candidates who guessed 7.5 without the half degree a minute rule freeze on 9:45 or 2:20. Learn the one line formula and the times take seconds.

    What the interviewer asks next

    • What is the angle at 9:45?
    • How many times a day do the hands overlap exactly, and why is it not 24?
    • At what exact time after 3:00 do the hands first overlap?