Summary of key results · M06B Funds and ETFs

Result	Formula
NAV	$(A - L)/N$ , with fees accrued daily: $E_t = c\,NA_t^{\text{pre}}/365$ (6B.1)–(6B.2)
Stale-NAV purchase transfer	$T_{\text{buy}} = S\varepsilon/(1 + f)$ , $f = S/(Nv_s)$ (6B.3)
Stale-NAV redemption transfer	$T_{\text{red}} = -R\varepsilon$ (6B.4)
Swing price	$\text{NAV}(1 \mp s)$ ; $s = k$ leaves stayers whole (6B.5)
Total expense ratio	BER + brokerage + transaction costs + statutory levies (6B.6)
Fee drag	$\rho_T = \big[(1 + r - c_1)/(1 + r - c_2)\big]^T$ , between $e^{-T\Delta c/(1 + r - c_1)}$ and $e^{-T\Delta c/(1 + r - c_2)}$ (6B.7)–(6B.8)
SIP fee drag	$\sum_m\omega_m\gamma^m$ , between $\gamma^n$ and $\gamma$ (6B.9)
AMC revenue	$\sum_s y_s\bar A_s$ (6B.10)
AMC profit beta	$\beta_\Pi = \beta_A(1 - v)y_eA_e/\Pi$ ; all-equity $\beta_A(1 + F/\Pi)$ (6B.12)
PV of a SIP customer's fees	$y_mP\,\frac{q}{1-q}\,\frac{1 - a^n}{1 - a}$ , $a = \frac{1-\lambda}{1+k}$ , $q = (1+g)a$ (6B.13)
SIP average cost	harmonic mean $\le$ arithmetic mean; $H \approx A(1 - \sigma_\delta^2)$ (6B.14)–(6B.15)
Lump sum versus averaging	$\mathbb E[W_L] - \mathbb E[W_S] > 0$ when $\mu > r_f$ (6B.16)
Performance measures	Sharpe, Jensen, TE and IR, active share (6B.17)–(6B.20)
Luck	expected false discoveries $N(1 - \Phi(z))$ (6B.21)
Berk–Green	$q^* = (\phi - f)/b$ , zero expected net alpha; flow $= \kappa\eta/b$ (6B.22)–(6B.23)
Tracking	$TD = R_I - R_F$ , $TE = \sqrt{252}\,\sigma(r_F - r_I)$ (6B.24)
TD decomposition	$c + w_c(R_I - r_c) + \kappa - \ell + \delta - \sum_i d_iR_i$ (6B.25)
ETF premium band	$I(1 - C_r) \le P \le I(1 + C_c)$ (6B.26)
Crossover horizon	$T^* = \ln[(1 - k_I)/(1 - k_E)]/\ln[(1 + r - c_E)/(1 + r - c_I)] \approx (k_E - k_I)/(c_I - c_E)$ (6B.27)
Debt NAV change	$-D_p\Delta y + \frac12C_p\Delta y^2$ ; horizon return and break-even (6B.28)–(6B.29)
Run dilution	$D(\rho) = \rho k/[(1 - k)(1 - \rho)]$ ; redeem if $\rho > b(1 - k)/[k + b(1 - k)]$ (6B.30)–(6B.31)
Portfolio overlap	$\sum_i\min(w^A_i, w^B_i)$ (6B.32)
Active expense ratio	$[c - (1 - AS)c_I]/AS$ (6B.33)