Portfolio Optimization and Risk Management

Diyaa Khurana

Abstract

This paper focuses on how mathematics supports investment analysis, specifically on portfolio optimization and risk management under Modern Portfolio Theory. We begin with compound interest, then develop the probabilistic foundations of quantitative finance, including expected value, variance, covariance, the Law of Large Numbers, the Central Limit Theorem, and Bayes’ Theorem. Building on these, we study diversification, the Sharpe ratio, Markowitz optimization, the efficient frontier, and the Capital Allocation Line. Risk is assessed through Value at Risk and the distinction between systematic and unsystematic risk, illustrated by major historical market crashes. Following the Capital Asset Pricing Model (CAPM), we relate an asset’s expected return to the expected market return and beta. We also review common technical indicators, including moving averages, the relative strength index, and the VIX fear index. Finally, we apply linear regression and random forest models to stock return prediction and algorithmic trading, and discuss their advantages and limitations. Keywords: expectation, covariance, volatility, CAPM, Modern Portfolio Theory, return rate, prediction, Bayes’ Theorem, random forest, linear regression

Source: semanticscholar · PDF

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