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…
Hypothesis Testing in Finance: Significance, Type I and Type II Errors
Why Hypothesis Testing Matters Investment professionals frequently need to decide whether an observed result is meaningful or merely the product of sampling variation. A fund may outperform its benchmark over three years. A new systematic strategy may show higher average…
What is the Central Limit Theorem in Finance?
Why Estimation Matters in Investment Analysis Investment decisions are usually made with incomplete information. An analyst may observe five years of daily returns, several hundred trades, a sample of client accounts, or a limited number of fund managers. The true…
Conditional Expectations, Variances, and Covariances in Financial Forecasting
From Unconditional Measures to Conditional Measures The first learning outcome focused on unconditional mean, variance, and covariance. Those measures describe a random variable before any additional information is imposed. In investment practice, however, analysts rarely stop there. New information arrives…
Key Statistical Distributions in Finance: Mean, Variance, Skewness, and Kurtosis
Why Distributions Matter in Investment Work The first learning outcome introduced the basic machinery of random variables, probability functions, expected value, variance, and covariance. This one, on the other hand, moves from general probability tools to the specific statistical distributions…
