NMV (normalized minimum variance)
Normalized Minimum Variance (NMV) is a portfolio optimization technique used in finance to construct an investment portfolio with the aim of achieving the minimum level of risk for a given expected return. NMV combines the principles of Modern Portfolio Theory (MPT) with an additional step that normalizes the weights of the assets in the portfolio. By normalizing the weights, NMV aims to address potential biases and limitations of the traditional Minimum Variance (MV) approach.
In traditional MV optimization, the objective is to minimize the portfolio variance, which is a measure of the portfolio's total risk. The portfolio variance is calculated based on the covariance matrix of asset returns and the allocation weights of the assets. The MV optimization problem involves finding the set of weights that minimizes the portfolio variance while satisfying certain constraints, such as the sum of the weights equaling one.
However, the MV approach can be sensitive to the scale of the covariance matrix and the magnitude of the expected returns. This means that small changes in the expected returns or the covariance matrix can result in significant changes to the optimal weights. In practice, this sensitivity can lead to portfolios that are heavily concentrated in a few assets, which may not be desirable for diversification purposes.
NMV addresses this issue by introducing a normalization step that adjusts the weights of the assets based on their individual volatilities. The normalization process aims to equalize the risk contributions of each asset to the portfolio's overall risk. By doing so, NMV attempts to create a more balanced and diversified portfolio.
The first step in the NMV process is similar to MV optimization, where the covariance matrix and expected returns of the assets are estimated. Next, the risk contribution of each asset is calculated by multiplying the asset's weight by its partial derivative with respect to the total portfolio variance. The partial derivative represents the sensitivity of the portfolio variance to changes in the asset's weight.
To normalize the weights, the risk contributions are divided by their sum. This normalization step ensures that each asset's contribution to the portfolio's total risk is proportional to its weight, regardless of the magnitude of its individual risk. The normalized risk contributions are then used as target weights in a second optimization step to find the final set of portfolio weights.
The second optimization step can be performed using various methods, such as quadratic programming or numerical optimization algorithms. The objective is to find the weights that minimize the sum of squared deviations between the current weights and the target weights (normalized risk contributions), subject to the same constraints as in MV optimization.
By normalizing the weights based on risk contributions, NMV aims to address the sensitivity issue of traditional MV optimization. The normalization process ensures that the portfolio is not excessively concentrated in a few assets, promoting diversification and potentially reducing the impact of extreme movements in individual asset returns.
It's important to note that NMV is just one of many portfolio optimization techniques available, and its effectiveness can vary depending on market conditions and the quality of input data. Like any optimization approach, NMV relies on assumptions and estimates, which may introduce uncertainties and potential limitations.
In summary, NMV is a portfolio optimization technique that extends the traditional MV approach by normalizing the weights of assets based on their risk contributions. By equalizing risk contributions, NMV aims to construct a more balanced and diversified portfolio that is less sensitive to changes in the expected returns and covariance matrix. However, it's crucial to consider the specific characteristics of the investment universe and the limitations of the methodology when applying NMV or any other portfolio optimization technique in practice.