---
title: "Mean-Variance Efficient Frontier"
method: POST
path: "/portfolios/analysis/mean-variance/efficient-frontier"
tags: ["Portfolio Analysis / Mean-Variance"]
---

# Mean-Variance Efficient Frontier

`POST /portfolios/analysis/mean-variance/efficient-frontier`

Compute the discretized mean-variance efficient frontier associated to a list of assets, optionally subject to:
* Minimum and maximum weights constraints
* Minimum and maximum group weights constraints
* Minimum and maximum portfolio exposure constraint

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

## Request body

- object
  - `assets` integer, required — The number of assets
  - `assetsMeanReturns` number[], required — assetsMeanReturns[i] is the arithmetic (expected) return of asset i
  - `assetsCovarianceMatrix` array[], required — assetsCovarianceMatrix[i][j] is the covariance between the asset i and the asset j arithmetic returns
    - number[]
  - `portfolios` integer — The number of portfolios to compute on the mean-variance efficient frontier
  - `constraints` object
    - `minimumAssetsWeights` number[] — minimumAssetsWeights[i] is the minimum weight of the asset i in the portfolio, in percentage
    - `maximumAssetsWeights` number[] — maximumAssetsWeights[i] is the maximum weight of the asset i in the portfolio, in percentage
    - `assetsGroups` array[]
      - integer[] — assetsGroups[k] is the indexes of the assets belonging to the assets group k in the portfolio; exclusive with assetsGroupsMatrix
    - `assetsGroupsMatrix` array[] — assetsGroupsMatrix[k][i] is the weight of the asset i in the group of assets k; exclusive with assetsGroups
      - number[]
    - `maximumAssetsGroupsWeights` number[] — maximumAssetsGroupsWeights[k] is the maximum weight of the assets group k in the portfolio, in percentage between 0 and 1 if assetsGroups is provided
    - `minimumPortfolioExposure` number — The minimum portfolio exposure; must be lower than or equal to maximumPortfolioExposure
    - `maximumPortfolioExposure` number — The maximum portfolio exposure; must be higher than or equal to minimumPortfolioExposure

## Response `200`

OK

- object
  - `portfolios` object[], required
    - `assetsWeights` number[], required — assetsWeights[i] is the weight of the asset i in the portfolio, in percentage
    - `portfolioMeanReturn` number, required — The mean return of the portfolio
    - `portfolioVolatility` number, required — The volatility (standard deviation) of the portfolio

---

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