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  1. 022An institutional investor asks for an 8% expected annual return with 10% volatility. Assuming returns are normal, what is the chance of a losing year, and what volatility would keep that chance below 10%?Probability and base ratesCoreMSCIAnonymous interview candidate in · 2013

    Try it first

    Roughly how often does this portfolio lose money in a year?

    Show the worked solution

    About 21%, and volatility would need to fall to about 6.2%. A loss means a return below zero, which is 8 points, or 0.8 standard deviations, under the mean. About 21.2% of a normal distribution lies below that, roughly one year in five. For a 10% chance, zero must sit 1.28 standard deviations below the mean, so volatility must be 8 divided by 1.28, about 6.2%.

    How do a return target and a volatility target fix the chance of loss?

    Think of a commute that takes 40 minutes on average but varies from day to day. Whether you are ever late for a 50 minute deadline depends on how much it varies, not only on the average. The chance of a losing year depends on how many standard deviations the expected return sits above zero: here 8 divided by 10, which is 0.8. Look up 0.8 in the normal table and about 21.2% of years fall below zero. The investor who hears 8% and thinks losses are rare is wrong one year in five.

    Same 8% target, two volatilities: the area below zero is the chance of a losing year-20%-10%0%8%20%30%Annual returnloss | gaintarget 8%Volatility 10%P(loss) = 21.2%Volatility 6.2%P(loss) = 10.0%Loss chance = N(-mean / vol)
    With an 8% expected return and 10% volatility, 21.2% of the return distribution falls below zero, while cutting volatility to 6.2% narrows the curve until exactly 10% of years show a loss.
    The relationship
    P(R<0)=N ⁣(−μσ)=N(−0.8)≈21.2%σmax⁡=μ1.2816≈6.2%P(R < 0) = N\!\left(-\frac{\mu}{\sigma}\right) = N(-0.8) \approx 21.2\% \qquad \sigma_{\max} = \frac{\mu}{1.2816} \approx 6.2\%
    muthe expected annual return, 8%
    sigmathe annual volatility
    Nthe standard normal cumulative distribution
    1.2816the number of standard deviations that leaves 10% in the lower tail
    What it says in wordsDivide the expected return by the volatility, and the normal table tells you how often returns fall below zero.

    What would you actually set as targets, and what is wrong with this model?

    Set the targets as a pair, and state the trade-off. If the investor cannot tolerate losing more than one year in ten, then either volatility must come down to about 6.2%, which usually lowers the expected return too, or the loss tolerance must be stated over a longer horizon. Over five years the mean grows five times but the volatility only by the square root of five, so the chance of a losing five-year stretch is much lower. Asking about the horizon is the question a good risk manager raises first.

    Then name the model's limits. Real returns have fatter left tails than a normal curve, so the chance of a large loss is understated; returns are not independent from year to year; and the 8% expected return is an assumption, not a promise. A drawdown limit, such as no more than a 15% fall from peak, is often more useful to an institution than a probability of a losing year.

    Where candidates lose it

    The trap is assuming that a positive expected return makes losing years rare. At 0.8 standard deviations above zero, they happen about one year in five.

    The second miss is solving for volatility with the wrong number from the normal table. For a 10% tail you need 1.28 standard deviations, not 1.645, which is the 5% tail.

    What the interviewer asks next

    • What is the chance of a negative return over five years with the same targets, assuming independent years?
    • The investor adds a limit of no more than a 15% loss in any year. What volatility does that imply at 99% confidence?
    • Why might a pension fund care more about a drawdown limit than a volatility target?

    Asked at MSCI, Risk Management, Anonymous interview candidate in, 2013 (Wall Street Oasis): What risk-return targets would you set for an institutional investor?