Summary of key results · M02 Statistics of returns

Result	Formula
Simple and log returns	$R_t = P_t/P_{t-1} - 1$ , $r_t = \ln(1+R_t)$ ; $r$ adds over time, $R$ adds across assets
Portfolio log-return gap	$r_p - \sum w_i r_i \approx \frac12\sum w_i (r_i - \bar r)^2$
Volatility drag	$E\ln(1+R) \approx \mu - \sigma^2/2$ ; exact $m = \ln(1+\mu) - s^2/2$ under lognormality
GBM	$\ln(S_T/S_0) \sim N((\mu - \sigma^2/2)T, \sigma^2 T)$ ; median $S_0e^{(\mu-\sigma^2/2)T}$ ; $P(S_T < ES_T) = \Phi(\sigma\sqrt T/2)$
AM–GM gap	$\ln(1+G) \approx \ln(1+A) - s^2/2$
$k$ -period variance	$\sigma^2[k + 2\sum_{j=1}^{k-1}(k-j)\rho_j]$
Variance ratio	$VR(q) = 1 + 2\sum_{j=1}^{q-1}(1 - j/q)\rho_j$ ; AR(1): $1 + \frac{2\phi(q(1-\phi) - 1 + \phi^q)}{q(1-\phi)^2} \to \frac{1+\phi}{1-\phi}$
VR tests	$z = (\widehat{VR}-1)/\sqrt{2(2q-1)(q-1)/(3qT)}$ ; $z^* = \sqrt T(\widehat{VR}-1)/\sqrt{\hat\theta}$
Mean estimation	$\operatorname{sd}(\hat\mu) = \sigma/\sqrt T$ (span, not frequency); $t \approx S\sqrt T$ ; $T_{t=2} \approx 4/S^2$ years
Moment s.e. under normality	$\operatorname{sd}(\hat S) \approx \sqrt{6/n}$ , $\operatorname{sd}(\hat K) \approx \sqrt{24/n}$ ; $JB = n(\hat S^2/6 + \hat K^2/24) \sim \chi^2_2$
Power-law moments	$E\lvert X\rvert^p < \infty \iff p < \alpha$
Hill estimator	$\hat\alpha_k = [\frac1k\sum_{i\le k}\ln X_{(i)} - \ln X_{(k+1)}]^{-1}$ , s.e. $\approx \alpha/\sqrt k$
Stochastic volatility kurtosis	$\kappa_r = \kappa_z E\sigma^4/(E\sigma^2)^2 \ge \kappa_z$
VaR / ES	Normal: $\text{VaR}_c = -\mu + \sigma\Phi^{-1}(c)$ , $\text{ES}_c = -\mu + \sigma\phi(\Phi^{-1}(c))/(1-c)$ ; $t_\nu$ : $-\mu + \sigma\sqrt{(\nu-2)/\nu}\,t_\nu^{-1}(c)$
Diversification floor	$\sigma_p^2 = \sigma^2(\rho + (1-\rho)/n) \to \rho\sigma^2$
Forbes–Rigobon	$\rho^* = \rho\sqrt{(1+\delta)/(1+\delta\rho^2)}$

Notation used in later modules

$R_t$ simple return, $r_t$ log return; $\mu$ , $\sigma$ annualised arithmetic mean and volatility; $S$ annual Sharpe ratio; $VR(q)$ variance ratio; $\alpha$ tail index; $\kappa$ kurtosis (not excess) and $K$ excess kurtosis.