Diversified Mean-Variance Efficient Portfolio

v1.0.0

Compute the asset weights of a diversified mean-variance efficient portfolio, as defined in the first reference, optionally subject to:

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

The diversification measure used in the optimization procedure is the of the assets weights.

A diversified mean-variance efficient portfolio does NOT belong to , but is close to this frontier.

References

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

Input

{
  "type": "object",
  "required": [
    "assets",
    "assetsCovarianceMatrix",
    "assetsReturns",
    "constraints"
  ],
  "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"
            }
          }
        },
        "deltaReturn": {
          "type": "number",
          "description": "The relative tolerance over the mean-variance efficient portfolio return"
        },
        "deltaVolatility": {
          "type": "number",
          "description": "The relative tolerance over the mean-variance efficient portfolio volatility"
        },
        "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"
        },
        "portfolioReturn": {
          "type": "number",
          "description": "The portfolio return; exclusive with portfolioVolatility and riskTolerance"
        },
        "portfolioVolatility": {
          "type": "number",
          "description": "The portfolio volatility; exclusive with portfolioReturn and riskTolerance"
        },
        "riskTolerance": {
          "type": "number",
          "description": "The portfolio risk tolerance; exclusive with portfolioReturn and portfolioVolatility"
        }
      }
    }
  }
}

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"
      }
    }
  }
}