We define a theoretical framework to exactly measure the additive risk-driver contributions to the performance of an investment portfolio. The approach is based on first principles, is non local and is especially suitable for non-linear portfolios. We consider a single-period return and assume that the generated cash flows are reinvested in the portfolio itself. We find that the portfolio performance can be exactly split into a calendar component and a risk-driver contribution. For the risk-driver contribution we define the projection schema, i.e. we split it into the sum of terms rising from each risk driver separately and from the combination of multiple risk drivers (the cross terms). The results are obtained computing the portfolio return projections on the risk-driver axes and they are consistent with a full Taylor expansion of the approximating pricing function. Remarkably, the performance-contribution schema thus defined works well even for long evaluation periods and can be used in fixed-income attribution. Finally, we provide two practical examples: an equity option, where we find an important contribution for the compound equity/volatility component, and a portfolio of fixed-rate coupon bonds, for which we find that many cross contributions are identically zero.
Keywords: projection contributions, performance contributions, performance-contribution schema, fixed-income attribution, risk drivers, pricing functions, operator algebra, linear operators, projection operators.
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