**Abstract**

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.

**Pdf file**: projection-performance-contributions.pdf

## Gulnaz says:

Hi Marco,

Thanks for sharing your research.

I have a question concerning the calculations from “Projection performance contributions of non-linear portfolios”, page 10. Two equations without reference between 30 and 31 for time contribution. From calculation point, I notice that the last term in the second has a type, it supposed to be y_n instead of y_1 (or y_i for each i-th term), but from pricing view, y_1 has a sense as a forward rate to recalculate the discount factors for a delta-t rolling down the old (t-time) curve.

What would you think of it?

The second question: do you make a difference between the pull-to-par and the roll-down effects or you consider them just as a sum representing convergence term?

Thanks for the help!

## Marco Marchioro says:

Thank you for your interest in the paper.

You are correct there is a typo in the equation you mention. I will fix it in the next version of the paper.

I do not distinguish between pull-to-par and roll-down since I did not find a generic way to do it. However I have a way to do it for bonds: if I have time I will write it in the next version of the paper.