Methodology — MAISNER

Portfolio Optimization — Mean-Variance

MAISNER implements Markowitz Mean-Variance Optimization (MVO) with several institutional-grade enhancements. The core problem maximizes the Sharpe ratio subject to linear constraints.

Objective Function

Max Sharpe Portfolio

\[ \max_{\mathbf{w}} \frac{\mathbf{w}^\top \boldsymbol{\mu} - r_f}{\sqrt{\mathbf{w}^\top \boldsymbol{\Sigma} \mathbf{w}}} \]

where w = portfolio weights, μ = expected return vector, Σ = covariance matrix, r_f = risk-free rate (3.80%)

Constraints

Standard Constraints

\[ \sum_i w_i = 1, \quad w_i \geq 0, \quad w_i \leq w_{max} \]

Full investment, long-only, individual weight cap (default 35%)

Three portfolios are computed simultaneously: Max Sharpe, Minimum Variance, and Equal Weight. This allows direct comparison of optimization strategies against a naive benchmark.

Parameter	Value	Description
RISK_FREE	3.80%	Risk-free rate (ECB rate proxy)
WEIGHT_MAX	35%	Maximum weight per asset
FREQ	252 / 12	Daily if ≥400 obs, else monthly

Constrained Optimization

The Constrained Optimizer allows portfolio managers to impose explicit structural requirements on individual positions and sectors, then optimizes the remaining free budget using the same RMT-MVO engine. This is the institutional standard for managing client mandates, regulatory constraints, and conviction-driven overrides.

Asset Partition

All assets in the portfolio are classified into one of three sets before optimization begins.

Partition of Assets

\[ \mathcal{A} = \mathcal{L} \cup \mathcal{B} \cup \mathcal{F} \]

𝓛 = locked set (fixed weight: w_i = w_i^lock)
𝓑 = bounded set (per-ticker bounds: lb_i ≤ w_i ≤ ub_i)
𝓕 = free set (standard bounds: 0 ≤ w_i ≤ w_max)

Remaining Budget

Free Budget After Locking

\[ c_{\mathcal{F}} = 1 - \sum_{i \in \mathcal{L}} w_i^{\text{lock}} \]

The optimizer distributes c_𝓕 across 𝓑 ∪ 𝓕. If locked weights sum to more than 1, the problem is infeasible and an error is raised before optimization.

Full Optimization Problem

Constrained Max Sharpe

\[ \max_{\mathbf{w}} \frac{\mathbf{w}^\top \boldsymbol{\mu} - r_f}{\sqrt{\mathbf{w}^\top \boldsymbol{\Sigma} \mathbf{w}}} \] \[ \text{subject to} \quad \sum_i w_i = 1 \] \[ w_i = w_i^{\text{lock}}, \quad i \in \mathcal{L} \] \[ \mathrm{lb}_i \leq w_i \leq \mathrm{ub}_i, \quad i \in \mathcal{B} \] \[ 0 \leq w_i \leq w_{\max}, \quad i \in \mathcal{F} \] \[ \sum_{i \in \text{sector}_k} w_i \leq s_k^{\max}, \quad \forall k \]

Locked positions enter the covariance matrix in full — they affect the risk structure of the free portfolio. Sector caps are applied to the entire weight vector, including locked positions.

Locked positions shift the covariance structure of the optimization subproblem. Adding a zero-weight locked asset changes the T/N ratio in the RMT filter and the Cholesky decomposition — this is mathematically correct behavior, not a bug.

Constraint Types

Type	Syntax	Description
Locked weight	w_i = c	Position fixed at exact percentage. Optimizer cannot move it.
Locked shares	n_i = k	Fixed number of shares. Weight derived from current market price.
Min/max per ticker	lb_i ≤ w_i ≤ ub_i	Optimizer chooses within the specified range.
Sector floor	Σ_sector ≥ s_min	Forces minimum allocation to a sector.
Sector ceiling	Σ_sector ≤ s_max	Caps total allocation to a sector (default 40%).

Infeasibility Handling

If the set of constraints is infeasible (e.g. locked weights sum to more than 1, or sector caps conflict with locked positions), MAISNER raises an explicit error identifying which constraint is violated. It never silently relaxes user-defined constraints.

Random Matrix Theory — Covariance Cleaning

Raw sample covariance matrices are notoriously noisy for typical portfolio sizes. MAISNER applies Marchenko-Pastur filtering to separate signal eigenvalues from noise eigenvalues.

Marchenko-Pastur Distribution

Noise Eigenvalue Bounds

\[ \lambda_{\pm} = \sigma^2 \left(1 \pm \sqrt{\frac{T}{N}}\right)^2 \]

T = observations, N = assets. Eigenvalues within [λ₋, λ₊] are noise and replaced with their mean.

The ratio Q = T/N determines the noise floor. For small portfolios with limited history, RMT cleaning provides substantial improvement in out-of-sample covariance estimates. The cleaned matrix retains the information eigenvalues (genuine risk factors) while suppressing estimation error.

MAISNER uses 2-year daily returns for the covariance matrix. A hybrid approach (10-year monthly + 2-year daily) was tested but produced spurious correlations between unrelated assets — pure 2-year daily provides cleaner signal.

Quality Tilt

Pure Sharpe maximization often overweights assets with recent momentum. MAISNER applies a quality penalty to the MVO objective, tilting the optimizer toward fundamentally strong companies.

Quality Score

Quality-Adjusted Objective

\[ \max_{\mathbf{w}} \; S(\mathbf{w}) - \lambda_Q \sum_i w_i \cdot \mathbb{1}[q_i < q_{thresh}] \]

λ_Q = quality penalty weight (0.05), q_thresh = minimum quality threshold (0.30)

Quality scores are computed from FMP fundamental data: ROE, gross margin, operating margin, and debt coverage ratio. Each metric is normalized and combined into a composite score in [0, 1].

Parameter	Value
LAMBDA_QUAL	0.05
QUALITY_THRESH	0.30
Metrics	ROE, Gross Margin, Op. Margin, Debt Coverage

CVaR Optimization

For multi-asset portfolios including options and non-linear instruments, mean-variance optimization is inappropriate. MAISNER uses Conditional Value-at-Risk (CVaR) minimization via linear programming.

CVaR Definition

Expected Shortfall

\[ \text{CVaR}_\alpha = \mathbb{E}[L \mid L \geq \text{VaR}_\alpha] \]

α = 0.95. CVaR is the expected loss given that losses exceed the 95th percentile VaR.

Correlated Scenario Generation

Cholesky Decomposition

\[ \mathbf{R}_{sim} = \mathbf{L} \mathbf{Z}, \quad \mathbf{\Sigma} = \mathbf{L} \mathbf{L}^\top \]

L = Cholesky factor of the RMT-cleaned covariance matrix. Z = matrix of independent standard normal draws. Produces correlated return scenarios preserving cross-asset dependencies.

Parameter	Value
Scenarios	5,000
Alpha	0.95
Method	LP (linear programming)
Correlation	Cholesky on RMT-cleaned Σ

Monte Carlo Simulation

Long-horizon wealth projections are generated via Geometric Brownian Motion with correlated asset paths.

GBM Asset Path

\[ S_t = S_0 \exp\left[\left(\mu - \frac{\sigma^2}{2}\right)t + \sigma W_t\right] \]

μ = annualised return, σ = annualised volatility, W_t = Wiener process. Portfolio paths use correlated draws via Cholesky decomposition.

Parameter	Value
Paths	10,000
Horizon	10 years
Output	5th / 50th / 95th percentile
Metrics	P(loss), P(2×), Median@10y, Range

Stress Testing

Historical scenario analysis replays actual crisis return distributions against the current and optimised portfolios. Custom stress mode allows arbitrary macro shock specification.

Historical Scenarios

Scenario	Period	S&P 500 Drawdown
2008 GFC	Sep 2008 – Mar 2009	−56.8%
2020 COVID	Feb 2020 – Mar 2020	−33.9%
2022 Rate Shock	Jan 2022 – Oct 2022	−25.4%
2000 Dot-com	Mar 2000 – Oct 2002	−49.1%

Advanced Stress — Custom Shocks

Users can define simultaneous shocks across equity markets (by region/sector), interest rates, FX rates, and individual ticker overrides. Shocks are applied as multiplicative return adjustments to portfolio weights.

Leverage Analysis

The Leverage Analyzer quantifies the risk and return impact of applying financial leverage to a portfolio. It combines the Kelly Criterion for optimal leverage sizing with continuous-time volatility decay, margin call analytics, and Monte Carlo liquidation probability.

Kelly Criterion — Optimal Leverage

Kelly Optimal Leverage

\[ f^* = \frac{\mu_p - r_f}{\sigma_p^2} \]

μ_p = annualised portfolio return, r_f = risk-free rate (3.80%), σ_p = annualised portfolio volatility. f* is the leverage ratio that maximises the long-run geometric growth rate. Half Kelly (f*/2) is the standard practical optimum.

Leveraged Return and Volatility

Leveraged Portfolio Metrics

\[ \mu_L = L \cdot \mu_p - (L - 1) \cdot r_m \] \[ \sigma_L = L \cdot \sigma_p \] \[ S_L = \frac{\mu_L - r_f}{\sigma_L} \]

L = leverage ratio, r_m = margin borrowing rate. Net return subtracts the financing cost on the borrowed fraction (L−1) of portfolio value. Volatility scales linearly with leverage.

Volatility Decay (Continuous-Time Drag)

Leverage Decay

\[ \text{decay} = -\frac{1}{2}(L^2 - L)\,\sigma_p^2 \]

Arises from the Itô correction in continuous-time GBM. At L=1 decay is zero; at L=2 it equals −½σ². Grows quadratically — a 3× levered position on a 20% vol portfolio loses ~3.6% per year to decay alone before financing costs.

Margin Call Threshold

Margin Call Drop

\[ \Delta_{MC} = \frac{1 - m}{L} \]

m = maintenance margin fraction (US RegT: 0.25, EU ESMA: 0.50). Δ_MC is the portfolio drawdown from entry that triggers a margin call. At 2× leverage under RegT: Δ_MC = 37.5%.

Monte Carlo Liquidation Probability

MAISNER simulates 5,000 independent GBM paths over a 252-trading-day horizon. Each path uses portfolio drift μ_L and volatility σ_L. A path is classified as a liquidation event if its running minimum touches the margin call threshold at any point during the year.

GBM Path (Daily)

\[ V_{t+1} = V_t \cdot \exp\!\left[\left(\mu_L - \frac{\sigma_L^2}{2}\right)\frac{1}{252} + \sigma_L \sqrt{\frac{1}{252}}\, Z_t\right] \]

Z_t ~ N(0,1), seeded deterministically (seed=42) for reproducibility. Liquidation probability = fraction of 5,000 paths where min(V) ≤ V₀(1 − Δ_MC).

Historical Stress Scenarios

Scenario	S&P 500 Drawdown	Applied to Leveraged Portfolio
2008 GFC	−56.8%	DD × L, then check MC threshold
2020 COVID	−33.9%	DD × L, then check MC threshold
2022 Rate Shock	−25.4%	DD × L, then check MC threshold
2000 Dot-com	−49.1%	DD × L, then check MC threshold
1987 Black Monday	−22.6%	DD × L, then check MC threshold

Kelly f* is an upper bound, not a recommendation. Leverage above Half Kelly introduces path-dependency risk where adverse sequences can permanently impair capital even when expected returns are positive. MAISNER displays Half Kelly and Quarter Kelly alongside f* for this reason.

Parameter	Value	Description
MC_PATHS_LEV	5,000	GBM paths for liquidation probability
MC_HORIZON_LEV	252 days	1-year daily simulation
US Maintenance	25%	RegT maintenance margin requirement
EU Maintenance	50%	ESMA margin requirement
Leverage range	1× – 3×	Table computed at 1.0, 1.5, 2.0, 2.5, 3.0

Stop Loss Calculator

The Stop Loss Calculator derives position-level and portfolio-level exit thresholds from the volatility structure of the portfolio. The methodology follows Giuseppe Paleologo's Advanced Portfolio Management (2021), which frames stop-loss placement as a statistical inference problem rather than an ad-hoc rule.

Position-Level Volatility Stop

Each position's stop is calibrated to its realised volatility (from the RMT-cleaned covariance diagonal) and the chosen horizon. The parameter k controls the width — how many volatility-adjusted standard deviations separate entry from exit.

Vol-Based Stop (Paleologo APM)

\[ \sigma_i^{\text{daily}} = \sqrt{\frac{\Sigma_{ii}}{252}} \] \[ \text{stop\%}_i = k \cdot \sigma_i^{\text{daily}} \cdot \sqrt{T} \] \[ S_i^{\text{stop}} = S_i^{\text{entry}} \cdot (1 - \text{stop\%}_i) \] \[ \text{Dollar Risk}_i = w_i \cdot V \cdot \text{stop\%}_i \]

Σ_ii = annualised variance (RMT covariance diagonal), T = horizon in trading days, k ∈ {1.5, 2.0, 2.5}, V = portfolio value. The square-root scaling places the stop at the k-σ level of the price distribution over the horizon.

Portfolio Drawdown Limit

The portfolio-level stop integrates correlation structure via the full covariance matrix, capturing diversification effects that position-level stops ignore.

Portfolio Drawdown and VaR

\[ \sigma_p^{\text{daily}} = \sqrt{\frac{\mathbf{w}^\top \boldsymbol{\Sigma}\, \mathbf{w}}{252}} \] \[ \text{MaxDD\%} = k \cdot \sigma_p^{\text{daily}} \cdot \sqrt{T} \] \[ \text{VaR}_\alpha = \sigma_p \cdot \sqrt{T/252} \cdot \Phi^{-1}(\alpha) \]

σ_p = annualised portfolio volatility, α ∈ {0.90, 0.95, 0.99}. VaR is the parametric normal-distribution estimate; MaxDD is the k-σ loss bound over the horizon.

Sharpe-Adjusted Stop (Paleologo Key Insight)

Paleologo's central contribution is showing that a strategy with genuine edge (positive Sharpe ratio) should use a tighter stop than a zero-edge strategy. The intuition: if returns are reliably positive, an observed loss of k-σ is stronger evidence of a regime change than it would be for a random walk.

Sharpe-Adjusted Stop

\[ \text{adj\_stop} = \sigma_p^{\text{daily}} \cdot \sqrt{T} \cdot \left( k - \frac{SR \cdot \sqrt{T/252}}{k} \right) \]

T = horizon in trading days (same as position-level formula). SR = annualised Sharpe ratio. The SR term is scaled by √(T/252) — converting the Sharpe to the same horizon as the vol term — while σ^daily·√T gives the k-σ bound in the same units. Non-binding when SR·√(T/252) ≥ k²; a tighter condition than it appears — at T=30 days, k=2.0 requires SR ≥ 2.31 to suppress the stop. Binding stop (adj_stop > 0) tightens the threshold relative to the pure-volatility stop.

Non-binding stop interpretation: if the Sharpe-adjusted stop is non-binding, it means the expected daily drift dominates the volatility over the horizon. Paleologo's prescription is to not exit on pure-vol grounds — only a structural signal justifies exiting a high-edge strategy.

GBM First-Passage Barrier Probability (Analytical)

The probability that a position touches its stop before the horizon is computed analytically using the reflection principle for Geometric Brownian Motion — no Monte Carlo required.

First-Passage Probability (Shreve 2004, reflection principle)

\[ P\!\left(\min_{0 \le t \le T} S_t \le S_0(1-\text{stop\%})\right) = \mathcal{N}(-d_1) + e^{-2\nu\lambda/\sigma^2}\,\mathcal{N}(-d_2) \] \[ \lambda = -\ln(1 - \text{stop\%}), \quad \nu = \mu - \tfrac{\sigma^2}{2}, \quad d_1 = \frac{\lambda + \nu T}{\sigma\sqrt{T}}, \quad d_2 = \frac{\lambda - \nu T}{\sigma\sqrt{T}} \]

λ = log-distance to barrier. ν = μ − σ²/2 is the log-space (physical) drift. Positive drift reduces the barrier probability: the exponent −2νλ/σ² < 0 when ν > 0, dampening the reflection term. The exponent is clamped to [−40, 40] to prevent overflow.

Kelly Connection

The Kelly Criterion provides a natural consistency check: a stop that implies a maximum loss exceeding the Kelly-optimal risk budget is oversized relative to the strategy's edge.

Kelly Fraction and Stop Consistency

\[ f^* = \frac{\mu_p - r_f}{\sigma_p^2} \] \[ \text{Max Loss}_i^{\text{Kelly}} = f^* \cdot V \cdot \text{stop\%}_i \cdot w_i \]

The total Kelly max-loss across positions should not exceed the portfolio drawdown limit. If it does, the stop widths are inconsistent with Kelly-optimal sizing and the positions are oversized.

Re-Entry Level

After a stop is triggered, the re-entry level accounts for the short-term mean-reversion noise band estimated as 0.5-σ over 5 trading days.

Re-Entry Price

\[ S_i^{\text{reentry}} = S_i^{\text{stop}} \cdot \left(1 - 0.5\,\sigma_i^{\text{daily}}\,\sqrt{5}\right) \]

Provides a slightly lower re-entry point after exit, accounting for the expected noise band at the 0.5-σ level over a 5-day window (roughly one trading week).

Parameter	Values	Description
k_multiplier	1.5 / 2.0 / 2.5	Stop width in daily-vol units × √T
horizon_days	30 / 60 / 90	Horizon for √T scaling
confidence α	0.90 / 0.95 / 0.99	VaR confidence level (parametric)
RF	3.80%	Risk-free rate (Kelly denominator)
Covariance source	RMT diagonal	Per-ticker variance from Marchenko-Pastur cleaned Σ

Options Pricing

European Options — Black-Scholes

Black-Scholes Call Price

\[ C = S_0 N(d_1) - K e^{-rT} N(d_2) \] \[ d_1 = \frac{\ln(S_0/K) + (r + \sigma^2/2)T}{\sigma\sqrt{T}}, \quad d_2 = d_1 - \sigma\sqrt{T} \]

American Options — Barone-Adesi Whaley (BAW)

American-style options cannot be priced with Black-Scholes due to early exercise. MAISNER implements the Barone-Adesi Whaley approximation, which provides a closed-form solution by decomposing the American option into a European component and an early exercise premium.

Implied Volatility

When live market prices are available, MAISNER back-calculates implied volatility using Brent's method — a bracketed root-finding algorithm that is guaranteed to converge. If the IV calculation fails (e.g., deep ITM/OTM with negligible time value), the platform falls back to historical volatility.

Bond Analytics

Duration and Convexity

Modified Duration

\[ D_{mod} = \frac{1}{P} \sum_{t=1}^{T} \frac{t \cdot C_t}{(1+y)^{t+1}} \]

P = bond price, C_t = cash flow at time t, y = yield to maturity

Vasicek Interest Rate Model

Bond returns in stress scenarios and portfolio projections use the Vasicek mean-reverting short rate model for interest rate simulation.

Vasicek SDE

\[ dr_t = \kappa(\theta - r_t)\,dt + \sigma_r\,dW_t \]

κ = mean reversion speed, θ = long-run rate, σ_r = rate volatility

Parameter	Value
BOND_MAX	60% of portfolio
CONVEXITY_THRESH	0.05
Credit ratings	AAA → CCC (FMP data)

Factor Analysis

The Factor Explorer computes Information Coefficient (IC) and related statistics for 11 cross-sectional factors.

Information Coefficient

IC — Rank Correlation

\[ IC_t = \text{Spearman}\left(\text{rank}(f_{t}),\; \text{rank}(r_{t+1})\right) \]

Rank correlation between factor scores at time t and forward returns at t+1. Robust to outliers; range [−1, +1].

Information Ratio

ICIR

\[ ICIR = \frac{\bar{IC}}{\sigma_{IC}} \]

Annualised ratio of mean IC to IC volatility. ICIR > 0.5 indicates a reliably predictive factor.

Factor	Category
momentum_12m	Momentum (12-1)
momentum_1m	Short-term reversal
value_composite	P/E, P/B, EV/EBITDA
quality_composite	ROE, margins, debt
low_volatility	12-month realised vol
fcf_yield	FCF / Market cap
earnings_revision	EPS estimate changes
size	Log market cap
dividend_yield	Trailing dividend yield
growth_composite	Revenue + EPS growth
short_interest	Short % of float

Backtesting & Deflated Sharpe Ratio

Walk-Forward Validation

All strategy backtests use walk-forward analysis: the strategy is fitted on a training window and evaluated on an out-of-sample test window that rolls forward in time. This prevents look-ahead bias and provides a realistic estimate of live performance.

Deflated Sharpe Ratio

Multiple testing bias is a major source of overfitting in quantitative strategies. MAISNER computes the Deflated Sharpe Ratio (DSR), which adjusts the observed Sharpe Ratio for the number of trials tested and the non-normality of returns.

Deflated Sharpe Ratio

\[ DSR = \Phi\left(\frac{(\hat{SR} - SR^*)\sqrt{T-1}}{\sqrt{1 - \gamma_3 \hat{SR} + \frac{\gamma_4-1}{4}\hat{SR}^2}}\right) \]

SR* = expected maximum Sharpe Ratio under repeated testing. γ₃ = skewness, γ₄ = kurtosis. DSR > 0.95 indicates the strategy survives multiple-testing correction.

DSR < 0.5 typically indicates the strategy is overfitted to the training data. MAISNER flags strategies with low DSR and warns users before live deployment.

Performance Attribution

MAISNER implements the Brinson-Hood-Beebower (BHB) three-effect decomposition, which explains the difference between portfolio return and benchmark return as the sum of allocation, selection, and interaction effects. The benchmark is a synthetic SPY blend constructed from sector ETF returns weighted by approximate S&P 500 sector weights.

Brinson-Hood-Beebower Decomposition

Let w^p = portfolio sector weight, w^b = benchmark sector weight, R^p = portfolio sector return, R^b = benchmark sector return, R^b_total = total benchmark return.

Allocation Effect

\[ A_s = (w_s^p - w_s^b)(R_s^b - R_{\text{total}}^b) \]

Measures the value added by over- or under-weighting a sector relative to the benchmark. Positive when the portfolio over-weights a sector that outperforms the overall benchmark.

Selection Effect

\[ S_s = w_s^b (R_s^p - R_s^b) \]

Measures the value added by selecting securities that outperform the sector benchmark. Uses benchmark weight to isolate pure security selection skill.

Interaction Effect

\[ I_s = (w_s^p - w_s^b)(R_s^p - R_s^b) \]

Cross-effect from simultaneously over-weighting and outperforming a sector. Represents the combined effect of active allocation and active selection.

Active Return Decomposition

\[ R_{\text{active}} = R^p - R^b = \sum_s (A_s + S_s + I_s) \]

The total active return equals the sum of all sector-level effects. Effects are expressed in basis points (bp = 0.01%).

Benchmark Construction

The sector benchmark uses approximate SPY sector weights (Technology 30%, Financials 13.2%, Healthcare 12.8%, …) applied to sector ETF returns (XLK, XLF, XLV, XLY, XLI, XLC, XLP, XLE, XLB, XLU, XLRE). SPY actual return is also fetched for reference.

Sector resolution uses a two-layer lookup with a 24-hour in-process cache: (1) FMP Professional /stable/profile is queried first for accurate GICS sector classification (e.g. AMZN → Consumer Discretionary, JPM → Financials, JNJ → Healthcare); (2) the internal Stocks.xlsx sector column (normalised from Russian to English) is used as fallback for tickers not found in FMP or for EU-listed securities. The same enriched sector map is applied consistently across the Optimizer, Constrained Optimizer, Stress Tests, and Attribution engine. Tickers that remain unresolved after both lookups are excluded from the BHB decomposition and do not affect total effects — all sector-level effects still sum exactly to active return over the resolved weight subset.

Sector	ETF	SPY Weight
Technology	XLK	30.0%
Financials	XLF	13.2%
Healthcare	XLV	12.8%
Consumer Discretionary	XLY	10.1%
Industrials	XLI	8.7%
Communication	XLC	8.6%
Consumer Staples	XLP	5.9%
Energy	XLE	3.8%
Materials	XLB	2.4%
Utilities	XLU	2.3%
Real Estate	XLRE	2.2%

Portfolio Monitor

The Portfolio Monitor provides real-time P&L tracking, currency exposure breakdown, and option expiry management for user-defined portfolios. Unlike the Analyzer and Optimizer, the Monitor supports multi-asset portfolios (stocks, ETFs, bonds, options, futures, crypto) for all subscription tiers.

Position-Level P&L

P&L Calculation

\[ \text{Cost}_{i,\text{USD}} = Q_i \cdot P_i^{\text{purchase}} \cdot \text{FX}_{C_i \to \text{USD}} \] \[ \text{Value}_{i,\text{USD}} = Q_i \cdot P_i^{\text{live}} \cdot \text{FX}_{C_i \to \text{USD}} \] \[ \text{P\&L}_i = \text{Value}_{i,\text{USD}} - \text{Cost}_{i,\text{USD}} \] \[ \text{P\&L\%}_i = \left(\frac{\text{Value}_{i,\text{USD}}}{\text{Cost}_{i,\text{USD}}} - 1\right) \times 100 \]

Q = quantity, P = price, FX = exchange rate to USD. Live prices are fetched from FMP Professional with per-instrument currency conversion; Yahoo Finance is used as fallback.

Option Valuation

For live options (expiry > today), the Monitor computes an estimated current value as intrinsic value plus an approximate time value derived from a simplified Black-Scholes formula with σ = 25% assumed volatility. Expired options (expiry ≤ today) are marked to zero (conservative) and displayed with an EXPIRED badge — they should be removed from the portfolio.

Option Monitor Value (approximate)

\[ V_{\text{call}}^{\text{intrinsic}} = \max(0, S - K) \] \[ V_{\text{put}}^{\text{intrinsic}} = \max(0, K - S) \] \[ V_{\text{time}} \approx S \cdot N(d_1) \cdot \sigma \cdot \sqrt{T} \cdot 0.4 \] \[ V_{\text{option}} = V^{\text{intrinsic}} + V_{\text{time}} \]

S = underlying spot, K = strike, T = years to expiry. The time value factor 0.4 is a rough ATM approximation. For precise option analytics use the dedicated Options Pricing engine.

Currency Exposure

Currency is inferred from the exchange suffix of each ticker (.DE/.PA → EUR, .L → GBP, .ST → SEK, etc.; no suffix → USD). The currency breakdown shows the percentage of total portfolio value held in each currency. FX rates are fetched from FMP at runtime.

Long/Short Optimizer

The Long/Short Optimizer extends the standard MVO framework to allow negative portfolio weights (short positions). The mathematical structure is identical to the long-only case — the same Sharpe objective, the same RMT-cleaned covariance matrix — with two key adaptations: a borrow cost term subtracted from the expected return of shorted positions, and relaxed weight bounds that permit negative values.

Portfolio Modes

Two net-exposure configurations are available:

Long/Short (net = 1): Σwᵢ = 1. Standard fully-invested portfolio allowing some short positions. Long gross > net; short gross < long gross.
Market Neutral (net = 0): Σwᵢ = 0. Dollar-neutral — long exposure equals short exposure. A minimum gross constraint (Σ|wᵢ| ≥ 0.5) prevents the degenerate all-zero solution.

Borrow Cost Adjustment

Effective Expected Return

\[ \mu_i^{\text{eff}} = \mu_i - b \cdot \mathbf{1}[w_i < 0] \] \[ \mu_p^{\text{eff}} = \sum_i w_i \mu_i^{\text{eff}} = \sum_i w_i \mu_i - b \sum_{w_i < 0} |w_i| \]

b = annual borrow rate (default 0.5%). The borrow cost is subtracted from the expected return contribution of each short position. Long positions are unaffected. This makes high-borrow-rate environments naturally discourage shorting when the alpha is insufficient to cover the cost.

Objective Modes

Mode	Objective	L/S Adaptation
Standard	max (μ_eff − r_f) / σ	Same Sharpe with borrow-adjusted μ; bounds [-max_short, max_long]
Conservative	min wᵀΣw	Minimise portfolio variance; negative weights allowed; tends toward fewer shorts
Balanced	max Sharpe s.t. σ ≤ vol_cap	SLSQP vol constraint; falls back to min-variance if vol_cap is infeasible
Aggressive	max μ_eff	Pure return maximisation; bounds widened to [−1.5×max_short, 1.5×max_long]
Dividend	max Σ(wᵢ>0)×ROIᵢ − 0.3σ	Yield only from long leg; vol penalty prevents pure high-yield concentration

Weight Constraints

Feasible Region

\[ -m_s \le w_i \le m_l \quad \forall i, \qquad \sum_i w_i = \eta, \qquad \text{(MN only:)} \quad \sum_i |w_i| \ge 0.5 \]

mₛ = max_short (default 0.30), mₗ = max_long (default 0.35), η = 1 (long/short) or 0 (market neutral). Aggressive mode uses 1.5× widened bounds. The market-neutral gross constraint prevents the trivial zero-weight solution.

Portfolio Statistics

Gross and Net Exposure

\[ \text{Gross} = \sum_i |w_i|, \qquad \text{Net} = \sum_i w_i, \qquad \text{L/S Ratio} = \frac{\sum_{w_i > 0} w_i}{\sum_{w_i < 0} |w_i|} \]

Gross exposure > 1 indicates leverage. L/S ratio = 1 for market-neutral. Borrow cost = borrow_rate × short_gross.

Volatility and Sharpe (identical to long-only)

Portfolio Variance — unchanged with negative weights

\[ \sigma_p^2 = \mathbf{w}^\top \Sigma \mathbf{w} \] \[ \text{Sharpe} = \frac{\mu_p^{\text{eff}} - r_f}{\sigma_p} \]

The covariance matrix wᵀΣw is always non-negative regardless of the sign of individual weights. RMT-cleaned covariance is used identically to the standard optimizer. Sharpe is computed using the borrow-adjusted expected return.

Upper Barrier Probability for Short Stop-Losses

When L/S portfolio weights are passed to the Stop Loss Calculator, short positions (w < 0) use the upper barrier formula (stop triggered when price rises above entry):

Upper Barrier — Short Position Stop-Out

\[ P\!\left(\max_{0 \le t \le T} S_t \ge S_0(1+\text{stop\%})\right) = \mathcal{N}(-d_2) + e^{+2\nu\lambda/\sigma^2}\,\mathcal{N}(-d_1) \] \[ \lambda = \ln(1 + \text{stop\%}), \quad d_1 = \frac{\lambda + \nu T}{\sigma\sqrt{T}}, \quad d_2 = \frac{\lambda - \nu T}{\sigma\sqrt{T}} \]

Positive drift ν > 0 increases upper-barrier probability (stock trends toward short stop). Note: d₁ and d₂ are swapped vs the long (lower-barrier) formula, and the exponent sign is positive. Stop price for short = entry × (1 + stop_pct).

Long/Short Analyzer

The Long/Short Analyzer evaluates an existing L/S portfolio given the user's actual positions — ticker, quantity, direction, and purchase price. Live prices are fetched at run-time to compute current mark-to-market values, realised P&L, and forward-looking risk metrics using the same RMT covariance infrastructure as the optimizer.

Position Valuation and P&L

Mark-to-Market P&L per Position

\[ \text{PnL}_i = \text{sign}_i \times (\text{qty}_i \times P_i^{\text{live}} - \text{qty}_i \times P_i^{\text{purchase}}) \] \[ \text{sign}_i = \begin{cases} +1 & \text{LONG} \\ -1 & \text{SHORT} \end{cases} \]

Long positions profit when price rises; short positions profit when price falls. Total portfolio P&L = Σ PnL_i.

Portfolio Exposure

Gross, Net, and L/S Ratio

\[ V_i^{\text{live}} = \text{qty}_i \times P_i^{\text{live}}, \qquad V_{\text{gross}} = \sum_i V_i^{\text{live}} \] \[ V_{\text{net}} = V_{\text{long}} - V_{\text{short}}, \qquad w_i = \frac{\text{sign}_i \times V_i^{\text{live}}}{V_{\text{gross}}} \] \[ \text{L/S Ratio} = \frac{V_{\text{long}}}{V_{\text{short}}}, \qquad \text{Net Exposure} = \frac{V_{\text{net}}}{V_{\text{gross}}} \]

Weights are normalised by gross value. Long weights are positive; short weights are negative. Net exposure = 1 for a long-only book; = 0 for market-neutral.

Portfolio Risk (RMT Covariance)

Volatility, Sharpe, and VaR

\[ \sigma_p = \sqrt{\mathbf{w}^\top \Sigma_{\text{RMT}} \,\mathbf{w}}, \qquad \mu_p^{\text{eff}} = \sum_i w_i \mu_i - b \sum_{w_i < 0} |w_i| \] \[ \text{Sharpe} = \frac{\mu_p^{\text{eff}} - r_f}{\sigma_p} \] \[ \text{VaR}_{95\%} = \sigma_p \times z_{0.95} = \sigma_p \times 1.645 \] \[ \text{CVaR}_{95\%} = \sigma_p \times \frac{\phi(z_{0.95})}{1 - 0.95} \approx \sigma_p \times 2.063 \]

Parametric VaR and CVaR under normality. σ_p uses RMT-cleaned covariance (identical to optimizer pipeline). Borrow cost b is deducted from expected return of short positions only. VaR and CVaR are expressed as a percentage of gross portfolio value, annualised.

Stress Scenarios (Beta-Adjusted)

Scenario Impact

\[ \text{Impact} = \sum_i \text{sign}_i \times \beta_i \times r_{\text{mkt}} \times V_i^{\text{live}} \]

For each historical scenario (GFC 2008, COVID 2020, Rate Shock 2022, Dot-com 2000, Black Monday 1987), the position-level impact is computed using the stock's beta (from Stocks.xlsx; default β = 1) and the scenario's market return. Long positions lose when the market falls; short positions gain (hedge effect). The sum shows how well shorts offset long-side losses under each scenario.

Hedging Optimizer

Implements the Yu & Sun (2017) model: "Optimal Hedging with Options and Futures against Price Risk and Background Risk", Math. Comput. Appl. 22(1):5. The model simultaneously optimises a put option hedge and a futures hedge for a commodity or equity position subject to correlated background risk (weather, operational, yield).

Model Structure

The terminal profit from the hedged position is (Equation 1 of the paper):

Π_T = (1 + β·Z_T) · { P_T·Q + [(K − F_T)⁺ − Φ·e^rτ]·h·Q + (F₀·e^rτ − F_T)·H·Q } + (1 − β)·Z_T

Symbol	Meaning
P_T	Spot commodity price at horizon T (lognormal)
F_T	Futures price at horizon T (lognormal, corr ρ₁ with P_T)
F₀	Current futures price
Q	Quantity held
K	Put option strike — grid-searched over [0.7⋅F₀, 1.3⋅F₀]
Φ	Black-Scholes European put price at K
h	Option hedge ratio (units of puts per unit of Q)
H	Futures hedge ratio — grid-searched over [0, 2]
Z_T	Background risk ~ N(0, σ_z²), corr ρ₂ with P_T
β	Background mix: β=0 pure additive, β=1 pure multiplicative
r	Risk-free rate (3.80%)
τ	Hedging horizon in years

Budget Constraint

The option budget is a fraction d of the position value P₀·Q:

h = d · P₀ / Φ

When Φ is near zero (deep in-the-money or very short expiry), h is clamped to zero to avoid a degenerate hedge.

Simulation

N = 10,000 correlated paths are drawn via a factored Cholesky structure. Three independent N(0,1) variates W₁, W₂, W₃ give:

log(P_T/P₀) = (μ_s − σ_s²/2)τ + σ_s√τ · W₁

log(F_T/F₀) = (μ_f − σ_f²/2)τ + σ_f√τ · (ρ₁W₁ + √(1−ρ₁²)·W₂)

Z_T = σ_z · (ρ₂W₁ + √(1−ρ₂²)·W₃)

This exactly reproduces corr(P_T, F_T) = ρ₁, corr(P_T, Z_T) = ρ₂, corr(F_T, Z_T) = ρ₁·ρ₂.

Risk Measure: ES_α (TCE)

The risk measure minimised is the Expected Shortfall (ES) of the normalised loss, which equals the Tail Conditional Expectation (TCE) by Proposition 1 of the paper. For a loss distribution L = −(Π_T − W₀) / W₀:

ES_α = E[L | L ≥ VaR_α(L)] = E[L | L ≥ q_1−α(L)]

All results are expressed as a fraction (or percentage) of initial wealth W₀ = P₀·Q, making them comparable across different position sizes.

Grid Search

The joint optimum (H*, K*) is found by exhaustive grid search:

(H*, K*) = argmin_{H ∈ [0,2], K ∈ [0.7F₀, 1.3F₀]} ES_α(−Π_T(H,K) / W₀)

Grid: H — 21 points (0.0, 0.1, …, 2.0); K — 20 points uniformly spaced. The optimal h* follows from the budget constraint: h* = d·P₀/Φ(K*).

Over-Hedge

When H* > 1 the futures short exceeds the underlying exposure. This can be ES-optimal when β is high (multiplicative background risk dominates) or when ρ₁ is low (futures are a poor spot proxy so a larger position is needed to hedge the residual). The UI flags over-hedge with an amber badge.

Sensitivity Analysis

Series	What it shows
ES vs β	How additive vs multiplicative background risk changes the hedge benefit (11 points, [0,1]).
ES vs d (budget)	Marginal value of increasing the option budget from 1% to 50% of position value (10 points).
ES vs α	How the optimal ES varies with the tail probability from 1% to 10% (10 points).
ES vs σ_spot	Hedge effectiveness across volatility regimes: 10%, 16%, 20%, 40%, 60% (5 points; paths re-simulated for each).

Futures Diagnostic Curve

At the optimal strike K*, ES_α is computed for H ∈ [0, 2] (41 points) with options at d fixed. The resulting U-shaped (or monotone) curve shows whether futures add value beyond options alone (it will be non-trivial when ρ₁ < 1 and σ_spot ≠ σ_futures). H* is annotated with a dashed vertical line.

Three Comparison Scenarios

Scenario	H	h	Purpose
No Hedge	0	0	Baseline unhedged position
Options Only	0	d⋅P₀/Φ	Pure put hedge without futures
Optimal	H*	h*	Joint optimum from grid search

Portfolio-Native Mapping

The abstract Yu & Sun model is mapped to a real portfolio by treating the portfolio as Q = W₀ / P₀ units of the hedge underlying (e.g. SPY). Portfolio statistics are estimated from price_history.xlsx (1-year monthly log-returns).

Model param	Portfolio quantity	Notes
Q	W₀ / P₀	Portfolio value in units of hedge underlying
σ_spot	σ_portfolio	Annualised vol from monthly returns (12-month window)
σ_futures	σ_hedge	Vol of hedge underlying — used for BS put pricing
ρ₁	corr(port returns, hedge returns)	Hedge effectiveness; higher = better hedge
ρ₂	√(1 − ρ₁²) × 0.5	Conservative approximation for background-spot correlation
σ_z	σ_portfolio × √(1 − ρ₁²)	Idiosyncratic (unhedgeable) vol
μ_spot	r_f + max(0, β) × 5%	CAPM estimate; β = cov(port, hedge) / var(hedge)
μ_futures	r_f	Risk-neutral futures drift
F₀	P₀ × e^rτ	Fair futures price (continuous compounding, no dividends)

Hedge Strategies Evaluated

Strategy	Structure	Free params optimised
Protective Put	Long put on hedge underlying, H = 0	K (strike)
Collar	Long put + short call, same expiry, H = 0	K_put, K_call
Put + Futures	Long put + short futures (full Yu & Sun 2017)	K, H
Futures Only	Short futures only, h = 0	H

All strategies are ranked by ES_α. The best strategy and trade instructions (number of contracts = ⌈h × Q / 100⌉, estimated premium, cost as % of AUM) are shown in the results panel alongside ES vs Budget and ES vs Horizon sensitivity charts.

Option Pricing

The put premium Φ is priced with the Black-Scholes formula using the futures price F₀ as the underlying:

Φ = K·e^−rτ·Φ(−d₂) − F₀·Φ(−d₁), d₁ = [ln(F₀/K) + (r + σ_f²/2)τ] / (σ_f√τ), d₂ = d₁ − σ_f√τ

The futures volatility σ_f is used for this pricing step, not the spot volatility, as the put is written on the futures contract.

Platform Constants

Constant	Value	Description
RISK_FREE	3.80%	Risk-free rate (annualised)
WEIGHT_MAX	35%	Max weight per linear asset
CRYPTO_MAX	15%	Max total crypto allocation
FUTURES_MAX	20%	Max total futures allocation
BOND_MAX	60%	Max total bond allocation
OPTION_MAX	30%	Max total options allocation
LAMBDA_QUAL	0.05	Quality tilt penalty weight
QUALITY_THRESH	0.30	Min quality score threshold
MC_PATHS	10,000	Monte Carlo simulation paths
MC_HORIZON	10 years	Monte Carlo time horizon
CVAR_SCENARIOS	5,000	CVaR scenario count
CVAR_ALPHA	0.95	CVaR confidence level
DATA_HISTORY	10 years	Price history for analysis
STALENESS	30 days	Max data age before refresh

Data sources: FMP Professional (primary — fundamentals, prices, dividends), Polygon (secondary — US equities), yfinance (EU tickers, fallback). All data fetched over HTTPS with API key authentication.

PRIIPs KID Generator

Implements EU Regulation 1286/2014 (PRIIPs) and Commission Delegated Regulation 2017/653 as amended by 2021/2268. Generates a Key Information Document using Method 1 (historical block bootstrap) for Category 3 products (equities / ETFs) with at least 24 months of price history. Five years (60 months) is recommended per ESMA guidance.

Summary Risk Indicator — VEV, MRM, SRI

Value-at-Risk Equivalent Volatility (Method 1)

\[\text{VEV} = \hat{\sigma}_{\text{monthly}} \times \sqrt{12}\]

σ̂_monthly = sample standard deviation (ddof=1) of monthly portfolio log-returns over available history. VEV is the annualised volatility used as the PRIIPs Method 1 market risk proxy.

MRM from VEV

\[\text{MRM} = \begin{cases} 1 & \text{VEV}\leq0.5\%\\ 2 & \leq1.2\%\\ 3 & \leq2.0\%\\ 4 & \leq3.0\%\\ 5 & \leq8.0\%\\ 6 & \leq16.0\%\\ 7 & >16.0\% \end{cases}\]

MRM 1–7 derived from VEV bounds. Most diversified equity portfolios fall at MRM 5–6 (VEV 8–16%).

SRI Composition — PRIIPs Annex IV Table 4

\[\text{SRI} = \max(\text{MRM},\;\text{CRM}-1)\]

Exact implementation of the 7×6 regulatory lookup table. CRM = 1 for directly-held equities/ETFs (ring-fenced custody), so SRI = MRM in the standard case. CRM 2–6 applies to structured products, bonds with issuer credit risk, or positions with counterparty exposure.

Performance Scenarios — Block Bootstrap

Block bootstrap, 12-month blocks, 10,000 paths

\[V_T^{(i)} = V_0 \cdot \exp\!\Bigl(\sum_{j=1}^{T\times12} r_j^*\Bigr), \quad \{r_j^*\} = \text{blocks sampled with replacement from } \{r_t\}_{t=1}^{n}\]

Overlapping 12-month blocks are drawn with replacement, preserving return autocorrelation and volatility clustering (GARCH effects). Block length falls back to min(12, n÷2) when history is shorter than 24 months.

Net-of-cost terminal values

\[V_T^{\text{net}} = V_T^{\text{gross}} \cdot (1 - f_{\text{entry}}) \cdot (1 - f_{\text{ong}})^T \;-\; V_0\bigl(f_{\text{exit}} + f_{\text{perf}}\cdot T\bigr), \quad V_T^{\text{net}} \geq 0\]

Scenarios are shown net of all costs per PRIIPs Annex V. Terminal values are floored at zero.

Annualised scenario returns

\[\bar{r}_T = \left(\frac{V_T^{\text{net}}}{V_0}\right)^{1/T} - 1\]

PRIIPs requires "Average return each year" — geometric annual return over holding period T. Scenario percentiles: Stress = P₁, Unfavourable = P₁₀, Moderate = P₅₀, Favourable = P₉₀.

Cost Disclosure — Reduction in Yield (RIY)

Cost components

\[C_{\text{entry}} = I \cdot f_{\text{entry}} \qquad C_{\text{ongoing}}(T) = I\Bigl[1-\bigl(1-f_{\text{ong}}\bigr)^T\Bigr]\] \[C_{\text{exit}} = V_T^{\text{net,pre-exit}} \cdot f_{\text{exit}} \qquad C_{\text{perf}} = \max\!\bigl(V_T^{\text{net,pre-perf}} - I_{\text{invested}},\;0\bigr)\cdot f_{\text{perf}}\]

Entry fee deducted upfront from initial investment. Ongoing: compound annual drag on I. Exit fee: percentage of the net terminal value before exit deduction. Performance fee: percentage of positive gain above invested amount (after entry and ongoing deductions). I_invested = I × (1 − f_entry).

Reduction in Yield (RIY) — bootstrap-derived

\[\text{RIY}(T) = \left(\frac{V_T^{\text{gross,mod}}}{I}\right)^{1/T} - \left(\frac{V_T^{\text{net,mod}}}{I}\right)^{1/T}\]

RIY is the difference between the annualised gross return and annualised net return at the moderate (P₅₀) bootstrap scenario. Unlike simplified analytic approximations, this correctly captures the interaction of all four cost types at the actual scenario terminal value. Reported in % per year.

Multi-Asset Optimizer

Extends the core MVO engine to portfolios containing equities, ETFs, futures, crypto, bonds, and options simultaneously. Each asset class is handled by a dedicated sub-optimizer; results are stitched into a single set of weights subject to class-level allocation caps.

Asset Class Constraints

Class	Cap	Treatment
Equities / ETFs / Crypto	35% per ticker · 15% total crypto	Standard MVO with RMT cleaning
Bonds	60% total	Vasicek synthetic returns → MVO
Futures	20% total	Linear payoff → included in MVO
Options	30% total	Scenario-based CVaR-LP

Bond Sub-Optimizer — Vasicek Synthetic Returns

Vasicek mean-reverting short rate

\[dr_t = \kappa(\theta - r_t)\,dt + \sigma_r\,dW_t\]

κ = mean-reversion speed, θ = long-run mean, σᵣ = yield volatility (7 bp/yr for government, 12 bp/yr for corporate, 30 bp/yr for high yield). Initial rate r₀ = risk-free + credit_spread(rating). 504 daily steps synthesize bond price paths. Modified duration and convexity control price sensitivity to rate moves.

Bond price return approximation

\[\frac{\Delta P}{P} \approx -D_{\text{mod}}\,\Delta y + \frac{1}{2}\,\mathcal{C}\,(\Delta y)^2\]

D_mod = modified duration, C = convexity. Second-order Taylor expansion captures the asymmetric (convex) price response. Used to construct 2-year daily synthetic return series for bond tickers before feeding into MVO.

Options Sub-Optimizer — CVaR Linear Programme

CVaR-LP (Rockafellar & Uryasev, 2000)

\[\min_{w,z,\zeta}\; \zeta + \frac{1}{(1-\alpha)S}\sum_{s=1}^{S} z_s \quad \text{s.t. } z_s \geq -r_s^\top w - \zeta,\; z_s \geq 0,\; \sum w_i = \text{opt\_alloc}\]

S = 5,000 scenarios, α = 0.95 (user-configurable). Each scenario r_s is computed from the Black-Scholes option P&L over the horizon. The LP minimises the expected shortfall of the option sub-portfolio while keeping total option weight at opt_alloc.

Combined Portfolio Metrics

Sharpe — linear + bond weights only

\[\text{Sharpe} = \frac{\mu_p - r_f}{\sigma_p}, \quad \text{where } \mu_p, \sigma_p \text{ use RMT-cleaned covariance of linear + bond assets}\]

Options are excluded from the Sharpe denominator (non-linear payoffs invalidate Gaussian variance). Portfolio Greeks (Δ, Γ, Θ, V) are aggregated across all option legs and displayed separately.

Portfolio Analyzer

Holdings-based workflow: enter current positions as (ticker, shares) pairs, and the engine computes current portfolio metrics, runs a full MVO re-optimization, and produces a rebalancing plan with trade-level cost estimates.

Holdings → Weights

Current weight computation

\[w_i^{\text{cur}} = \frac{n_i \cdot S_i}{\sum_j n_j \cdot S_j}\]

nᵢ = shares held, Sᵢ = current price. Weights are recomputed at analysis time using live prices from FMP (primary) → Polygon → Yahoo fallback chain. The denominator is the total portfolio market value.

Rebalancing Plan

Trade quantity and cost

\[\Delta n_i = \frac{(w_i^* - w_i^{\text{cur}}) \cdot V_{\text{total}}}{S_i}, \quad C_{\text{tx}} = |{\Delta n_i}| \cdot S_i \cdot c\]

w*ᵢ = MVO-optimal weight, V_total = total portfolio value, c = transaction cost (default 0.10%). ΔnΔn > 0 → BUY, ΔnΔn < 0 → SELL. Fractional shares are rounded to the nearest integer; residual cash is tracked.

Comparison Metrics

Metric	Formula
Sharpe	(μ − r_f) / σ
Sortino	(μ − r_f) / σ_down, σ_down = √(E[min(r,0)²]×252)
Beta	Cov(r_p, r_SPY) / Var(r_SPY)
Alpha (Jensen)	μ_p − r_f − β(μ_SPY − r_f)
Max Drawdown	max(V_peak − V_trough) / V_peak over 2-year daily log

Advanced Stress Test — Custom Scenarios

Extends the historical scenario library with a fully configurable shock constructor. The user specifies a base crisis (optional), market-wide shock, per-sector shocks, and per-ticker overrides. A GBM path is generated to visualise how portfolio value evolves through the scenario.

Shock Priority Chain

Per-asset shock resolution

\[r_i = \begin{cases} \text{ticker\_shocks}[i] & \text{if ticker override set} \\ \text{sector\_shocks}[\text{sector}(i)] & \text{elif sector shock set} \\ \text{base\_impact}[\text{sector}(i)] & \text{elif base crisis loaded} \\ \text{market\_shock} & \text{otherwise} \end{cases}\]

Four-level priority ensures fine-grained control. Sector shocks override the base crisis for specific sectors; ticker shocks are the highest priority and override everything else for individual positions.

Spillover between sectors

\[r_i' = r_i + \phi \sum_{s \neq \text{sector}(i)} \text{sector\_shocks}[s]\]

φ = spillover coefficient (0–1). When φ > 0, stress from other sectors bleeds into all positions, modelling contagion. φ = 0 (default) produces fully independent sector shocks.

Portfolio scenario return

\[R_p = \sum_i w_i \cdot r_i'\]

Weighted sum of per-asset shocked returns. Compared against the same portfolio under the current weights and under the MVO-optimal weights to highlight divergence.

GBM Path Generation

Constrained GBM

\[\text{innovations}_t \sim \mathcal{N}(0,\,\sigma\sqrt{\Delta t}), \quad \text{innovations rescaled s.t. } \sum_t \text{innov}_t = \ln(1 + R_p)\]

50-point path over T years. Random normal increments are generated then linearly rescaled to hit the target log-return exactly. This preserves the specified volatility while matching the scenario endpoint — giving a realistic visual trajectory rather than a straight line.

Options Strategy Builder

Assembles multi-leg options strategies from a ticker, spot price, implied volatility, and expiration date. Each strategy is defined as a set of legs with direction (+1 long / −1 short), option type, and a delta target used to select the appropriate strike.

Supported Strategies

Strategy	Legs	View
Covered Call	Long stock + short OTM call (Δ≈0.30)	Neutral-bullish, income
Protective Put	Long stock + long OTM put (Δ≈−0.25)	Bullish with downside insurance
Collar	Long stock + short OTM call + long OTM put	Bounded risk and reward
Long Straddle	Long ATM call + long ATM put	Volatility long, direction neutral
Long Strangle	Long OTM call (Δ≈0.25) + long OTM put (Δ≈−0.25)	Vol long, cheaper than straddle
Bull Call Spread	Long ATM call + short OTM call (Δ≈0.30)	Bullish, defined risk/reward
Bear Put Spread	Long ATM put + short OTM put (Δ≈−0.30)	Bearish, defined risk/reward
Short Straddle	Short ATM call + short ATM put	Sell volatility, range-bound
Iron Condor	Short OTM call spread + short OTM put spread	Low-vol income, 4 legs

Strike Selection via Delta Targeting

Delta-implied strike

\[K = S \cdot \exp\!\left(\sigma\sqrt{\tau} \cdot d_1^{-1}(\Delta_{\text{target}})\right)\]

ATM legs target |Δ| = 0.50, OTM legs target |Δ| = 0.25–0.30. The strike is solved by inverting the Black-Scholes delta formula. BAW adjustment is applied for American-style options. The resulting legs are priced using the full BS/BAW engine.

Portfolio Greeks Aggregation

Aggregate greeks

\[\Delta_p = \sum_i q_i \cdot \text{dir}_i \cdot \Delta_i, \quad \Gamma_p = \sum_i q_i \cdot |\text{dir}_i| \cdot \Gamma_i\]

qᵢ = position size (contracts × 100 shares/contract), dirᵢ = ±1. Theta and Vega sum linearly. Portfolio delta-dollar = Δ_p × S (exposure to 1% move in spot). Greeks are recomputed live on price or IV change.

Tax Loss Harvesting

Identifies unrealised losses in a portfolio that exceed a user-defined threshold and estimates the potential tax saving from crystallising those losses. Accounts for country-specific rates, holding periods (US), and wash-sale restrictions. Suggests substitute securities to maintain market exposure during the exclusion window.

Core Calculation

Unrealised P&L and tax saving

\[\text{pnl}_i = (S_i - B_i) \cdot n_i, \quad \text{tax\_saving}_i = |\text{pnl}_i| \cdot \tau_i\]

Sᵢ = live price (Polygon → Yahoo → FMP), Bᵢ = cost basis per share, nᵢ = shares held. Position qualifies for harvesting when pnl_pct = pnl/cost ≤ −threshold (default 5%). τᵢ = applicable tax rate (see below).

Net harvestable loss

\[\text{usable\_loss} = \max(\text{total\_loss},\; -\text{realised\_gains})\]

If the user has already realised gains this year, harvestable losses are bounded by those gains (you cannot create a net loss greater than your total gains for the purpose of this estimate).

Country Tax Rates

Country	Rate	Wash-sale	Notes
US	37% ST / 20% LT	30 days	Dual rate; holding > 1 year qualifies for LT
DE	26.375%	None	Abgeltungsteuer 25% + 5.5% Solidaritätszuschlag
FR	30%	None	PFU flat tax (prélèvement forfaitaire unique)
UK	20%	30 days	Bed-and-breakfasting rule; CGT higher rate
NL	N/A	—	Box 3 taxes fictitious return on net wealth; TLH not applicable
CH	N/A	—	No CGT for private investors (Privatvermögen)

Substitute Selection

After crystallising a loss, the position is replaced with a similar-but-not-identical security to maintain sector exposure during the wash-sale window. Substitutes are sourced from: (1) a curated ticker map (~60 tickers); (2) same-sector ETFs; (3) FMP screener peers (same sector, similar market cap). Up to 4 substitutes are returned per position.

Signal Analysis

Evaluates each factor from the Factor Library as a standalone trading signal over forward periods of 1, 3, 6, and 12 months. Produces Information Coefficient, hit rate, long/short spread, and t-statistics. Supports ensemble composite construction via OLS signal combination.

Information Coefficient (IC)

Spearman rank IC

\[\text{IC} = \rho_S(\text{factor scores},\; r_{\text{fwd}})\]

ρ_S = Spearman rank correlation between cross-sectional factor scores and subsequent forward returns. IC ∈ [−1, 1]; IC > 0 means the factor predicts future returns positively. Significance assessed by p-value from exact Spearman distribution.

Hit Rate & Long/Short Spread

Long/Short classification and spread

\[\text{Long} = \{i : s_i \ge 0.67\}, \quad \text{Short} = \{i : s_i \le 0.33\}\] \[\text{Spread} = \bar{r}_{\text{long}} - \bar{r}_{\text{short}}, \quad \text{AnnSpread} = (1 + \text{Spread})^{12/T} - 1\]

Top/bottom tercile split. Hit rate = fraction of long positions with positive forward return. Spread = mean forward return difference between top and bottom tercile. T-statistic from Welch's two-sample t-test (unequal variance).

Ensemble Composite

OLS signal combination

\[\hat{r} = \sum_k \omega_k \cdot s_k^{(i)}, \quad \hat{\omega} = \arg\min \|r_{\text{fwd}} - X\omega\|^2\]

X = matrix of standardised factor scores (z-scores), r_fwd = forward returns. OLS weights ω̂ are estimated from the cross-section. The composite score ŝᵢ = Xᵢω̂ is used to rank stocks for the combined signal. Weights are clipped to [0, 1] and re-normalised to avoid factor cancellation.

Efficient Frontier

Generates the mean-variance efficient frontier by Monte Carlo sampling of random portfolios, then renders an interactive scatter chart. Clicking any point populates weight sliders; dragging a slider instantly recomputes portfolio metrics for the custom allocation.

Random Portfolio Generation

Dirichlet sampling (long-only)

\[\mathbf{w} \sim \text{Dirichlet}(\mathbf{1}_n), \quad \sum_i w_i = 1,\; w_i \ge 0\]

1,500 random portfolios are drawn from a symmetric Dirichlet distribution (concentration = 1, uniform over the simplex). Each portfolio's expected return (μ_p = w⊤μ) and volatility (σ_p = √(w⊤Σw)) are computed using the same RMT-cleaned covariance matrix as the main optimizer.

Sharpe colour axis

\[\text{Sharpe}_p = \frac{\mu_p - r_f}{\sigma_p}\]

Each point on the scatter chart is coloured by its Sharpe ratio. The maximum-Sharpe portfolio (tangency portfolio) appears at the rightmost tip of the upper envelope. The minimum-variance portfolio sits at the leftmost point of the frontier.

Interactive Metrics

Metric	Formula	Notes
Sharpe	(μ − r_f) / σ	Uses RMT covariance
Sortino	(μ − r_f) / σ_down	σ_down = downside deviation × √252
Beta	Σ wᵢ βᵢ	Weighted sum of individual betas vs SPY
Quality	Σ wᵢ qᵢ	Weighted quality score (0–1)

How MAISNER Works

Portfolio Optimization — Mean-Variance

Objective Function

Constraints

Constrained Optimization

Asset Partition

Remaining Budget

Full Optimization Problem

Constraint Types

Infeasibility Handling

Random Matrix Theory — Covariance Cleaning

Marchenko-Pastur Distribution

Quality Tilt

Quality Score

CVaR Optimization

CVaR Definition

Correlated Scenario Generation

Monte Carlo Simulation

Stress Testing

Historical Scenarios

Advanced Stress — Custom Shocks

Leverage Analysis

Kelly Criterion — Optimal Leverage

Leveraged Return and Volatility

Volatility Decay (Continuous-Time Drag)

Margin Call Threshold

Monte Carlo Liquidation Probability

Historical Stress Scenarios

Stop Loss Calculator

Position-Level Volatility Stop

Portfolio Drawdown Limit

Sharpe-Adjusted Stop (Paleologo Key Insight)

GBM First-Passage Barrier Probability (Analytical)

Kelly Connection

Re-Entry Level

Options Pricing

European Options — Black-Scholes

American Options — Barone-Adesi Whaley (BAW)

Implied Volatility

Bond Analytics

Duration and Convexity

Vasicek Interest Rate Model

Factor Analysis

Information Coefficient

Information Ratio

Backtesting & Deflated Sharpe Ratio

Walk-Forward Validation

Deflated Sharpe Ratio

Performance Attribution

Brinson-Hood-Beebower Decomposition

Benchmark Construction

Portfolio Monitor

Position-Level P&L

Option Valuation

Currency Exposure

Long/Short Optimizer

Portfolio Modes

Borrow Cost Adjustment

Objective Modes

Weight Constraints

Portfolio Statistics

Volatility and Sharpe (identical to long-only)

Upper Barrier Probability for Short Stop-Losses

Long/Short Analyzer

Position Valuation and P&L

Portfolio Exposure

Portfolio Risk (RMT Covariance)

Stress Scenarios (Beta-Adjusted)

Hedging Optimizer

Model Structure

Budget Constraint

Simulation

Risk Measure: ESα (TCE)

Grid Search

Over-Hedge

Sensitivity Analysis

Futures Diagnostic Curve

Three Comparison Scenarios

Portfolio-Native Mapping

Hedge Strategies Evaluated

Risk Measure: ES_α (TCE)