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  1. 047An equity fund has a 25% chance of a negative year, independently each year. What is the chance that an investor who holds it for five years sees at least one losing year?Probability and expected valueWarm upFund research and ratingsIndian AMCs

    Try it first

    Your first guess: the chance of at least one losing year in five?

    Show the worked solution

    About 76%. The easy route is the opposite event. Five positive years in a row needs a 75% chance to come up five times: 0.75 to the fifth is about 23.7%. Every other outcome includes at least one losing year, so the chance is 1 minus 0.237, about 76.3%. A small yearly chance of loss becomes close to a sure thing over an ordinary holding period.

    Why work with the opposite event?

    Ask a cricket fan the chance that a bowler concedes at least one boundary in an over, and the quick way is to ask the chance he concedes none. At least one is messy to count directly, because it covers one loss, two losses and every combination; none is a single clean path, so compute none and subtract from one. Here none means five positive years, each with a 75% chance, independent of the others: 0.75 times itself five times is 23.7%.

    Each year keeps only 75% of the clean streaks that came before100%yr 075.0%yr 1x0.7556.2%yr 2x0.7542.2%yr 3x0.7531.6%yr 4x0.7523.7%yr 5x0.75Chance that every year so far was positiveAt least onelosing year76.3%Five positiveyears: 23.7%Over five years1 - 0.75^5
    The chance that every year so far has been positive falls by a quarter each year, from 75% after one year to 23.7% after five, so the chance of at least one losing year in five is about 76.3%.
    The relationship
    P(at least one loss)=1−(1−p)n=1−0.755≈0.763P(\text{at least one loss}) = 1 - (1-p)^n = 1 - 0.75^5 \approx 0.763
    pthe chance of a losing year, 25%
    nyears held, 5
    What it says in wordsThe chance of at least one losing year is one minus the chance that every year is positive.

    Why does an adviser care about this number?

    Because it sets the client's expectation before the first bad year arrives. Over five years a losing year is the likely case, not the unlucky one, so a client who has not been told this reads the first negative year as something gone wrong. On the same assumptions the expected number of losing years in five is 1.25, the chance of exactly one is about 40%, and over ten years the chance of at least one rises to about 94%. A losing year is a calendar-year label; it says nothing about whether the five-year return was positive.

    State the two assumptions. The 25% is an illustration, not a measured figure for any market, and real yearly returns are not fully independent: bad years cluster in some periods. Clustering lowers the chance of at least one losing year slightly while making the losing stretches longer. The method survives both caveats; the exact figure does not.

    Where candidates lose it

    The trap is answering 25%, as if holding longer did not create more chances to see a loss, or adding 25% five times and reaching 125%. Both come from treating at least one as a simple sum.

    The quieter loss is forgetting the independence assumption. Say it in one breath with the answer; it is what makes 0.75 to the fifth valid.

    What the interviewer asks next

    • What is the chance of exactly two losing years in five?
    • How many years would you have to hold for the chance of at least one losing year to pass 90%?
    • Why might the chance of a losing five-year period be far lower than the chance of a losing year?