Modified duration in multi-currency portfolio

Question

I was thinking about how to figure aut duration for portfolio of bonds denominated in different currencies… I would like to compare sensitivity of portfolio to shift of yield with competitive portfolios, which have only EUR denominated bonds and the only indicator I know is modified duration of portfolios.

The duration of bond portfolio is equal to the weighted average of bond durations. Modified duration does measure the sensitivity of changes in bond price to changes in yield. But, if I counted modified duration (MD) of the portfolio as a weighted average of bond MD aside from denominated currency of bonds, I'd get measure for sensitivity of portfolio to parallel shift of all yield curves (in all currencies) at the same time, by assuming that curves are perfectly correlated. But they are not.

So, I would like to measure sensitivity of multi-currency bond portfolio to changes in EUR yield. I have modified duration of bond in denominated currency (for example USD), let’s name it Modified duration (USD). We can say, that Bond Price Change (in %) is approximately –Modified Duration x Yield Change

So, for that bond (bond in USD):

Bond Price Change (in %) is approximately –Modified Duration(USD) x Yield Change(USD)

First idea was that if I had correlation, or Beta from regression, where dependent variable would be yield (USD) and independent variable would be yield (EUR), I could write: Bond Price Change (in %) is approximately –Modified Duration(USD) x Beta x Yield Change(EUR) *

But can I? Or is there any other way how to do it correctly? Thank you very much for your comments.

Note*: If Modified duration (USD) of USD bond was 5.2, I would seek dependence of 5-year USD swap rate from 5-year euro swap rate ...

Answer 1

An approach that I also use, would be to find the currency exposures for each currency, stratify them in tenors and compute the duration in each case as you would normally do in a single currency. You would have $ Duration, Euro Duration, GBP duration, etc.

This is similar in a way to the key-rate durations, the approach for sensitivities to benchmark rates for each tenor (maturity). Instead of using, one big block duration, you use partial durations (as partial sensitivities/derivatives) wrt changes in a small portion of one interest rate curve.

If you have large portfolios of bonds, I would build one meta-bond for each currency first, with aggregated (net) cash-flows at each point in time, for each currency, and compute separately the sensitivities. This way, you don't have to measure the cross-effect of changes in interest rates from different curves.

The problem this way is of order O(n) for each currency. If you want to analyse the cross-effects of the movements in interest rate curves also, you will get an O(n^2) problem, where n=number of knots in each interest rate curve.