MPT – Modern Portfolio Theory (MPT), also known as mean-variance analysis, is a mathematical framework that aims to construct portfolios with maximum expected returns for a given level of risk. It builds upon the concept of diversification, which suggests that owning a mix of different assets is less risky than holding a single asset. MPT takes into account the interplay between an asset’s risk and return in the context of the overall portfolio.
It was developed by Sir Harry Markowitz in the 1950s and has become a cornerstone of portfolio management.
Maths Behind MPT –
In MPT, the first step is to quantify the risk and return characteristics of individual assets. This is typically done by analyzing historical data, such as the mean (average) return and standard deviation (a measure of volatility) of each asset. Let’s denote the mean return of asset i as μi and the standard deviation as σi.
The next step is to analyze the correlation or covariance between different assets. Correlation measures the linear relationship between two variables, while covariance measures how two variables move together. In MPT, we typically use the covariance matrix, denoted as Σ, which captures the pairwise covariances between all assets in the portfolio.
Given these inputs, MPT aims to construct an efficient frontier, which represents 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.
Mathematically, the efficient frontier can be obtained by solving an optimization problem. Let’s assume we have n assets in our portfolio. We can represent the portfolio weights for each asset as w = [w1, w2, …, wn], where wi represents the weight of asset i. The sum of all weights must be equal to 1, i.e., ∑wi = 1.
The expected return of the portfolio, denoted as μp, can be calculated as the weighted sum of the individual asset returns:
μp = ∑(μi * wi)
Similarly, the portfolio variance, denoted as σp^2, can be calculated as:
σp^2 = w * Σ * w^T
where Σ is the covariance matrix, w is the weight vector, and w^T represents the transpose of w.
The optimization problem in MPT is to find the set of portfolio weights that maximizes the expected return for a given level of risk or minimizes the risk for a given level of expected return, subject to the constraints mentioned earlier.
By applying MPT, investors can construct portfolios that achieve an optimal balance between risk and return based on their risk tolerance and investment objectives.
