Debt Capital Markets puzzles, solved step by step

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  1. 002Estimate 1.01 to the power 365 and 0.99 to the power 365 in your head. What do the two answers say about small daily edges?Mental maths and numeracyHardFixed income asset management

    Try it first

    Gut call first: roughly where does 1.01 to the power 365 land?

    Show the worked solution

    About 38 and about 0.026. A 1% daily gain doubles money roughly every 70 days, so a year holds just over five doublings: 2 to the power 5.2 is about 37.8. A 1% daily loss halves it about every 69 days, leaving about 0.026. The two outcomes end roughly 1,481 times apart: a tiny daily edge, held consistently, is enormous by year end.

    Why is 365 times 1% the wrong way to think about it?

    Picture a savings jar where every evening you add 1% of whatever is already inside. On day 1 that is one paisa per rupee; by day 200 the jar holds about seven times its start, so 1% of it is seven paise of the original rupee. Compounding means each day's 1% is earned on everything built so far, so the growth accelerates rather than adding up in a straight line. Simple addition, 365 times 1%, gives 4.65 and misses almost all of it.

    How do you get to 38 without a calculator?

    Use the rule of 70A shortcut: a quantity growing at r per cent a period doubles in about 70 divided by r periods.. At 1% a day, money doubles in about 70 days. A year of 365 days holds about 5.2 doublings, and 2 to the power 5 is 32 while 2 to the power 5.2 is about 37, so the answer is in the high thirties. The exact figure is 37.78. The same shortcut run backwards gives the loss: 1% down a day halves value about every 69 days, a little over five halvings in a year, which is one part in about 40.

    One per cent a day, up or down, for a year (log scale)1001010.10.01Day 0Day 70Day 140Day 210Day 280Day 365x1.01 a day: doubles every 70 days, 5.2 timesx0.99 a day: halves every 69 days37.80.0261,481xapart
    On a log scale both paths are straight lines: growing 1% a day doubles about every 70 days and ends the year at 37.8, shrinking 1% a day halves about every 69 days and ends at 0.026, leaving the two roughly 1,481 times apart.
    The relationship
    1.01365=e365ln⁡1.01≈e3.63≈37.80.99365=e365ln⁡0.99≈e−3.67≈0.0261.01^{365} = e^{365\ln 1.01} \approx e^{3.63} \approx 37.8 \qquad 0.99^{365} = e^{365\ln 0.99} \approx e^{-3.67} \approx 0.026
    ln 1.01about 0.00995, a shade under 0.01
    ln 0.99about minus 0.01005, a shade beyond minus 0.01
    ethe base of natural growth, about 2.718
    What it says in wordsA small rate compounded many times is close to e raised to the rate times the number of periods.

    Why does a markets interviewer care about this?

    Because desks live on small edges repeated many times: a few basis points of carry every day, a slightly better fill on every trade, a small cost leak on every rebalance. An edge or a leak that looks trivial on one day decides the year, and it cuts both ways. Also notice that up 1% then down 1% does not get you back: 1.01 times 0.99 is 0.9999. Volatility on its own drags a compounding path downwards, which is worth one sentence after the numbers. The limit: real returns are not a steady 1% a day, so treat this as arithmetic about compounding, not a model of any market.

    Where candidates lose it

    The common loss is saying 4.65 for the first number, adding the percentages instead of compounding them. A close second is freezing, because nobody computes a 365th power by hand and the candidate does not reach for a doubling rule.

    The other miss is treating the down case as the mirror image. It is not symmetric: the up path gains about 37 times its start, the down path loses about 97% and can never go below zero. Say both numbers and then the asymmetry.

    What the interviewer asks next

    • What is 1.01 to the power 365 times 0.99 to the power 365, and why is it below 1?
    • How many days of 1% gains does it take to recover from 30 days of 1% losses?
    • Where does this compounding asymmetry show up in a bond portfolio's returns?