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  1. 013A stock index stands at 24,000 and its annual volatility is 16%. What is a reasonable one standard deviation range for where it closes four months from now?Data and statistics intuitionCoreMorgan StanleyTokyo · 2025

    Try it first

    What is one standard deviation of the index move over four months?

    Show the worked solution

    Roughly 21,800 to 26,200. Scale the 16% annual volatility by the square root of the time, not the time itself: four months is a third of a year, the square root of a third is about 0.58, and 16% times 0.58 is about 9.2%. That is about 2,217 points either side of 24,000, a band that holds the close about two times in three.

    Why can you not just say where the index will close?

    You cannot, and saying so is part of a good answer. The honest reply to a where-will-it-close question is a central point and a range, with the range doing most of the work. Today's level is the natural centre over a few months, since the expected drift is small next to the spread. The interviewer reported here asked a sales and trading candidate exactly this, and what they are testing is whether you can turn a volatility number into a range in your head.

    Why does volatility scale with the square root of time?

    Picture someone walking along a lane, taking each step forward or back at random. After four steps they are not usually four steps away; good and bad steps partly cancel, and the typical distance is about two steps, the square root of four. Price moves behave the same way: variance adds up with time, so the standard deviation grows with the square root of time. A third of a year is not a third of the annual volatility; it is the square root of a third, about 58% of it.

    The relationship
    σT=σT=16%×4/12=16%×0.577=9.24%\sigma_T = \sigma \sqrt{T} = 16\% \times \sqrt{4/12} = 16\% \times 0.577 = 9.24\%
    σannual volatility, 16%
    Ttime in years, 4/12
    σ_Tone standard deviation of the move over T
    What it says in wordsMultiply the annual volatility by the square root of the fraction of a year.
    Four months ahead: one standard deviation is 9.2%, not 5.3%19,600-2 sd21,800-1 sd24,000today26,200+1 sd28,400+2 sdabout 68%of outcomes1 sd = 16% x sqrt(4/12) = 9.2%= 2,217 index pointsRed bracket, wrong: 16% x 4/12 = 5.3%, 22,720 to 25,280
    Over four months one standard deviation is about 9.2%, so roughly two thirds of outcomes fall between about 21,800 and 26,200. Scaling the volatility by time instead of the square root of time gives a band of only 22,720 to 25,280, which is far too narrow.

    How do you check it, and what are you leaving out?

    Check by another route. Monthly volatility is 16% over the square root of 12, about 4.6%, and four months is the square root of 4, which is 2, times that: about 9.2%. Same answer. Two standard deviations, about 19,600 to 28,400, covers roughly 95% of outcomes, and naming that wider band shows you know a one standard deviation range will be wrong a third of the time. Say the limits too: prices compound, so the upper side is slightly wider, about 21,900 to 26,300 on a log basis; the drift is ignored; and real index returns have fatter tails than a bell curve.

    Where candidates lose it

    The common loss is scaling linearly: a third of 16% is 5.3%, which gives a band from about 22,700 to 25,300. It sounds precise and is far too narrow, because it ignores the way ups and downs cancel.

    The second loss is answering the literal question with a single number for the close. Give the centre, the range and the confidence that goes with it, then stop.

    What the interviewer asks next

    • What one standard deviation range would you give for one week ahead?
    • Where would you get a better volatility number than the historical one?
    • Why might the downside of the range be more likely to be breached than the upside?

    Asked at Morgan Stanley, Sales and Trading, Tokyo, 2025 (Wall Street Oasis): What do you think this index will close at by the end of the year (4 months from now)