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Predicted Values, Standard Error, Prediction Intervals, and Functional Forms in Regression
Why Prediction Requires More Than a Fitted Line Regression is often used because analysts want to predict something: next month’s asset return, a company’s sales growth, a credit spread, a fund’s factor exposure, or a macroeconomic variable. The fitted line…
Regression Assumptions, Residual Analysis, Goodness of Fit, Coefficients, and ANOVA in Finance
The first step in regression is estimating a line. The second, more important step is deciding whether the estimated line deserves the analyst’s confidence. A regression output can look mathematically precise while still being economically misleading. The slope can be…
Simple Linear Regression in Finance: Least Squares, Intercepts, Slopes, and Interpretation
Why Linear Regression Matters in Finance Investment professionals rarely observe relationships directly. They observe data: asset returns, earnings growth, inflation, interest rates, credit spreads, valuation multiples, factor exposures, transaction costs, fund flows, and other market variables. Linear regression gives the…
Monte Carlo Simulation in Finance: Modeling Asset Prices, Returns, and Investment Risk
Monte Carlo simulations are about producing many random variables based on specific probability distributions. This helps estimate the probability of various outcomes. We will give an example to illustrate the Monte Carlo Simulation implementation. Steps Involved in Project Appraisal Imagine…
Bootstrap Resampling in Finance: Estimating Uncertainty from Historical Data
Resampling Resampling refers to the process of repeatedly drawing samples from the original observed sample to make statistical inferences about population parameters. There are two common methods: Bootstrap and jackknife. Here, we’ll focus on the Bootstrap method. Bootstrap Resampling Bootstrapping…
Historical Simulation in Investment Analysis: Using Past Returns to Model Future Outcomes
Historical Simulation Simulation forecasts possible outcomes for a system by testing different scenarios. The process involves four steps: Three common simulation approaches differ mainly in how they source scenarios: A robust simulation system can handle scenarios from any of these…
Optimal Portfolio Selection: Risk Aversion, Capital Allocation Line, Market Portfolio, and Capital Market Line
Why Optimal Portfolio Selection Starts With the Investor Portfolio theory is sometimes presented as if there is one best portfolio for everyone. That is not the right intuition. The optimal portfolio depends on both the investment opportunity set and the…
Portfolio Return and Risk: Expected Return, Variance, Standard Deviation, Covariance, and Correlation
Portfolio Statistics Matter A portfolio is more than a list of securities. Each security contributes return, but each also contributes risk and interacts with the other holdings. A high-volatility asset may increase portfolio risk sharply if it moves with the…
Minimum-Variance Portfolio and Efficient Frontier: Calculation and Interpretation
From Feasible Portfolios to Portfolio Choice Once an analyst can calculate portfolio return and portfolio risk, the next question is more interesting: which portfolio should be chosen? A universe of available securities can be combined in many ways. Some combinations…
Parametric vs Non-Parametric Tests: When to Use Each in Financial Analysis
Why the Choice of Test Matters A hypothesis test is only as useful as the assumptions behind it. A polished t-statistic, p-value, or critical-value comparison can look authoritative even when the test is poorly matched to the data. The analyst…
