Summary of key results · M04 Systematic research

Result	Formula
MA crossover as return filter (Prop. 4.1)	$\mathrm{SMA}_m(p)-\mathrm{SMA}_n(p)=\sum_{i=0}^{n-2}w_ir_{t-i}$ , $w_i=\frac{n-1-i}{n}-\frac{(m-1-i)^+}{m}\ge0$ , $\sum w_i=\frac{n-m}2$
Linear rule expectation (Prop. 4.2)	$\mathbb E[x_tr_{t+1}]=\mu^2\sum w_i+\sum w_i\gamma_{i+1}$
RSI identity (Prop. 4.3)	$\mathrm{RSI}=50+50\,W(d)/W(\lvert d\rvert)$
Overlapping horizons (Prop. 4.4)	$\operatorname{Var}(\bar Y)\approx h^2\sigma^2/N$ vs naive $h\sigma^2/N$
Breakeven win rate (Prop. 4.5)	$p^*=(1+c)/(1+W)$ , $c$ in units of $R$
Lo–MacKinlay (Prop. 4.6)	$\mathbb E[\pi]=\frac{N-1}{N^2}\operatorname{tr}\Gamma_1-\frac1{N^2}(\mathbf 1^\top\Gamma_1\mathbf 1-\operatorname{tr}\Gamma_1)+\sigma^2_\mu$
Momentum beta (Prop. 4.7)	slope of $\beta_i$ on past return $=\sigma^2_\beta R_m/(\sigma^2_\beta R_m^2+\sigma^2_e)$ , sign of $R_m$
Vectorised backtest (Prop. 4.8)	$E^o_t=E^o_{t-1}(1+a_{t-1}(g_t-1))(1-k\lvert a_t-\tilde a_t\rvert)$ , $a_t=w_{t-1}$
Survivorship (Prop. 4.9)	$\mathbb E[R\mid R>-d]=\mu+\sigma\phi(a)/(1-\Phi(a))$ , $a=(-d-\mu)/\sigma$
Cost drag (Prop. 4.10)	$\mu_n=\mu_g-c\,\mathrm{TO}$ ; $c^*=\mu_g/\mathrm{TO}$
Sharpe SE (Prop. 4.11)	$\operatorname{SE}(\hat S)=\sqrt{(1-\gamma_3S+\frac{\gamma_4-1}4S^2)/T}$ ; normal: $\sqrt{(1+S^2/2)/T}$
Annualisation (Prop. 4.12)	$\eta(q)=q/\sqrt{q+2\sum_{k=1}^{q-1}(q-k)\rho_k}$
PSR	$\Phi\big((\hat S-S^*)\sqrt{T-1}/\sqrt{1-\hat\gamma_3\hat S+\frac{\hat\gamma_4-1}4\hat S^2}\big)$ ; PSR(0.1, 252) = 0.942987
MinTRL	$1+(1-\hat\gamma_3\hat S+\frac{\hat\gamma_4-1}4\hat S^2)\,(z_{1-\alpha}/(\hat S-S^*))^2$
Expected max (Prop. 4.13)	$\mathbb E[\max]\approx\sqrt V[(1-\gamma)\Phi^{-1}(1-\frac1N)+\gamma\Phi^{-1}(1-\frac1{Ne})]$ ; 1.5746, 2.5306, 3.2551 for $N=10,100,1000$
DSR	$\mathrm{PSR}(S_0)$ with $S_0$ the expected max
Bonferroni / Holm (Props. 4.14–4.15)	$p_i\le\alpha/m$ ; step-down $p_{(k)}\le\alpha/(m-k+1)$ ; FWER $\le\alpha$
BH (Prop. 4.16)	largest $k$ with $p_{(k)}\le k\alpha/m$ ; FDR $\le m_0\alpha/m$ (independence)
Bonferroni $t$ cut-offs ( $\alpha=5\%$ )	1.96 ( $m=1$ ), 2.81 (10), 3.48 (100), 3.76 (300)
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