Minimum Variance Portfolio

v1.0.0

Compute the asset weights of the minimum variance portfolio, optionally subject to:

Minimum and maximum weights constraints
Maximum group weights constraints
Minimum and maximum portfolio exposure constraints

References

Harry M. Markowitz, Portfolio Selection, Efficient Diversification of Investments, Second edition, Blackwell Publishers Inc.

Input

{
  "type": "object",
  "required": [
    "assets",
    "assetsCovarianceMatrix"
  ],
  "properties": {
    "assets": {
      "type": "integer",
      "description": "The number of assets"
    },
    "assetsCovarianceMatrix": {
      "type": "array",
      "description": "assetsCovarianceMatrix[i][j] is the covariance between the asset i and the asset j",
      "items": {
        "type": "array",
        "items": {
          "type": "number"
        }
      }
    },
    "assetsReturns": {
      "type": "array",
      "description": "assetsReturns[i] is the arithmetic return of asset i",
      "items": {
        "type": "number"
      }
    },
    "constraints": {
      "type": "object",
      "properties": {
        "assetsGroups": {
          "type": "array",
          "items": {
            "type": "array",
            "description": "assetsGroups[k] is the indexes of the assets belonging to the assets group k in the portfolio; exclusive with assetsGroupsMatrix",
            "items": {
              "type": "integer"
            }
          }
        },
        "assetsGroupsMatrix": {
          "type": "array",
          "description": "assetsGroupsMatrix[k][i] is the weight of the asset i in the group of assets k; exclusive with assetsGroups",
          "items": {
            "type": "array",
            "items": {
              "type": "number"
            }
          }
        },
        "maximumAssetsGroupsWeights": {
          "type": "array",
          "description": "maximumAssetsGroupsWeights[k] is the maximum weight of the assets group k in the portfolio, in percentage between 0 and 1 if assetsGroups is provided",
          "items": {
            "type": "number"
          }
        },
        "maximumAssetsWeights": {
          "type": "array",
          "description": "maximumAssetsWeights[i] is the maximum weight of the asset i in the portfolio, in percentage",
          "items": {
            "type": "number"
          }
        },
        "maximumPortfolioExposure": {
          "type": "number",
          "description": "The maximum portfolio exposure; must be higher than or equal to minimumPortfolioExposure"
        },
        "minimumAssetsWeights": {
          "type": "array",
          "description": "minimumAssetsWeights[i] is the minimum weight of the asset i in the portfolio, in percentage",
          "items": {
            "type": "number"
          }
        },
        "minimumPortfolioExposure": {
          "type": "number",
          "description": "The minimum portfolio exposure; must be lower than or equal to maximumPortfolioExposure"
        }
      }
    }
  }
}

Output

{
  "type": "object",
  "required": [
    "assetsWeights"
  ],
  "properties": {
    "assetsWeights": {
      "type": "array",
      "description": "assetsWeights[i] is the weight of the asset i in the portfolio, in percentage",
      "items": {
        "type": "number"
      }
    }
  }
}