Returns-based style analysis (RBSA) involves regressing what against the return series of securities indexes?

Returns-based style analysis (RBSA) focuses on examining how a portfolio's return series correlates with various securities indexes, specifically using the portfolio's own return series as the dependent variable in regressions against those indexes. By regressing the portfolio returns against the return series of different securities indexes, analysts can determine which indexes are most representative of the portfolio's investment style and asset allocation.

This approach allows for a comprehensive understanding of the drivers of portfolio performance, including how it varies with different asset classes. By evaluating the coefficients derived from this regression, analysts can infer the degree to which the portfolio's returns are influenced by certain types of securities or market segments, effectively portraying the portfolio’s style.

The other contextually relevant choices would not align with the focus of RBSA; investment data does not provide the specific return series needed, market average returns are too generalized, and risk-adjusted returns are a secondary measure that reflects the return in relation to the risk taken rather than a direct series for regression analysis. Thus, only the portfolio returns serve as the appropriate dataset for this analytical framework.