# Hedging bond with CDS of different maturity

Say I buy a 10-year bond with a notional of 100k. To hedge my credit risk entirely I could buy a 10-year CDS, also on a notional of 100k.

Now, if there are only 5-year CDS trading and no 10-year CDS, then I could still hedge the first 5 years of my bond, assuming that I do not "care" about the years 5 till 10 right now.

But the question is, on which notional should I buy the 5-year CDS. Intuitively I would say it should also be 100k. But I heard the following reasoning which I do not fully understand:

Making the simplying assumption that the risky annuities (RA) of the
two CDS contracts are 5 and 10 respectively one would need to buy a
5-year CDS with a notional of 200k. The reason being that (in its first
five years) a 5-year CDS with 2*100k notional and RA of 5 acts like a
10-year CDS with notional 100k and RA 2*5.


Could somebody explain this behaviour? Is the reasoning right or wrong? Basically, how would one try to cope with the fact that 10-year CDS are not currently traded, but 5-year CDS are?

It depends on how one is thinking about the hedge. One might be thinking of it as

1. A hedge against catastrophic risk (default of the issuer), or
2. A hedge against changes in (market-implied) default intensity or hazard rate

In the former case, which seems to be how you are considering it, the hedge is a static hedge, kept for up to 5 years, and insulates you against losses from default. In the (likely) case that no default happens, you will accrue PL at roughly the bond rate minus the CDS rate, and you will be left unhedged after 5 years.

In the latter case, a practitioner would be hedging our the CD01 (credit sensitivity) of the bond, effectively stripping it to a risk-free interest rate product for a small amount of time and small default intensity changes. The 2x multiplier is very approximate, and the actual hedge ratio would be taken as the quotient of the bond CD01 by the CDS CD01.

In this latter case, one would expect to trade small amounts of bond and/or CDS periodically (probably 1-4 times per year) in order to keep the position matched to the target hedge ratio.