MODEL MIXED INTEGER QUADRATIC PROGRAMMING UNTUK OPTIMISASI PORTOFOLIO

Meimanah Nur Baitillah, Dr. Noorma Yulia Megawati, M.Sc; Dr. Solikhatun

2026 | Skripsi | S1 ILMU AKTUARIA

Model Markowitz klasik mengukur risiko portofolio menggunakan varians yang bersifat simetris dan tidak secara langsung menggambarkan potensi kerugian ekstrem. Dalam skripsi ini diformulasikan model optimisasi portofolio Mean-Variance-VaR ke dalam kerangka Mixed Integer Quadratic Programming (MIQP) dengan mengintegrasikan Value-at-Risk (VaR) sebagai kendala risiko tambahan. Formulasi dilakukan melalui pendekatan chance-constrained programming sehingga menghasilkan model MIQP. Masalah multi–objektif yang melibatkan tiga kriteria simultan ditransformasikan menjadi masalah single–objective menggunakan metode ?–constraint dengan variansi sebagai fungsi objektif utama. Model diterapkan pada data harga penutupan harian lima saham, yaitu BBCA, BBRI, TLKM, ASII, dan ANTM dalam periode 3 Januari 2022 hingga 30 Desember 2025 dengan 959 skenario historis. Pada tingkat kepercayaan 95?n batas VaR 2% per hari, diperoleh portofolio optimal dengan komposisi BBCA 35,06%, ASII 27,48%, TLKM 15,57%, ANTM 13,92%, dan BBRI 7,97%. Portofolio tersebut menghasilkan expected return tahunan 6,35%, volatilitas tahunan 18,02%, dan VaR historis 1, 68% yang memenuhi batas risiko yang ditetapkan.

The classical Markowitz model measures portfolio risk using the variance, which is symmetric in nature and does not directly capture the potential for extreme losses. In this thesis, a Mean–Variance–Value-at-Risk (Mean–Variance–VaR) portfolio optimization model is formulated within the framework of Mixed Integer Quadratic Programming (MIQP) by incorporating Value-at-Risk (VaR) as an additional risk constraint. The formulation is developed using a chance-constrained programming approach, resulting in an MIQP model. The resulting multi-objective optimization problem, which simultaneously considers three criteria, is transformed into a single-objective optimization problem using the ?–constraint method, with portfolio variance serving as the primary objective function. The proposed model is applied to the daily closing prices of five Indonesian stocks, namely BBCA, BBRI, TLKM, ASII, and ANTM, over the period from January 3, 2022, to December 30, 2025, comprising 959 historical scenarios. At a confidence level of 95% and a daily VaR limit of 2%, the optimal portfolio consists of BBCA (35.06%), ASII (27.48%), TLKM (15.57%), ANTM (13.92%), and BBRI (7.97%). The resulting portfolio yields an annual expected return of 6.35%, an annual volatility of 18.02%, and a historical VaR of 1.68%, satisfying the prescribed risk threshold.

Kata Kunci : Portfolio optimization, Asset allocation, Value-at-risk, Multi objective optimization, MIQP