Modern Portfolio Theory (MPT) is a framework for constructing investment portfolios that seeks to maximize expected return for a given level of risk or minimize risk for a given level of expected return. MPT considers the risk and return of a portfolio as a whole, rather than looking at individual assets in isolation. It emphasizes the importance of diversification, stating that by investing in a mix of assets with low correlation, an investor can reduce overall portfolio risk without sacrificing returns. The optimal portfolio, according to MPT, is one that lies on the efficient frontier - the set of all portfolios that provide the highest expected return for a given level of risk. MPT is a cornerstone of modern finance and is widely used by investors, portfolio managers, and financial analysts.

Mean-variance optimization is a quantitative technique used in MPT to construct an investment portfolio that maximizes expected returns for a given level of risk or minimizes risk for a given level of expected returns. It considers the expected returns and variances of individual assets to identify the optimal portfolio weights for each asset in the portfolio. The goal is to create a portfolio that provides the highest possible return for a given level of risk.

The efficient frontier is a graphical representation of the set of portfolios that offer the highest expected return for a given level of risk or the lowest risk for a given level of expected return. It is a fundamental concept in modern portfolio theory, which assumes that investors are risk-averse and seek to maximize their returns for a given level of risk.

The efficient frontier is created by plotting the expected return and standard deviation (or other risk measures) of a set of portfolios on a graph. Each point on the graph represents a different portfolio that can be constructed using a combination of assets with different expected returns and risks. The efficient frontier represents the boundary of the set of feasible portfolios, where any portfolio lying on or above the efficient frontier is considered to be an efficient portfolio.

Investors can use the efficient frontier to identify the optimal portfolio that meets their risk and return objectives. This can be done by selecting a point on the efficient frontier that corresponds to the investor's desired level of risk and expected return. The optimal portfolio can then be constructed using a combination of assets that lie on the efficient frontier. By constructing an efficient portfolio, investors can maximize their expected return for a given level of risk or minimize their risk for a given level of expected return.

A weighted portfolio is constructed by assigning weights to securities based on their expected returns and risk, as well as their correlations with other securities in the portfolio. The objective of a weighted portfolio in the context of Modern Portfolio Theory is to achieve the optimal balance between risk and return by combining assets that are not perfectly correlated with each other. This is done by calculating the expected returns and risk of each security and assigning weights based on their contribution to the overall portfolio's risk and return. The weights are adjusted periodically to maintain the optimal balance as market conditions change. By using a weighted portfolio approach, investors can potentially achieve better risk-adjusted returns than they would by holding a portfolio of individual securities without considering their correlations and risk-return profiles.

A custom-weighted portfolio is an investment portfolio that is constructed by assigning custom weights to securities, rather than using pre-determined weights based on market capitalization or other factors. Custom-weighted portfolios are often used by investors who have specific investment goals or preferences that cannot be easily achieved through traditional weighting methods.

An equal-weighted portfolio is a type of investment portfolio in which each security or asset is given an equal weight or allocation. In other words, each security in the portfolio contributes the same percentage of the total portfolio value. For example, if an equal-weighted portfolio consists of 10 stocks, each stock would make up 10% of the portfolio's total value. This is in contrast to a market-cap weighted portfolio, where the weights of individual securities are determined based on their market capitalization. An equal-weighted portfolio can be a simple and straightforward way to achieve diversification across a range of securities or asset classes. However, it may not always be the most appropriate approach, as it can lead to overexposure to certain assets or sectors. Return, volatility and correlation between assets are not taken into account in this method.

An equal inverse-risk-weighted portfolio is a type of portfolio in which each asset is assigned a weight proportional to its inverse volatility. This means that assets with lower volatility are assigned a higher weight in the portfolio, while assets with higher volatility are assigned a lower weight. The goal of this approach is to achieve a more balanced portfolio by reducing the impact of high volatility assets on the overall portfolio performance. The equal inverse-risk-weighted portfolio is a form of naive risk parity portfolio, which aims to distribute volatility equally across all assets in the portfolio. These methods do not take into account expected return and correlation between assets.

Equal-Risk-Contribution (ERC) is a portfolio optimization technique that aims to create a portfolio where each asset contributes equally to the portfolio's overall risk. This means that the portfolio is diversified across all assets, not just in terms of the number of assets, but also in terms of the risk they contribute to the portfolio.

The ERC approach calculates the contribution of each asset to the overall portfolio risk based on the asset's volatility and correlation with other assets. The weights of each asset are then adjusted to ensure that the contribution of each asset to the portfolio risk is equal. This results in a portfolio where each asset contributes equally to the overall risk, and therefore has an equal risk contribution.

The Minimum Variance Portfolio is a portfolio optimization strategy that aims to construct a portfolio with the lowest possible risk. It does this by allocating weights to assets based on their covariance with other assets in the portfolio, with the goal of minimizing the portfolio's overall variance. The idea behind the Minimum Variance Portfolio is that by reducing the variance of the portfolio, the risk of the portfolio can also be reduced. The strategy is commonly used by investors who prioritize risk reduction over achieving high returns.

Maximum decorrelation portfolio is a portfolio optimization technique that aims to minimize the correlation between assets in a portfolio. This approach is based on the concept that diversification of assets with low correlation can help reduce overall portfolio risk. The technique involves constructing a portfolio that allocates more weight to assets that have low correlation with other assets in the portfolio, and less weight to those with high correlation. The goal is to achieve a portfolio with the lowest possible correlation between assets, while still achieving the desired level of return.

The maximum diversification portfolio is a portfolio optimization strategy that tries create the portfolio that is a diversified as possible by maximize the diversification ratio. Diversification ratio is a risk management metric used in portfolio management to assess the effectiveness of diversification. It measures the ratio of the portfolio's weighted average volatility to the volatility of the equally weighted portfolio of the same assets. A higher diversification ratio indicates a better diversification, as the portfolio's volatility is relatively lower compared to the equally weighted portfolio.