Debt Capital Markets puzzles, solved step by step

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  1. 083If a given date is a Monday this year, what day of the week will the same date be one year from now? When does the answer change?Mental maths and numeracyWarm upOaktree Capital ManagementLos Angeles · 2022

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    Tuesday, or Wednesday if 29 February falls in between. A normal year is 365 days, which is 52 weeks and 1 day, so every date moves on one weekday. When the year you step across contains 29 February, it is 366 days, 52 weeks and 2 days, and the date moves on two. The remainder after dividing by 7 does all the work.

    Why does the weekday move by exactly one?

    A clock that runs for 25 hours ends up one hour past where it started, because 24 of those hours bring it round in a full circle. Weeks are the same circle with seven stops. Only the remainder after dividing the day count by 7 moves the weekday, and 365 divided by 7 leaves a remainder of 1. The 52 full weeks bring you back to Monday; the one extra day carries you to Tuesday.

    52 full weeks change nothing; only the leftover days move the weekday365 days = 52 weeks + 1 day+1364 days: every weekday comes round exactly 52 timesTuesday366 days = 52 weeks + 2 days+1+1364 days: every weekday comes round exactly 52 timesWednesdayCount forward from MondayMonTueWedThuFriSatSunnormal year: +129 Feb in between: +2
    A normal year of 365 days is 52 full weeks plus one day, so a Monday date falls on a Tuesday the next year; a year containing 29 February has 366 days and moves the same date to a Wednesday.

    When exactly does the answer become Wednesday?

    When 29 February sits between the two dates. That depends on the date as well as the year. 15 March 2027 is a Monday, and 15 March 2028 is a Wednesday, because 29 February 2028 falls in between: 366 days. Step on again and 15 March 2029 is a Thursday, only 365 days later, because the next 29 February is still years away. Ask whether 29 February lies inside the year you are stepping across, not whether either year is a leap year. Leap years are those divisible by 4, except century years, which count only when divisible by 400.

    The relationship
    365=52×7+1366=52×7+2365 = 52 \times 7 + 1 \qquad 366 = 52 \times 7 + 2
    52 x 7364 days, which leaves the weekday unchanged
    +1, +2the leftover days that move the weekday on
    What it says in wordsDivide the day count by seven and the remainder is how far the weekday moves.

    Why would a credit interviewer bother with this?

    Because bonds and loans live on calendars. A coupon due on the 15th of March lands on a different weekday every year, and when it lands on a weekend or holiday the documents say whether it moves to the next business day and whether interest accrues for the extra days. Day count conventions such as actual/365 turn the leap day into a slightly larger coupon in some years. Quick calendar arithmetic is how you catch a payment schedule that has been rolled wrongly.

    Where candidates lose it

    The instinctive answer is Monday, because it is the same date. It takes one sentence about 52 weeks being 364 days to fix, but candidates who say Monday first rarely recover the room.

    The second loss is the leap-year rule stated too broadly. What matters is whether 29 February falls between the two dates, so a date in January of a leap year and a date in March of the year before can both move by two.

    What the interviewer asks next

    • If today is a Monday, what day is it 100 days from now?
    • A 5-year bond pays annual coupons on 15 March. How many of those five coupon dates can fall on a weekend?
    • Why was 1900 not a leap year but 2000 was?

    Asked at Oaktree Capital Management, Generalist, Los Angeles, 2022 (Wall Street Oasis): If [a date] is a Monday what week of the day would [same date] be a year from now?