Skewness
Whether a return series' extreme days lean more toward big gains or big losses.
Skewness measures the lopsidedness of a distribution's tails. Zero means the upside and downside tails are mirror images of each other; positive means the rare, extreme days lean toward big gains; negative means they lean toward big losses.
A single stock's returns are usually POSITIVELY skewed — most days are ordinary, but the occasional huge up-day (a takeover bid, a blowout result) outweighs the occasional huge down-day. A diversified portfolio or a market index tends to flip to NEGATIVE skew instead: the upside jumps of individual holdings average out across many companies, while a genuine market-wide crash still hits everything at once. Seeing the sign flip between a single holding and a whole portfolio is expected, not a bug.
Skewness is one of the two shape measures (with excess kurtosis) that feed the Jarque-Bera test and the Cornish-Fisher VaR adjustment.
In StockProof Shown on the Return Distribution section of the portfolio risk page, computed from the same daily-return series as the Value at Risk table above it.
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Excess kurtosis
How much fatter — or thinner — a return distribution's tails are than a normal distribution's.
Kurtosis measures how much of a distribution's variance comes from rare, extreme values rather than ordinary ones. 'Excess' kurtosis subtracts 3, the value a normal distribution always has, so a normal distribution reads exactly 0 on this scale.
Positive excess kurtosis means fatter tails and a sharper peak than a normal distribution: more days clustered near zero, but also more days far more extreme than a normal distribution would predict. This is the single most common way real daily returns depart from normality — a −5 standard-deviation day should happen roughly once every 3.5 million trading days under a true normal distribution, and in practice it happens far more often than that.
Negative excess kurtosis (rarer in real return data) means thinner tails and a flatter shape — fewer extreme days than a normal distribution would predict.
In StockProof Shown beside skewness in the Return Distribution section, and it is the direct reason the parametric Value-at-Risk figure above it can be exceeded more often than its stated confidence promises — see the Kupiec backtest entry.
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Jarque-Bera test
A statistical test for whether a set of returns looks like it came from a normal distribution.
The Jarque-Bera test combines skewness and excess kurtosis into a single number: the further either one sits from the 0-and-0 a true normal distribution would have, the larger the statistic. A small p-value (conventionally below 5%) means the test rejects the idea that these returns are normally distributed.
With a large sample — StockProof typically has several hundred to a few thousand daily returns on file for a holding — this test rejects normality for almost every real return series. That is the expected, ordinary outcome of testing real market data, not evidence that something is unusual about one particular holding.
In StockProof Shown as the first of three normality tests on the Return Distribution section — alongside Shapiro-Wilk and Anderson-Darling, because the three tests weigh evidence differently and can disagree on a shorter window.
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Shapiro-Wilk test
A second, independently-constructed test for whether returns look normally distributed.
Shapiro-Wilk works differently from Jarque-Bera: it compares the actual sorted returns against where a perfectly normal sample of the same size would be expected to fall, and reports how closely they line up as a single statistic, W, between 0 and 1. A W close to 1 looks close to normal; the further below 1, the less the sample resembles one. As with Jarque-Bera, a p-value below 5% rejects normality.
It is generally regarded as one of the more powerful normality tests at spotting a genuine departure from normal with a limited sample — which is why StockProof shows it alongside, rather than instead of, Jarque-Bera and Anderson-Darling: three tests built on different logic are less likely to all be fooled the same way.
In StockProof Shown as the second of three normality tests on the Return Distribution section, computed on the exact same daily-return series as the other two.
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Anderson-Darling test
A third normality test that weighs a distribution's tails more heavily than its middle.
Anderson-Darling is built to be especially sensitive to what happens in the tails of a distribution — the rare, extreme days — rather than treating every part of the distribution equally. That makes it a natural complement to Jarque-Bera and Shapiro-Wilk here: the tails are exactly the part of the distribution that matters most for something like Value at Risk, which is trying to describe how bad an unusually bad day could be.
Like the other two tests, it produces a statistic and a p-value, and a p-value below 5% rejects the hypothesis that the returns are normally distributed.
In StockProof Shown as the third of three normality tests on the Return Distribution section. Its p-value is computed from a purpose-built calibration rather than a lookup table with only a handful of fixed points, so it reads as a genuine number rather than 'less than 1%' at the extreme end.
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Kupiec backtest (VaR backtest)
A check on whether a Value-at-Risk model's own stated confidence actually held up.
A 99% Value at Risk figure is a promise: the actual loss should exceed that threshold on only about 1% of days. A Kupiec backtest counts how many days the loss genuinely DID exceed the VaR figure over some window, compares that count to how many the stated confidence promised, and runs a statistical test on the difference. A model whose real exceedance rate is far from its promised rate fails the backtest — the number was framed with the wrong confidence for this data, whatever the reason.
This is a check on the VAR MODEL's own calibration, not on the portfolio. A 'not consistent' backtest result does not mean a portfolio is riskier or safer — it means one particular way of estimating its risk (say, assuming normally distributed returns) does not match its own track record on this data as well as a different method does.
An exceedance is counted correctly only one way: a day's ACTUAL LOSS worse than the VaR threshold. A large GAIN is never an exceedance, however large — VaR is a one-sided statement about the downside.
In StockProof Shown as the Backtest table beneath the Value at Risk table on the portfolio risk page, for all three VaR methods (historical, parametric, Cornish-Fisher) at both confidence levels, over the exact same return series the VaR figures above it use.
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Sharpe ratio standard error
How much uncertainty surrounds an estimated Sharpe ratio, given only a limited stretch of history.
A Sharpe ratio computed from, say, five years of daily returns is an ESTIMATE, not an exact measurement — a different five-year window for the same strategy would produce a somewhat different number, purely from sampling noise. The standard error quantifies how much that estimate could plausibly move, and a 95% confidence interval built from it shows the range the true, underlying Sharpe ratio is likely to sit in.
It is common for that interval to comfortably span zero even when the point estimate itself looks clearly positive or negative — meaning the data on hand cannot actually rule out the possibility that the strategy's true risk-adjusted return is close to zero. A single Sharpe ratio number, read without its own uncertainty, can overstate how confidently it distinguishes a strategy from an ordinary one.
In StockProof Shown directly beneath the Sharpe ratio itself on the portfolio risk page, using the same i.i.d.-returns assumption the normality tests above it check — and often reject, which is the same caveat that applies here.
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