{
  "recorded_at": "2026-09-18",
  "chapter": "portfolio-optimization.md",
  "related_chapter": "black-litterman.md",
  "delivery": "Optimization Problem and Black–Litterman are now separate lessons with independent executed Notebooks.",
  "source": {
    "provided_filename": "JA252.2 Notes1.pdf",
    "source_location": "User-provided local PDF; not included in the repository.",
    "title": "Fundamentals of Optimization and Application to Portfolio Selection",
    "author_metadata": "CQF",
    "creator_metadata": "LaTeX with Beamer class",
    "producer_metadata": "pdfTeX-1.40.20",
    "creation_date_metadata": "2025-02-13T17:00:29+07:00",
    "physical_pages": 145,
    "internal_slide_numbering": "Slides 1-143 align with physical pages 1-143. The two bibliography pages at physical pages 144-145 also display 143/143 in the footer.",
    "page_size": "A4 landscape, 841.89 x 595.276 points",
    "sha256": "eebd6f73231ea0c249afb91aaf551dd45d9e54f3a2898fac88d5e8bf66106c20",
    "access": "User-provided PDF read locally as source material. The PDF, extracted text, rendered pages, portraits and decorative source images are not copied into this repository.",
    "method": "Metadata checked with pdfinfo; all pages extracted with pdftotext -layout; embedded raster objects inventoried with pdfimages -list; formula, numerical-example and suspected-error pages rendered and visually inspected with pdftoppm. Numerical examples were independently recomputed before use."
  },
  "source_page_map": [
    {
      "pages": [1, 3],
      "topics": ["title", "lecture roadmap"]
    },
    {
      "pages": [4, 8],
      "topics": ["optimization problem formulation", "objective function", "decision variables", "constraints", "min-max and affine transformations"]
    },
    {
      "pages": [9, 10],
      "topics": ["unconstrained optimization", "gradient", "Hessian", "first- and second-order conditions"]
    },
    {
      "pages": [11, 22],
      "topics": ["mean-variance notation", "covariance decomposition", "risk-free asset", "unconstrained mean-variance allocation"]
    },
    {
      "pages": [23, 47],
      "topics": ["CAPM regression motivation", "ordinary least squares", "residuals", "multifactor regression"]
    },
    {
      "pages": [48, 54],
      "topics": ["generalized least squares", "Mahalanobis objective", "Cholesky whitening"]
    },
    {
      "pages": [55, 60],
      "topics": ["equality constraints", "Lagrangian", "first-order system"]
    },
    {
      "pages": [61, 76],
      "topics": ["minimum variance for a target return", "four-asset numerical example", "A-B-C scalars"]
    },
    {
      "pages": [77, 84],
      "topics": ["minimum-variance frontier", "global minimum-variance portfolio", "four-asset GMV example"]
    },
    {
      "pages": [85, 100],
      "topics": ["risk-free funding", "target return with a risk-free asset", "tangency portfolio", "four-asset numerical example"]
    },
    {
      "pages": [101, 107],
      "topics": ["Black-Litterman motivation", "Bayesian roadmap", "Bayes formula background"]
    },
    {
      "pages": [108, 118],
      "topics": ["reverse optimization", "market-implied prior", "market risk aversion", "tau", "prior numerical example"]
    },
    {
      "pages": [119, 123],
      "topics": ["absolute and relative views", "P-Q-Omega representation", "view uncertainty"]
    },
    {
      "pages": [124, 131],
      "topics": ["Black-Litterman posterior", "posterior expected excess returns", "risk-aversion allocation examples"]
    },
    {
      "pages": [132, 135],
      "topics": ["inequality constraints", "Kuhn-Tucker and Fritz John form", "130-30 heading without a worked example"]
    },
    {
      "pages": [136, 137],
      "topics": ["benchmark-relative optimization", "active weights", "active mean-variance objective"]
    },
    {
      "pages": [138, 139],
      "topics": ["lecture recap"]
    },
    {
      "pages": [140, 143],
      "topics": ["conditional probability", "multiplication rule", "independence", "total probability"]
    },
    {
      "pages": [144, 145],
      "topics": ["bibliography"]
    }
  ],
  "corrected_source_issues": [
    {
      "pages": [3, 55, 101],
      "source_issue": "Equality-constrained optimization and Black-Litterman are both labelled Part III.",
      "chapter_treatment": "Use semantic section titles and a fresh chapter order instead of copying the source part numbers."
    },
    {
      "pages": [49],
      "source_issue": "The slide says GLS does not assume regressors are uncorrelated with residuals, then imposes E[epsilon|X]=0, and prints Var[epsilon_i|X]=0 while naming Omega as the variance.",
      "chapter_treatment": "State E[epsilon|X]=0 and Var[epsilon|X]=Omega; do not claim that GLS removes the exogeneity condition."
    },
    {
      "pages": [50],
      "source_issue": "The printed GLS Mahalanobis objective uses Omega rather than Omega inverse, while the estimator below uses Omega inverse.",
      "chapter_treatment": "Use (Y-X beta)' Omega^-1 (Y-X beta)."
    },
    {
      "pages": [52, 53],
      "source_issue": "The whitening covariance expression omits the transpose on the right inverse Cholesky factor, refers to the untransformed residual, and equates the transformed OLS objective to a quadratic form containing Omega rather than Omega inverse.",
      "chapter_treatment": "Use Omega=C C', epsilon*=C^-1 epsilon, Var[epsilon*|X]=C^-1 Omega C^-T=I, and run OLS on X*=C^-1 X and Y*=C^-1 Y."
    },
    {
      "pages": [60],
      "source_issue": "The slide suggests checking the Hessian of the objective is generally enough after solving an equality-constrained problem.",
      "chapter_treatment": "Limit the conclusion to the convex quadratic portfolio examples with affine constraints; avoid presenting it as a general constrained second-order test."
    },
    {
      "pages": [67, 69],
      "source_issue": "The derivation inverts the covariance matrix and states AC-B^2>0 without stating the required nonsingularity and linear-independence assumptions.",
      "chapter_treatment": "Require a positive-definite covariance matrix and independent budget and target-return constraints; solve linear systems in code instead of forming an explicit inverse."
    },
    {
      "pages": [74, 75],
      "source_issue": "Gamma is reported as -0.0105013 on page 74, but the substitution line on page 75 changes it to +0.0105013. The displayed final weights correspond to the negative value.",
      "chapter_treatment": "Retain gamma=-0.0105013 and explicitly use Sigma^-1(lambda mu + gamma 1)."
    },
    {
      "pages": [106],
      "source_issue": "The product P(I|E)P(E) is labelled the likelihood.",
      "chapter_treatment": "Treat P(I|E) as the likelihood and the product with the prior as the unnormalized posterior or joint density."
    },
    {
      "pages": [110, 111, 125, 127],
      "source_issue": "Equation references and numbers are duplicated or refer to slide numbers rather than stable equation numbers.",
      "chapter_treatment": "Use unnumbered semantic equations and local prose references."
    },
    {
      "pages": [108, 110, 112],
      "source_issue": "The market-cap prior is described as neutral and uninformed and the reverse optimization is tied loosely to CAPM.",
      "chapter_treatment": "Describe it as an equilibrium prior implied by a selected market proxy, covariance estimate and risk-aversion parameter; disclose those assumptions."
    },
    {
      "pages": [117, 118],
      "source_issue": "Tau=1/T is presented as a convenient heuristic even though tau and Omega are convention-dependent.",
      "chapter_treatment": "Use tau=1/120 only as the reproducible teaching convention and state that no unique calibration follows from the PDF."
    },
    {
      "pages": [120, 121],
      "source_issue": "Omega is called confidence although larger diagonal entries mean more uncertainty, and the conditional distribution is written as though P(I|E) itself were a random variable distributed N(Q,Omega).",
      "chapter_treatment": "Call Omega view-error covariance and write Q=P mu+epsilon_v with epsilon_v~N(0,Omega), equivalently Q|mu~N(P mu,Omega)."
    },
    {
      "pages": [128],
      "source_issue": "The slide says three allocations and three risk-aversion levels but lists four.",
      "chapter_treatment": "Describe four sensitivity cases only when all four are shown."
    },
    {
      "pages": [128, 129],
      "source_issue": "Lambda=1 is called the Kelly portfolio and the lowest risk aversion anyone should use, without the assumptions needed for that identification or normative threshold.",
      "chapter_treatment": "Treat lambda only as a mean-variance sensitivity parameter and remove the universal Kelly and minimum-safe-lambda claims."
    },
    {
      "pages": [130],
      "source_issue": "The rounded risky weights shown sum to 76.63%, inconsistent with the stated 23.27% risk-free weight; independent recomputation under the chapter inputs gives 23.4140% because the chapter consistently uses lambda_mkt=2.24.",
      "chapter_treatment": "Compute every displayed weight from the deterministic chapter implementation and reconcile risky plus risk-free weights to one."
    },
    {
      "pages": [133, 134],
      "source_issue": "The displayed conditions include a lambda_0 multiplier on the objective, which is a Fritz John form, while the slide calls them Kuhn-Tucker conditions; the sufficiency discussion is narrower than the general convex case.",
      "chapter_treatment": "Present standard KKT stationarity, primal feasibility, dual feasibility and complementary slackness, with convexity and a stated constraint qualification."
    },
    {
      "pages": [135],
      "source_issue": "The 130-30 slide repeats the generic inequality problem and contains no 130-30 formulation or numerical example.",
      "chapter_treatment": "Define 130% long, 30% short, 100% net and 160% gross directly; do not attribute a worked 130-30 example to the source."
    },
    {
      "pages": [137],
      "source_issue": "The zero-net active-weight constraint is printed as Delta w'1=1, and the benchmark return term omits the transpose on w_B.",
      "chapter_treatment": "Use 1' Delta w=0 and R_B=w_B'R when both benchmark and portfolio weights sum to one."
    }
  ],
  "examples": {
    "data_kind": "Hypothetical teaching example; the source names assets only X1-X4 and provides no instrument identifiers, dates or raw observations.",
    "period_and_units": "Expected returns and volatilities are decimals for one common holding period. Covariances are return-squared. Percentages are display formatting.",
    "shorting": "Closed-form risky-only target, frontier and GMV examples permit negative weights unless long-only is stated explicitly.",
    "frictions": "No taxes, fees, turnover, margin, borrow spread, short-borrow cost or integer holdings.",
    "expected_returns": [0.05, 0.07, 0.15, 0.27],
    "volatilities": [0.07, 0.12, 0.3, 0.6],
    "correlation": [
      [1, 0.8, 0.5, 0.4],
      [0.8, 1, 0.7, 0.5],
      [0.5, 0.7, 1, 0.8],
      [0.4, 0.5, 0.8, 1]
    ],
    "covariance": [
      [0.0049, 0.00672, 0.0105, 0.0168],
      [0.00672, 0.0144, 0.0252, 0.036],
      [0.0105, 0.0252, 0.09, 0.144],
      [0.0168, 0.036, 0.144, 0.36]
    ],
    "target_return_risky_only": {
      "target": 0.1,
      "A": 239.34404679746285,
      "B": 9.618456078083405,
      "C": 0.5502809819580006,
      "return_multiplier": 0.36527936942728567,
      "budget_multiplier": -0.010501299717695724,
      "weights": [0.5284121083377582, 0.17288807520660476, 0.15976434270310244, 0.13893547375253518],
      "variance": 0.02602663722503291,
      "volatility": 0.1613277323495031
    },
    "global_minimum_variance": {
      "weights": [1.2748867225044624, -0.26311272827131676, 0.016339420568447043, -0.028113414801592743],
      "expected_return": 0.040186736235067966,
      "variance": 0.004178085953590554,
      "volatility": 0.06463811533136277
    },
    "risk_free_target": {
      "risk_free_return": 0.025,
      "target": 0.1,
      "risky_weights": [0.8873524831610342, 0.08126325071765012, 0.1548434304583574, 0.12164862380157235],
      "risky_weight_sum": 1.245107788138614,
      "risk_free_weight": -0.2451077881386141,
      "volatility": 0.16028414561637885
    },
    "tangency": {
      "risk_free_return": 0.025,
      "weights": [0.7126712173952348, 0.06526603679761364, 0.12436146647981546, 0.09770127932733609],
      "expected_return": 0.08523574883594776,
      "variance": 0.016571706535229463,
      "volatility": 0.12873114050310222,
      "sharpe": 0.46791901788904083
    },
    "black_litterman": {
      "market_weights": [0.05, 0.4, 0.45, 0.1],
      "market_sharpe_assumption": 0.5,
      "market_volatility": 0.22352304131789186,
      "market_risk_aversion_used": 2.24,
      "tau": 0.008333333333333333,
      "P": [
        [-1, 0, 1, 0],
        [0, 1, 0, 0]
      ],
      "Q": [0.1, 0.03],
      "omega_rule": "diag(P (tau Sigma) P')",
      "omega": [
        [0.0006158333333333334, 0],
        [0, 0.00012]
      ],
      "prior_excess_returns": [0.02091712, 0.04712064, 0.1467312, 0.2599296],
      "posterior_excess_returns": [0.016781811019405336, 0.03755243557399479, 0.12484270485942535, 0.22717373844727903],
      "posterior_risky_weights_at_lambda_2_24": [0.09869571476911322, 0.16585951221494336, 0.4013042852308861, 0.1],
      "posterior_risk_free_weight_at_lambda_2_24": 0.23414048778505725
    },
    "long_only_target_20_percent": {
      "target": 0.2,
      "binding_constraint": "w1 = 0",
      "unconstrained_w1": -0.7195963887413726,
      "weights": [0, 0.02631578947368496, 0.5394736842105264, 0.4342105263157895],
      "variance": 0.16307763157894745,
      "volatility": 0.40382871564432793
    },
    "exposure_definitions": {
      "portfolio_130_30_long": 1.3,
      "portfolio_130_30_short_absolute": 0.3,
      "portfolio_130_30_net": 1,
      "portfolio_130_30_gross": 1.6,
      "active_weight_sum_when_fully_invested": 0
    }
  },
  "editorial_notes": [
    "The chapter is a new Thai explanation and reconstruction, not a verbatim slide translation. no-ai-slop is applied to remove generic setup, unsupported importance claims and repeated recap.",
    "The existing Portfolio Theory chapter already covers mean-variance geometry, GMV, tangency and CAPM. This chapter emphasizes the optimization machinery, source corrections, implementability constraints and Black-Litterman update.",
    "The source's statement that the covariance came from 120 monthly returns is retained only as the teaching convention behind tau=1/120. Without asset identities, dates and raw observations, the example is not presented as empirical market evidence.",
    "All model weights are reconciled with budget, target-return and risk-free residual identities. Negative risk-free weight is borrowing, not a free gain.",
    "Posterior mean uncertainty and the covariance of asset returns are distinct. The displayed allocation convention uses Sigma as return covariance and states that alternative Black-Litterman conventions exist.",
    "Source portraits, the compass illustration and PDF screenshots are not reused."
  ],
  "visuals": {
    "design_system": "QuantCorner / QuantSeras; preserve the current light book layout, purple and teal accents, readable Thai typography, accessible dark theme and offline operation.",
    "route": "no-image-generator",
    "reason": "Every illustration communicates a mathematical relationship, calculated example or model flow and is generated as an editable or deterministic diagram rather than decorative artwork.",
    "image_generator_used": false,
    "generator": "scripts/make_portfolio_optimization_figures.py generates 13 SVGs; scripts/render_optimization_roadmap.py renders the editable Excalidraw roadmap.",
    "revision": "2026-09-18: all 14 figures redesigned from mathematical inputs. OLS matches the Notebook, the Lagrange contour is exactly tangent, and exposure bars use a common scale. The interactive Black–Litterman chart is also redesigned.",
    "svg_files": [
      "assets/images/optimization-types.svg",
      "assets/images/optimization-curvature.svg",
      "assets/images/optimization-covariance.svg",
      "assets/images/optimization-ols.svg",
      "assets/images/optimization-gls.svg",
      "assets/images/optimization-lagrange.svg",
      "assets/images/optimization-target-allocation.svg",
      "assets/images/optimization-frontier.svg",
      "assets/images/optimization-target-weights.svg",
      "assets/images/optimization-black-litterman-roadmap.svg",
      "assets/images/optimization-black-litterman-beliefs.svg",
      "assets/images/optimization-black-litterman-weights.svg",
      "assets/images/optimization-kkt.svg",
      "assets/images/optimization-active.svg"
    ],
    "editable_diagram": {
      "rendered_file": "assets/images/optimization-black-litterman-roadmap.svg",
      "source_file": "assets/diagrams/optimization-black-litterman-roadmap.excalidraw",
      "spec": {
        "layout": "Two input branches: market to prior; prior and views to posterior; posterior to allocation",
        "stages": [
          "market weights and covariance",
          "reverse optimization prior",
          "P-Q-Omega views",
          "posterior expected excess returns",
          "portfolio allocation"
        ],
        "requirements": [
          "Each arrow names the object passed to the next stage.",
          "Market inputs, investor views and allocation outputs remain visually distinct.",
          "The SVG and Excalidraw source express the same five stages.",
          "Text remains legible at 320 CSS pixels and in both site themes.",
          "No PDF artwork or source portrait is embedded."
        ]
      }
    },
    "labels": "SVGs include explicit titles, descriptions, axes, units and source-aligned numerical values. Chapter alt text and figure captions explain the takeaway without relying on color alone.",
    "editable": "Thirteen deterministic calculated SVGs plus one Excalidraw roadmap SVG and its editable source."
  },
  "notebook": {
    "files": ["notebooks/portfolio-optimization.ipynb", "notebooks/black-litterman.ipynb"],
    "generator": "scripts/make_portfolio_optimization_notebook.py",
    "dependencies": "Python standard library",
    "scope": [
      "covariance construction",
      "target-return minimum variance",
      "GMV and tangency checks",
      "GLS whitening",
      "Black-Litterman prior, views, posterior and weights",
      "unconstrained versus long-only target-return allocation"
    ],
    "execution": "All non-empty code cells are executed by the generator and must finish without an error output."
  },
  "verification_commands": [
    "python3 scripts/make_portfolio_optimization_figures.py",
    "python3 scripts/render_optimization_roadmap.py",
    "python3 scripts/make_portfolio_optimization_notebook.py",
    "node qa/optimization-figure-review.cjs",
    "node qa/portfolio-optimization-checks.mjs",
    "npm run build:pages",
    "npm test",
    "node qa/portfolio-optimization-browser.cjs",
    "node qa/portfolio-optimization-page-checks.cjs",
    "git diff --check"
  ],
  "learning_revision": {
    "display_title": "Optimization Problem",
    "url_preserved": true,
    "editorial": "New introduction; covariance inputs moved after regression and Lagrange; equations and quote retained; production details removed from reader prose.",
    "labs": [
      "2D quadratic constraints with KKT diagnostics",
      "Four-asset target frontier with long-only active-set enumeration and explicit infeasibility"
    ],
    "portraits": [
      {
        "path": "assets/images/lagrange-portrait.jpg",
        "source": "https://commons.wikimedia.org/wiki/File:Lagrange_portrait.jpg",
        "rights": "Public domain; unknown creator; original file retained",
        "sha256": "5e37505cc5140246bd020079f17f59604b42e1789b9fc7998c66b7289f88c6db"
      }
    ],
    "date": "2026-09-18"
  }
}