Learning outcomes

After this module you can:

1. 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.
2. 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.
3. 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.
4. 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 $S$.
5. 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.
6. Show that a martingale-difference return with stochastic volatility is serially uncorrelated yet has autocorrelated squares and excess kurtosis.
7. Compute Normal and Student-$t$ Value-at-Risk at matched volatility, and explain why the two agree at 99% and disagree at 99.9%.
8. Derive the Forbes–Rigobon bias in conditional correlation and separate it from genuine correlation breakdown in crises.