> ## Documentation Index > Fetch the complete documentation index at: https://docs.fincept.in/llms.txt > Use this file to discover all available pages before exploring further. # Black-Litterman Posterior Returns & Weights > Computes posterior (updated) returns by combining market equilibrium with investor views using Bayesian inference. Returns optimal portfolio weights based on the posterior distribution. **Use Cases:** - Incorporating proprietary views into portfolio construction - Blending quantitative signals with market equilibrium - Creating more stable, diversified portfolios than MVO - Tactical asset allocation with strategic baseline **Mathematical Background:** Posterior returns: E[R] = [(τΣ)^-1 + P^T Ω^-1 P]^-1 [(τΣ)^-1 Π + P^T Ω^-1 Q] Where Π is equilibrium returns, P is the picking matrix, Q is views, and Ω is view uncertainty. **Credits:** 5 per request [Tier: PRO, Credits: 5] ## OpenAPI ````yaml api-specs/portfolio.json post /quantlib/portfolio/black-litterman/posterior openapi: 3.1.0 info: title: FinceptQuantLib API - Portfolio description: >- Portfolio optimization and risk management module for institutional-grade portfolio construction, analysis, and monitoring. Includes mean-variance optimization, Black-Litterman, risk parity, and comprehensive risk metrics. **Tier:** Pro (5 credits per request) **Key Features:** - Mean-variance optimization (minimum variance, maximum Sharpe, target return) - Efficient frontier construction with tangency portfolio identification - Black-Litterman model for Bayesian view incorporation - Risk parity strategies (ERC, HRP, inverse volatility) - Comprehensive risk metrics (VaR, CVaR, tracking error, information ratio) - Advanced portfolio analytics (risk contribution, drawdown, diversification) version: 3.0.0 contact: name: Fincept API Support url: https://fincept.in servers: - url: https://api.fincept.in description: Fincept API Production Server security: - APIKeyHeader: [] tags: - name: quantlib-portfolio description: Portfolio optimization and risk management endpoints x-displayName: Portfolio paths: /quantlib/portfolio/black-litterman/posterior: post: tags: - quantlib-portfolio summary: Black-Litterman Posterior Returns & Weights description: >- Computes posterior (updated) returns by combining market equilibrium with investor views using Bayesian inference. Returns optimal portfolio weights based on the posterior distribution. **Use Cases:** - Incorporating proprietary views into portfolio construction - Blending quantitative signals with market equilibrium - Creating more stable, diversified portfolios than MVO - Tactical asset allocation with strategic baseline **Mathematical Background:** Posterior returns: E[R] = [(τΣ)^-1 + P^T Ω^-1 P]^-1 [(τΣ)^-1 Π + P^T Ω^-1 Q] Where Π is equilibrium returns, P is the picking matrix, Q is views, and Ω is view uncertainty. **Credits:** 5 per request [Tier: PRO, Credits: 5] operationId: bl_posterior requestBody: required: true content: application/json: schema: type: object required: - covariance_matrix - market_caps properties: covariance_matrix: type: array items: type: array items: type: number description: Asset covariance matrix (annualized) example: - - 0.04 - 0.006 - 0.008 - 0.01 - - 0.006 - 0.09 - 0.012 - 0.015 - - 0.008 - 0.012 - 0.0625 - 0.018 - - 0.01 - 0.015 - 0.018 - 0.16 market_caps: type: array items: type: number description: Market capitalizations for each asset example: - 5000000000 - 3000000000 - 2000000000 - 1000000000 risk_aversion: type: number description: Market risk aversion parameter default: 2.5 example: 2.5 views: type: array items: type: object properties: type: type: string enum: - absolute - relative description: >- View type: 'absolute' for single asset return view, 'relative' for relative performance view asset: type: integer description: Asset index for absolute view (0-based) asset_long: type: integer description: Long asset index for relative view asset_short: type: integer description: Short asset index for relative view value: type: number description: >- Expected return (absolute) or outperformance (relative) confidence: type: number description: >- View confidence level (0.0 to 1.0). Higher values mean stronger conviction. minimum: 0 maximum: 1 default: 0.5 description: >- List of investor views to incorporate. Can be empty for pure equilibrium. nullable: true example: - type: absolute asset: 0 value: 0.09 confidence: 0.7 - type: relative asset_long: 1 asset_short: 2 value: 0.03 confidence: 0.6 responses: '200': description: Black-Litterman posterior successfully computed content: application/json: schema: type: object properties: success: type: boolean example: true data: type: object properties: posterior_returns: type: array items: type: number description: >- Updated expected returns incorporating views (annualized) example: - 0.085 - 0.12 - 0.098 - 0.142 optimal_weights: type: array items: type: number description: Optimal portfolio weights based on posterior returns example: - 0.32 - 0.28 - 0.25 - 0.15 '401': $ref: '#/components/responses/UnauthorizedError' '402': $ref: '#/components/responses/InsufficientCreditsError' '422': $ref: '#/components/responses/ValidationError' components: responses: UnauthorizedError: description: Authentication information is missing or invalid content: application/json: schema: type: object properties: detail: type: string example: Invalid API key InsufficientCreditsError: description: Insufficient API credits content: application/json: schema: type: object properties: detail: type: string example: Insufficient credits. This endpoint requires 5 credits. ValidationError: description: Request validation error content: application/json: schema: type: object properties: detail: type: array items: type: object properties: loc: type: array items: type: string msg: type: string type: type: string example: detail: - loc: - body - expected_returns msg: field required type: value_error.missing securitySchemes: APIKeyHeader: type: apiKey in: header name: X-API-Key description: >- API key for authentication. Get your key at https://api.fincept.in/auth/register ````