Appendix A · Quick-reference formula sheet

How to use this formula sheet. This appendix condenses every operational formula in the course into two dense tables — one per area — using the exact notation of modules 7 through 9. Each row gives the concept, the formula, the symbols to fix before computing, and the usage caution; the most frequently consulted formulas also appear as display blocks, outside the table, so they can be copied at a glance. Global conventions: losses are positive unless stated otherwise; $r_t$ = log return; $\alpha$ = tail level (0.01 for 99% VaR); $z_\alpha = \Phi^{-1}(1-\alpha)$; $\varphi, \Phi$ = standard normal density and CDF; $w$ = weights, $\Sigma$ = covariances, $\mathbf{1}$ = vector of ones. And the golden rule of the course: no formula is executed by the LLM — the LangChain agent orchestrates, explains, and alerts; the number is always produced by a deterministic tool.

13.1 Risk and performance

13.1.1 Formula table: delta-normal/historical/Monte Carlo VaR, ES/CVaR (Rockafellar-Uryasev), realized volatility, EWMA λ=0.94, GARCH(1,1), MDD, Sharpe/Sortino/Omega/Calmar/Sterling/Ulcer, beta, TE/IR + Grinold's law, skew/kurtosis, Cornish-Fisher, Component/Marginal VaR (Euler), Kupiec/Christoffersen LR, PSR/DSR

Key formulas for quick reference:

$$\mathrm{VaR}_{\alpha}(h) = V\left(z_{\alpha}\,\sigma\sqrt{h} - \mu h\right), \qquad z_{0{,}05}=1{,}645,\;\; z_{0{,}01}=2{,}326$$ $$\mathrm{ES}_{\alpha}(L) = \frac{1}{\alpha}\int_{0}^{\alpha} \mathrm{VaR}_{u}(L)\,du = \mathbb{E}\!\left[L \mid L \ge \mathrm{VaR}_{\alpha}(L)\right], \qquad \mathrm{ES}_{\alpha}^{\text{normal}} = \mu + \sigma\,\frac{\varphi(z_{\alpha})}{\alpha}$$ $$\sigma_t^2 = \lambda\,\sigma_{t-1}^2 + (1-\lambda)\,r_{t-1}^2 \;\;(\text{EWMA},\ \lambda=0{,}94) \qquad\qquad \sigma_t^2 = \omega + \alpha\,\varepsilon_{t-1}^2 + \beta\,\sigma_{t-1}^2 \;\;(\text{GARCH}(1,1))$$ $$S = \frac{R_p - R_f}{\sigma_p}, \qquad S_{\mathrm{anual}} = S_{\mathrm{diario}}\sqrt{252} \;\;(\text{solo bajo i.i.d.})$$ $$z_{CF} = z + \frac{1}{6}(z^2 - 1)S + \frac{1}{24}(z^3 - 3z)K_{\mathrm{exceso}} - \frac{1}{36}(2z^3 - 5z)S^2$$ $$LR_{POF} = -2\ln\!\left[\frac{(1-p)^{T-N}p^{N}}{\left(1-\frac{N}{T}\right)^{T-N}\left(\frac{N}{T}\right)^{N}}\right] \;\sim\; \chi^2_1$$ $$\widehat{DSR} = \Phi\!\left( \frac{(\widehat{SR}-SR_0)\sqrt{T-1}}{\sqrt{\,1-\hat{\gamma}_3\,\widehat{SR}+\frac{\hat{\gamma}_4-1}{4}\widehat{SR}^{2}\,}} \right), \qquad SR_0 = \sqrt{\mathbb{V}\!\left[\{\widehat{SR}_k\}\right]}\left[(1-\gamma_{EM})\,\Phi^{-1}\!\left(1-\tfrac{1}{N}\right) + \gamma_{EM}\,\Phi^{-1}\!\left(1-\tfrac{1}{Ne}\right)\right]$$

Table A.1 — Risk and performance formula sheet: formula, key symbols, and usage cautions

Usage notes — risk and performance. When each formula fails: delta-normal VaR systematically underestimates risk under fat tails and negative skewness ($S<0$, $K_{\mathrm{exceso}}>0$ in equity) and does not capture derivative non-linearities; historical VaR relies on a single path and at 99% with 250 days has only ~2.5 tail observations; Monte Carlo shifts the risk to the stochastic generator — the comparative evidence shows that Gaussian MC underestimates tails relative to filtered historical simulation. The $\sqrt{h}$ rule and Sharpe annualization with $\sqrt{252}$ are valid only under i.i.d. returns: with serial autocorrelation, Lo (2002) documents annualized Sharpe ratios overstated by up to 65% (Alpha Vantage). Fixed course conventions: $\lambda=0{,}94$ daily (0.97 monthly; effective memory ≈ 17 days), per the RiskMetrics Technical Document (Godel Discount); EWMA does not revert to the mean — if reversion is needed, use GARCH(1,1) with $\alpha+\beta<1$. Sterling is the least standardized formula in the literature: the original version adds 10% to the denominator, but variants with $-10\%$ or with $R_f$ circulate; always state the convention used (±10% tolerance) (Alpha Vantage). Cornish-Fisher stops being monotonic for $\lvert S\rvert\gtrsim 2$ or $K\gtrsim 6$: verify the monotonicity of $z_{CF}$ in $\alpha$ before reporting. Kupiec has low power with small $T$ and does not detect time clustering — which is why it is complemented with Christoffersen. Calmar and MDD depend on the window: a crisis-free history inflates them. And no backtest Sharpe is evidence without DSR > 0.95 after declaring the $N$ trials (PyPI).

13.2 Portfolios and execution

13.2.1 Formula table: Markowitz MVO + GMV + tangency, CAPM, FF3/FF5, Black-Litterman (π=δΣw, posterior, Idzorek ω), Kelly (discrete/continuous/fractional), ERC (Spinu), Ledoit-Wolf δ*, HRP, Almgren-Chriss (trajectory, κ, E(X), V(X)), square-root law, signal cross-sectional z-score

Key formulas for quick reference:

$$\mathbf{w}_{\mathrm{GMV}} = \frac{\boldsymbol{\Sigma}^{-1}\mathbf{1}}{\mathbf{1}^{\top}\boldsymbol{\Sigma}^{-1}\mathbf{1}}, \qquad \mathbf{w}_{\mathrm{tan}} = \frac{\boldsymbol{\Sigma}^{-1}(\boldsymbol{\mu} - r_f\mathbf{1})}{\mathbf{1}^{\top}\boldsymbol{\Sigma}^{-1}(\boldsymbol{\mu} - r_f\mathbf{1})}$$ $$\boldsymbol{\pi} = \delta\,\boldsymbol{\Sigma}\,\mathbf{w}_{m}, \qquad \boldsymbol{\mu}_{BL} = \left[(\tau\boldsymbol{\Sigma})^{-1} + P^{\top}\boldsymbol{\Omega}^{-1}P\right]^{-1}\left[(\tau\boldsymbol{\Sigma})^{-1}\boldsymbol{\pi} + P^{\top}\boldsymbol{\Omega}^{-1}\mathbf{q}\right]$$ $$f^{*} = \frac{bp - (1-p)}{b} = p - \frac{q}{b} \;\;(\text{discreto}), \qquad f^{*} = \frac{\mu - r_f}{\sigma^{2}}, \quad \mathbf{f}^{*} = \boldsymbol{\Sigma}^{-1}(\boldsymbol{\mu} - r_f\mathbf{1}) \;\;(\text{continuo})$$ $$RC_i = \frac{w_i\,(\boldsymbol{\Sigma}\mathbf{w})_i}{\sigma_p} = \frac{\sigma_p}{n} \quad \forall\, i \;\;(\text{ERC}), \qquad \sigma_p = \sum_{i=1}^{n} RC_i$$ $$x_j = \frac{\sinh\!\big(\kappa\,(T-t_j)\big)}{\sinh(\kappa T)}\,X, \qquad \kappa \;\sim\; \sqrt{\frac{\lambda\,\sigma^{2}}{\eta}} \;\;(\text{Almgren--Chriss})$$ $$I(Q) = Y\,\sigma_{D}\left(\frac{Q}{V_{D}}\right)^{1/2}, \qquad Y \approx 0{,}3\!-\!1$$

Table A.2 — Portfolio and execution formula sheet: formula, key symbols, and usage cautions

Usage notes — portfolios and execution. The weak link of MVO is not the QP but $\mu$: errors in means weigh ~11× those of variances (Chopra–Ziemba) and the optimizer systematically maximizes them (Michaud); that is why GMV, ERC, and HRP — which do not use $\mu$ or do not invert $\Sigma$ — tend to win out of sample, and long-only constraints with bounds act as implicit shrinkage (Jagannathan–Ma). In production, never feed the raw sample $\Sigma$: Ledoit–Wolf with closed-form $\delta^{*}$ is the standard before any $\Sigma^{-1}$. In Black-Litterman, $\tau$ scales the confidence in the equilibrium prior: institutional practice is $\tau\in[0{,}025,\ 0{,}05]$ or $\tau = 1/T$, with $\Omega$ calibrated per Idzorek (Github); the views $(P,\mathbf{q},\Omega)$ are the auditable entry point for signals, including LLM-generated ones. Full Kelly maximizes growth but with intolerable drawdowns: in practice it is traded fractionally at 1/4–1/2 — half-Kelly retains ≈ 75% of the growth with ~50% of the volatility (Github). In execution: $\kappa T\ll1$ reproduces TWAP; $\kappa T\gg1$ liquidates front-loaded with half-life $1/\kappa$, and the naive strategy ($\lambda=0$) is never optimal for a risk-averse trader (Blockchain News). The square-root law is concave in $Q$ — "it costs ~1 day of volatility to move 1 day of volume" (🦜️🔗 LangChain): liquidating $X = 1{,}000{,}000$ shares with $Q/V_D = 0{,}25$, $\sigma_D = 2\%$ and $Y=0{,}6$ costs $\approx 0{,}6\times2\%\times0{,}5 = 60$ bp. The cross-sectional z-score requires winsorizing the signal by date before ranking; with LLM signals, explicitly handle the point mass in the neutral class.

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Appendix A of 3 — LangChain for Quantitative Trading. Data and versions verified as of July 2026; prices and market figures subject to change. LangChain 1.3.14 / langchain-core 1.5.2.

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