AGGREGATION OF RETURNS: HETEROSKEDASTIC MODELS, (GEOMETRIC) BROWNIAN MOTION, AND FORECASTING RETURN-AT-RISK - Elonasari
Aggregation of returns can be classified into two types: Type I and Type II. Type I involves summing returns over time to understand their volatility. Typically, the volatility of aggregated returns for assets is heteroscedastic, meaning it varies over time. Therefore, heteroskedastic models like ARCH(1)/GARCH(1,1) are required to handle this non-constant volatility. Furthermore, the aggregation of returns that follows a heteroskedastic model is linked with the Brownian motion (Bm) process, allowing Bm to model the aggregation of returns and geometric Brownian motion (gBm) to model asset prices. Type II, on the other hand, involves aggregating returns from multiple assets to examine the relationships among these assets. This type represents the sum of Type I aggregations of returns for several assets, considering dependencies (Pearson correlation). Since the results of aggregating returns from both types can yield negative values (losses), a risk measure is needed to assess these losses. One such method is Return-at-Risk (RaR), which measures the maximum loss of aggregated returns that can be tolerated at a specific confidence level.