Part I · Foundations · M02 · week 1 · 16 h

Statistics of returns

Returns as the object of study: aggregation rules, volatility drag, horizon scaling and variance ratios, what sampling error lets you estimate, the stylized facts, heavy tails and the Hill estimator, volatility clustering, VaR, and dependence between assets.

Start reading · Learning outcomes

Builds on M01 Market structure

Prerequisites M01 (order types, costs, how an index is quoted). Probability to the level of moment-generating functions, the delta method and the CLT; basic time series (stationarity, autocovariance). Time: ~16 hours (notes 7 h, problems 6 h, code 3 h). Primary texts: Campbell, Lo & MacKinlay, The Econometrics of Financial Markets (1997), ch. 1–2; Tsay, Analysis of Financial Time Series (3rd ed., 2010), ch. 1, 3, 7; Cont (2001), "Empirical properties of asset returns: stylized facts and statistical issues".

Throughout, numbers marked "Nifty sample" were computed for this module from daily closes of the Nifty 50 price index (Yahoo Finance ticker ^NSEI), 17 Sep 2007 to 4 Nov 2026, 4,693 daily returns, retrieved 6 Nov 2026. Yahoo data are unofficial; the closes used in the case studies were cross-checked against the press reports cited there. To reproduce the figures, put a Nifty 50 close series in the toolkit cache (NSE's daily index files ind_close_all_DDMMYYYY.csv carry it) and read it with toolkit.data.load("NIFTY", source="cache"), then apply the functions in this module's exercises (exercises/modules/M02-statistics-of-returns/). Without a cached file, load returns deterministic synthetic data, which is not the Nifty and must not be quoted.

Learning outcomes

Convert between simple and log returns, aggregate each correctly across time and across assets, and state the error made when you use the wrong one.
Prove that the expected log growth of a risky asset is approximately $\mu - \sigma^2/2$ , that it is exact under lognormality, and compute the gap between arithmetic and geometric mean returns for any volatility.
Derive the variance of a $k$ -period return under AR(1) dependence, hence the population variance ratio $VR(q)$ , and compute and interpret the Lo–MacKinlay statistics $z$ and $z^*$ on real data.
Derive the sampling standard errors of the mean, skewness ( $\sqrt{6/n}$ ) and excess kurtosis ( $\sqrt{24/n}$ ) under normality, and show that a $t$ -statistic of 2 on a mean return needs about $4/S^2$ years of data for an annual Sharpe ratio $S$ .
Derive the Hill estimator as a Pareto maximum-likelihood estimator, estimate the tail index of Nifty returns, and explain why the answer near 3 makes sample kurtosis unreliable.
Show that a martingale-difference return with stochastic volatility is serially uncorrelated yet has autocorrelated squares and excess kurtosis.
Compute Normal and Student- $t$ Value-at-Risk at matched volatility, and explain why the two agree at 99% and disagree at 99.9%.
Derive the Forbes–Rigobon bias in conditional correlation and separate it from genuine correlation breakdown in crises.

Sections ≈ 2 h 30 min of study

Foundation 3 sections · ≈ 25 min
Core 7 sections · ≈ 1 h 10 min
Advanced 5 sections · ≈ 55 min

Learning outcomes Foundation 5′
2.1 Prices, returns and the two aggregation rules Foundation 10′
2.2 Volatility drag: arithmetic versus geometric growth Core 10′
Figure Volatility drag: arithmetic mean versus compound growth
2.3 Horizon scaling and the variance ratio Advanced 15′
Figure Variance ratio against horizon
Lab Variance ratios and aggregational Gaussianity on the Nifty
2.4 Sampling error: what you can and cannot estimate Core 15′
Figure What T years of data can estimate
2.5 The stylized facts Core 10′
2.6 Heavy tails, power laws and the Hill estimator Advanced 10′
Figure Heavy tails: Student-t against the normal
2.7 Volatility clustering Advanced 10′
Lab Stylized facts of Nifty daily returns
2.8 Value-at-Risk: Normal versus Student- $t$ Advanced 10′
2.9 Dependence between assets Advanced 10′
2.10 Case studies Core 15′
2.11 A Brownian toolkit: results used before M08 Core 10′
Summary of key results Foundation 10′
Readings and resources Core 5′
Facts that change Core 5′