Returns, Statistical Measures and the Discipline of Inference

Learning objectives

You can:

1. Compute a holding period return from a price change and its income, chain HPRs across sub-periods, and say why chaining is multiplicative rather than additive.
2. Choose among the arithmetic, geometric, weighted and harmonic means by naming the question each one answers, and quantify the arithmetic-minus-geometric gap from the variance alone.
3. Compute a money-weighted return as an internal rate of return on dated cash flows, compute the Modified Dietz approximation to it, and state the conditions under which the approximation breaks.
4. Annualise a return of any period length correctly, show why prorating understates a sub-year return and overstates a multi-year one, refuse to annualise a period shorter than a year, and convert between simple and continuously compounded returns.
5. Separate the five subtractions that stand between a headline return and money a person keeps: fees, taxes, inflation, leverage and the gross-versus-net basis, and price each one.
6. Compute the full descriptive set by hand on a return series, from the median and the quartiles through the sample standard deviation, downside deviation and coefficient of variation to sample skewness and excess kurtosis, and show numerically how leptokurtosis makes a normal-based VaR understate its own promised loss frequency.
7. Run a bootstrap and a jackknife, build a percentile interval that a t-interval cannot honestly give, and name and detect the five sampling biases with a specific fix for each.
8. Fill an ANOVA table for a regression, compute F two independent ways, encode categories as dummies without falling into the dummy trap, and build a prediction interval that widens correctly away from the sample mean.
9. Fit and read a logistic regression, from the log-odds link and the odds ratio to the likelihood-ratio test and why McFadden's pseudo-R-squared must never be read on an ordinary R-squared scale, and compute Spearman rank correlation by hand with ties.
10. Correct a screen for the number of trials that produced it (Bonferroni, Holm, Sidak, Benjamini-Hochberg), deflate the winner's Sharpe ratio against the expected maximum from N attempts, derive the minimum backtest length that Sharpe implies, and (productivity objective: R10) reproduce all of it in Python while keeping the verification no tool performs.

The duality, stated once (R10). The gated skill is the arithmetic: a skewness computed from a column of twenty numbers, a percentile interval read off a sorted vector, a BH threshold ladder checked by eye. The productivity payoff is doing all of it in fifteen lines over a thousand series. The risk sits in between. A statistic is a single number with no internal evidence of its own correctness, so a wrong sample standard deviation and a right one look identical on the page, and the only defence is a second route to the same figure. Every worked example below carries one.

Prerequisites & connections

Builds on. M0.03 supplies the arithmetic of percentages and compounding, used here without re-teaching. M3.01 supplies present value and the internal-rate-of-return solve; the money-weighted return below is an XIRR and nothing more. QM1.01 supplies probability, the normal and lognormal distributions, the Central Limit Theorem and the confidence-interval logic that the resampling section extends; take the last seven sections only after it. QM1.02 supplies the regression build, the four diagnostics and the omitted-variable formula.

Feeds forward. QM1.05 takes the volatility models and the penalised regressions that the model-selection discipline here is a prerequisite for. QD1.02 builds factors and consumes the rank correlation taught here as its rank information coefficient. QD1.03 builds the backtest engine and consumes the multiple-testing and Sharpe-deflation work wholesale; its cross-link table already points here. QD2.03 implements the deflated Sharpe ratio in code against the theory derived here. AA1.03 takes the return definitions into performance measurement proper. DA1.05 is where all of it becomes vectorised.

Several neighbouring nodes own material that is deliberately not re-taught here (R1). QM1.01 owns probability, distributions and the construction of a confidence interval from a sampling distribution; the resampling section assumes all of it and adds only the alternatives. QM1.02 owns hypothesis testing, the regression build, heteroskedasticity, serial correlation, multicollinearity and misspecification; the ANOVA, dummy, interval and logistic sections read the output of a regression and never re-derive the fit. AA1.03 owns the time-weighted-versus-money-weighted fork, its worked example, and the whole risk-adjusted measure family (Sharpe, Treynor, Information Ratio, M-squared, Jensen); what follows borrows the money-weighted definition only, to make the Modified Dietz approximation error visible, and treats the Sharpe ratio purely as an estimator with a sampling distribution. M3.06 owns harmonic-mean aggregation of valuation multiples; the harmonic mean appears here for a different job, averaging purchase prices. QD1.03 owns the backtest engine, QD2.02 owns production VaR systems, and QD1.02 owns factor construction. Cross-link, never restate.