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For evaluating a hedge for a bond, I noticed that we often look at the yield return correlation between the two instruments, instead of the price return. Why is that?

To me, the price would make more sense, as the PNL is computed from the price move

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I would put the answer a bit differently:

  • In the end you care about the price, don't you? If you sell the bond then it is bad if you can sell it for less. No matter what the yield is. E.g. if you have assets in a mutual fund then investors enter and leave the fund and you probably have to sell and buy assets (and there are more clever ways of cash management, I know). But what does this price depend on? E.g. the US10Y bond and the US3Y note? The yield curve.

  • How does your hedge look? Most probably you hedge a bond with a bond-futures. For the US, Canadian, German or French government e.g. you can do this nicely. But the time-to-maturity of the hedge will not perfectly match your bond's TTM. What you do know are (keyrate) durations of your bond and the hedge. The duration measure sensitivity to the common risk factor: the yield curve (that matches your market as closely as possible). Thus thinking in yields or durations (based on the same market, such as US government bonds or German ones) makes you think and act on the common risk factor. Your bond will have smaller and smaller returns as it approaches maturity. Thinking in duration and yield changes you can generate a time series that is stationary.

In my view this is the reason why thinking in duration/yield(curve) is useful.

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If the aim of the hedge is to make the portfolio insensitive (as much as possible) to small movements in the yield, then the question that needs answering is the following:

If the yield of the hedge moves by $x$, by how much did the yield of the bond move?

The answer is given by the correlation between yield movements between bond and hedging instrument, and not by price returns.

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