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Quant Finance · Topic lesson

Statistics

The five statistics calculations quant interviews come back to: scaling risk with the square root of time, combining two assets, splitting one asset's risk into market and own, the standard error of a mean, and whether a Sharpe ratio is real.

5 chapters About 56 minutes0 of 5 complete
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Statistics questions in quant interviews are about risk and evidence. Risk: how volatility scales across time, adds across assets, and splits between the market and the stock. Evidence: how sure you can be of an average, and therefore of a Sharpe ratio, and how a search over many strategies manufactures false certainty. The square root appears in every chapter, and it is the one idea to hold onto: independent things add in variance, so their standard deviations grow with the root.

Two old lessons are replaced by this one. Expectation and variance, which the old moments lesson covered, are taught with a game in the Probability lesson's expected-value chapter. Fit, inference and overfitting, the old regression lesson, are chapters three, four and five here: beta and R², the standard error, and the multiple-testing arithmetic that makes a data-mined Sharpe suspect.

  • Annualizing volatility. √252, horizons, dollar moves, a one-day VaR, and when √t fails.
  • Two-asset variance. Covariance in %², the portfolio formula, the minimum-variance weight, the long-short spread, the implied correlation.
  • Beta and R². β = ρ σs ÷ σm, systematic against idiosyncratic, alpha on excess returns, the correlation a beta implies.
  • Standard error. s ÷ √n, the 95% interval, the t-statistic, days for a margin.
  • Sharpe significance. t = SR × √years, years needed, minimum Sharpe, √252, false positives from a search.
The rule that solves most questions

Independent things add in variance. Days add to a year, so σ scales with √time. Assets add with a cross term, so covariance decides whether a mix or a spread is calmer. Observations add to a mean, so the standard error falls with √n and a Sharpe's t-statistic is SR × √years. Every time a square root appears in this topic, that is why.

Risk, then evidence

Each step points to the chapter that practices it.

  1. 1
    Scale risk in time

    √252; horizons; VaR.

    Chapter 1
  2. 2
    Combine risk

    Covariance; the minimum; the spread.

    Chapter 2
  3. 3
    Split risk

    Beta, R², what a hedge leaves.

    Chapter 3
  4. 4
    Measure an average

    Standard error; the interval; t.

    Chapter 4
  5. 5
    Judge a track record

    SR × √years; how many were tried.

    Chapter 5

Chapters

1

The square root of time

10 min

Independent days add in variance, so volatility scales with the square root of time. Daily to annual, annual to a ten-day horizon, a standard deviation to a value at risk: one rule, run in every direction.

  • Annualize a daily volatility with √252, and go back
  • Scale volatility to any horizon and turn it into a dollar move
  • Compute a one-day 95% value at risk
  • Solve for the days a target move needs, and say when √t fails
2

Two assets: covariance, the minimum and the spread

12 min

The variance of a sum is the sum of the variances plus twice the covariance. Attach weights and every portfolio calculation is that identity, including the one where a weight is negative.

  • Turn a correlation into a covariance, in %²
  • Compute a two-asset portfolio's variance and volatility
  • Find the long-only minimum-variance weight, and say when it is a corner
  • Compute the volatility of a long-short spread and the correlation a portfolio's volatility implies
3

Beta, R-squared and what a hedge cannot touch

12 min

Regress a stock on the market and its risk splits in two. Beta is correlation rescaled by the ratio of the volatilities; R-squared is the correlation squared; the rest is idiosyncratic. Alpha is whatever is left after beta is paid.

  • Compute beta from correlation and the two volatilities, and see why it is not correlation
  • Split a stock's variance into systematic and idiosyncratic with R²
  • Compute alpha on excess returns
  • Recover the correlation a reported beta implies
4

How sure is the average? The standard error

11 min

A sample mean is a random number with its own volatility: the daily standard deviation over the square root of the days. Halving the uncertainty takes four times the data, which is why strategy returns take years to tell apart from zero.

  • Compute the standard error of a mean daily return
  • Build a 95% confidence interval and read it against zero
  • Compute a t-statistic and say whether it clears 1.96
  • Solve for the days a target precision needs, and the standard deviation an interval implies
5

Is a Sharpe ratio significant? t = Sharpe × √years

11 min

A Sharpe ratio is a mean over a volatility, so testing it is testing the mean, and the t-statistic is the annual Sharpe times the square root of the years. That one line explains why track records are so hard to judge.

  • Turn a Sharpe ratio and a track record into a t-statistic
  • Solve for the years a Sharpe needs, or the Sharpe a track record needs, to clear a hurdle
  • Annualize a daily Sharpe with √252 and say why sampling more often does not help
  • Count the false positives a search over many strategies produces