Quantitative Risk Management: A Practical Guide to Financial Risk, + Website

10.4 Replicating Portfolio

Representing a complex portfolio in terms of a simpler portfolio is both useful and relatively straightforward. For a single asset k, the best hedge position minimizes the portfolio variance when changing asset k only. This implies that the difference between the original and best hedge position is a mirror portfolio,

in the sense that the variance of the difference between the original and mirror portfolio is minimized:

It is simple to calculate the best single-asset mirror portfolio across all assets in the original portfolio: for each asset, calculate the volatility at the best hedge position and choose from among all assets that single asset with the lowest best-hedge volatility. It is natural to call this best mirror portfolio a replicating portfolio because it best replicates the portfolio (best using single assets chosen from the original portfolio).

The idea becomes clearer when we focus on an example. Table 10.10 shows the best hedges, both the holdings and the volatilities at those holdings, for the simple portfolio consisting of long $40 million of the 5-year Treasury, $20 million 10-year Treasury, and €7 million CAC futures. The best hedge using ...