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  1. 029Rs 1 was invested in a broad stock index 30 years ago. If yearly log returns are independent with mean 7% and standard deviation 16% (illustrative inputs), give a median and a 95% interval for what it is worth today.Statistics and estimationHardOld Mission CapitalChicago · 2025

    Try it first

    Which is the best central 95% range for the Rs 1 today?

    Show the worked solution

    Median about Rs 8.2; 95% interval roughly Rs 1.5 to Rs 45. Log returns add, so after 30 years the log of wealth has mean 30 x 0.07 = 2.1 and standard deviation 0.16 x √30 = 0.88. The median is e to the 2.1, about 8.2. The band is e to the power 2.1 plus or minus 1.96 x 0.88. In rupees it is lopsided, and the mean, about Rs 12, sits above the median.

    Why work in log returns rather than percentage returns?

    Pay rises compound: 10% and then another 10% is 21%, not 20%. Logs turn that multiplication into addition. Log returns add across years, so the 30-year log return is a sum of 30 yearly pieces, and a sum of independent pieces is close to normal. With mean 0.07 and standard deviation 0.16 a year, the sum has mean 2.1 and variance 30 x 0.16 squared, so a standard deviation of 0.16 x √30, about 0.876. The spread grows with the square root of time, not with time.

    The relationship
    ln⁡W30∼N(30μ, 30σ2)W30∈e 2.1 ± 1.96×0.876\ln W_{30} \sim N(30\mu,\ 30\sigma^2) \qquad W_{30} \in e^{\,2.1 \,\pm\, 1.96 \times 0.876}
    W_30value of the Rs 1 after 30 years
    mu = 0.07mean yearly log return, an illustrative input
    sigma = 0.16standard deviation of the yearly log return
    1.96the number of standard deviations that cuts off 2.5% in each tail of a normal
    What it says in wordsBuild the interval for the log of wealth, where it is symmetric, then exponentiate the two ends.
    Symmetric in the exponent, lopsided in rupees0.5125102050100Value of Rs 1 after 30 years, log scalemedian Rs 8.2mean Rs 12.0Rs 1.5Rs 45.5central 95%Each end is the median divided or multiplied by 5.6The same band on an ordinary rupee scale01020304050Rs 6.7 belowmedianRs 37.3 above the medianRs
    On a log scale the 95% band is symmetric around the median of Rs 8.2, running from Rs 1.5 to Rs 45.5; on an ordinary rupee scale the same band reaches Rs 6.7 below the median and Rs 37.3 above it, and the mean of Rs 12.0 sits right of the median.

    Why is the band so lopsided, and where does the mean sit?

    Symmetric in the exponent means lopsided in rupees. Going 1.96 standard deviations down divides the median by e to the 1.72, a factor of 5.6; going the same distance up multiplies by 5.6. Dividing and multiplying by the same factor leaves Rs 6.7 of room below the median and Rs 37.3 above it. The same skew separates mean from median. The mean of a lognormalA variable whose logarithm is normally distributed; it is always positive and skewed to the right. variable is e to the power (mean plus half the variance), about Rs 12.0 here, because a few very good paths pull the average up while most paths finish below it.

    Close with the limits. The 7% and 16% are illustrative inputs, not a claim about any real index. The calculation assumes independent years and constant volatility; real markets have fat tails and calm and stormy regimes, so treat the band as a floor on the true uncertainty. What the interviewer is testing is whether you scale the mean with t and the volatility with √t, and exponentiate only at the end. One useful extra: the chance the Rs 1 is worth less than Rs 1 is the chance the log falls below zero, about 0.8%.

    Where candidates lose it

    The most common slip is building the interval in rupees: take 8.2 and add and subtract a symmetric amount, which can even run below zero. Build it in logs and exponentiate the two ends.

    The second is scaling the 16% by 30 instead of √30, which gives a log standard deviation of 4.8 and a band from paise to crores. Variance adds across years; standard deviation grows with the square root.

    What the interviewer asks next

    • What is the probability the Rs 1 is worth less than Rs 1 today?
    • If you are given the average percentage return rather than the average log return, how do you convert?
    • How does the band change over a 10-year horizon?

    Asked at Old Mission Capital, Prop Trading, Chicago, 2025 (Wall Street Oasis): Confidence interval of portfolio value if you invested $1 in S&P 500 30 years ago

  2. 069Daily returns are drawn from a normal with mean zero and standard deviation 1% on 90% of days, and from a normal with mean zero and standard deviation 4% on the other 10%. What are the overall standard deviation and the kurtosis of daily returns?Statistics and estimationHardTwo SigmaNew York · 2025

    Try it first

    What is the kurtosis of the mixture?

    Show the worked solution

    The standard deviation is sqrt(2.5), about 1.58%, and the kurtosis is 12.72, against 3 for a normal. Moments of a mixture are weighted averages of the pieces' moments. The variance is 0.9 x 1 + 0.1 x 16 = 2.5. A normal's fourth moment is 3 sigma^4, so the fourth moment is 3 x (0.9 x 1 + 0.1 x 256) = 79.5. Kurtosis is 79.5 / 6.25 = 12.72.

    Why does mixing two normals create fat tails?

    Think of a city's daily traffic: most days are ordinary, and a few days a year there is a festival or a strike and everything is wild. Averaged across the year, the typical day looks calmer than the average suggests, and the extreme days are far more extreme than a single bell curve would allow. Mixing a calm regime with a rare wild one concentrates most days near zero and puts the rest far out, which is exactly what kurtosis measures: a high peak with heavy tails. Every piece is normal; the mixture is not.

    Same variance, very different tails: kurtosis 12.7 against 3-6%-3%0%3%6%mixture, peak 0.37normal,sd 1.58%daily return11e-21e-41e-60%2%4%6%8%10%mixture: 1 day in 75normal: 1 in 6,766Density on a log scale; beyond 6% shown
    With the same 1.58% standard deviation, the mixture is more peaked than the normal and its tail is far heavier: a daily move beyond 6% comes about 1 day in 75 under the mixture against about 1 day in 6,766 under the normal.

    How do you compute the moments without integrating?

    Condition on the regime. Any moment of a mixture is the weighted average of that moment in each regime, because the density itself is the weighted average of the two densities. The second moment is 0.9 x 1^2 + 0.1 x 4^2 = 2.5, so the standard deviation is about 1.58%. For the fourth, use the fact that a normal with standard deviation sigma has fourth moment 3 sigma^4: 3 x (0.9 x 1 + 0.1 x 256) = 79.5. Kurtosis is the fourth moment over the variance squared.

    The relationship
    σ2=∑iwiσi2=2.5κ=3∑iwiσi4(∑iwiσi2)2=3×26.56.25=12.72\sigma^2 = \sum_i w_i \sigma_i^2 = 2.5 \qquad \kappa = \frac{3\sum_i w_i \sigma_i^4}{\big(\sum_i w_i \sigma_i^2\big)^2} = \frac{3 \times 26.5}{6.25} = 12.72
    w_ithe regime weights, 0.9 and 0.1
    sigma_ithe regime standard deviations, 1% and 4%
    kappakurtosis, the fourth moment divided by the variance squared; 3 for any normal
    What it says in wordsAverage the variances and the fourth moments across regimes, then compare the fourth moment with the squared variance.

    What does the number mean for risk?

    Kurtosis is 3 times the ratio of the average of sigma^4 to the square of the average of sigma^2, so any variation in volatility pushes kurtosis above 3, and the more uneven the regimes, the further it goes. The practical cost shows in the tails. A risk model that fits one normal with a 1.58% standard deviation expects a move beyond 6% about once in 6,766 days, roughly once every 27 years of trading; the mixture produces one about once in 75 days, several times a year. This is the simplest model of volatility clustering, and the reason daily returns on real assets show kurtosis well above 3.

    Where candidates lose it

    The trap is answering 3 because each piece is normal, or averaging the two standard deviations to 1.3% and treating the mixture as one normal. Mixtures average densities and moments, not shapes or standard deviations.

    The second slip is forgetting the factor of 3 in a normal's fourth moment, or reporting excess kurtosis without saying so. Say which you mean: kurtosis 12.72, excess kurtosis 9.72.

    What the interviewer asks next

    • What mix of the two regimes maximises the kurtosis for a fixed overall variance?
    • If the two regimes had different means, what would happen to the skew?
    • How would you estimate the two regime volatilities from a year of daily returns?

    Asked at Two Sigma, Quantitative Research, New York, 2025 (Wall Street Oasis): They asked a couple questions involving Mixture Gaussians (e.g., probability density and moments).