Applied Mathematics \ Financial Mathematics \ Portfolio Theory
Description:
Portfolio Theory, a central topic in the field of Financial Mathematics, concerns the art and science of making investment decisions to optimize returns while managing risk. This theory provides a framework for assembling a collection of assets—known as a portfolio—such that the risk-return profile is efficient. It originated with Harry Markowitz’s groundbreaking work on Modern Portfolio Theory (MPT) in the 1950s, which revolutionized the way investors consider risk and return in their investment strategies.
Key Concepts:
Expected Return: This is the anticipated return from a portfolio considering all possible scenarios. Mathematically, for a portfolio comprising \(n\) assets, the expected return \(E(R_p)\) is given by
\[E(R_p) = \sum_{i=1}^{n} w_i E(R_i),\]
where \(w_i\) is the proportion of the portfolio’s total value invested in asset \(i\), and \(E(R_i)\) is the expected return of asset \(i\).Risk (Variance and Covariance): Risk is quantified by the variance and covariance of asset returns. The variance (\(\sigma_i^2\)) of an asset represents the volatility of its returns. When considering a portfolio, it’s essential to account for how assets interact, which is captured by the covariance (\(\sigma_{ij}\)) between asset returns \(i\) and \(j\). The portfolio variance \(\sigma_p^2\) is given by:
\[
\sigma_p^2 = \sum_{i=1}^{n} \sum_{j=1}^{n} w_i w_j \sigma_{ij},
\]
where \(\sigma_{ij}\) is the covariance between the returns of assets \(i\) and \(j\).Efficient Frontier: The efficient frontier is a set of optimal portfolios that offer the highest expected return for a defined level of risk or the lowest risk for a given level of expected return. These portfolios are considered efficient as they maximize return relative to risk.
Asset Diversification: The principle of diversification posits that by investing in a variety of assets, an investor can reduce the unsystematic risk (also known as idiosyncratic or firm-specific risk). This works because the negative performance of some investments will be offset by the positive performance of others, minimizing the overall portfolio risk.
Mean-Variance Optimization: This is the process of selecting the proportions of various assets to include in a portfolio to achieve the best possible balance between expected return and risk. The objective function is to maximize the portfolio’s expected return while minimizing its variance, often using the Lagrange multipliers method for constraint minimization.
Capital Market Line (CML) and Security Market Line (SML): The CML represents portfolios that optimally combine risk-free assets and the market portfolio, depicting the highest expected return per unit of risk. The SML, on the other hand, represents the return expected from an individual asset as a function of its systemic or market risk, described by the Capital Asset Pricing Model (CAPM).
In conclusion, Portfolio Theory equips investors with the quantitative tools to construct efficient portfolios, manage risk through diversification, and make informed investment decisions. It remains a fundamental element in both academic research and practical applications in investment management.