Summary of key results · M14 Portfolio and going live

Result	Formula
Frontier constants	$A = \mathbf 1^\top\Sigma^{-1}\mathbf 1$ , $B = \mathbf 1^\top\Sigma^{-1}\mu$ , $C = \mu^\top\Sigma^{-1}\mu$ , $D = AC - B^2$
GMV	$w = \Sigma^{-1}\mathbf 1/A$ , $\sigma^2 = 1/A$ , $\mu = B/A$
Frontier	$w(m) = [(C - Bm)\Sigma^{-1}\mathbf 1 + (Am - B)\Sigma^{-1}\mu]/D$ , $\sigma^2 = (Am^2 - 2Bm + C)/D$
Tangency	$w_T = \Sigma^{-1}(\mu - r_f\mathbf 1)/\mathbf 1^\top\Sigma^{-1}(\mu - r_f\mathbf 1)$ ; $S_T^2 = (\mu - r_f\mathbf 1)^\top\Sigma^{-1}(\mu - r_f\mathbf 1)$
Two-fund separation	frontier affine in $m$ ; with $r_f$ , everyone holds $r_f$ + $w_T$
Weight sensitivity	$\partial x/\partial\mu = \Sigma^{-1}$ ; two assets: amplification $1/(1-\rho^2)$
Optimal shrinkage	$\delta^* = \mathbb E\|S - \Sigma\|^2/(\mathbb E\|S - \Sigma\|^2 + \|\Sigma - F\|^2)$
Euler risk contributions	$RC_i = w_i(\Sigma w)_i/\sigma$ , $\sum RC_i = \sigma$
ERC	$\min \tfrac12y^\top\Sigma y - \sum b_i\ln y_i$ , $w = y/\mathbf 1^\top y$
Inverse vol = ERC	iff correlation-matrix row sums are equal
Vol targeting	$S_{VT} = \mu\mathbb E[1/\sigma]/\sqrt{1 + \mu^2\operatorname{Var}(1/\sigma)}$
Conditional correlation	$\rho_A = \rho\sqrt k/\sqrt{1 + \rho^2(k-1)}$
Two strategies	$S^* = \sqrt{(S_1^2 + S_2^2 - 2\rho S_1S_2)/(1-\rho^2)}$ ; 50/50 equal risk: $(S_1 + S_2)/\sqrt{2(1+\rho)}$
$N$ strategies	$S\sqrt{N/(1 + (N-1)\bar\rho)} \to S/\sqrt{\bar\rho}$
Drawdown bound	exposure $m(W - (1-d)M)$ ⇒ DD < $d$ for continuous paths
Prospect theory	$v(x) = x^{0.88}$ or $-2.25(-x)^{0.88}$ ; $w(p) = p^\gamma/(p^\gamma + (1-p)^\gamma)^{1/\gamma}$ , $\gamma = 0.61$ , $\delta = 0.69$
Disposition	concave gains ⇒ sell winners; convex losses ⇒ hold losers
Odean measures	PGR, PLR; 0.148 vs 0.098
Trades needed	$N \ge (2s/E)^2$ ; with power $1-\beta$ : $N \ge ((2 + z_{1-\beta})s/E)^2$
Familywise false positive	$1 - (1 - \alpha)^k$