So far, we have assumed that to hedge a bond against default, we would match the notional size of the CDS with the face value of the bond.

However, this is actually only true if the bond is trading at par.

So now we will consider how to adjust our hedge if the bond is trading away from par.

The reason the equal notional hedge only works for a par bond is that the payout on the default leg of the CDS is one minus recovery, which is another way of saying par minus recovery, but the loss on the bond is only par minus recovery.

If the bond was bought at par, if for example, the bond was bought at a price of 85%, the loss on the bond is 85% minus recovery, and therefore an equal notional hedge would lead to a net gain on default.

That is, we would be over hedged.

To compensate for this, we need to adjust the CDS notional so that its payout equals the loss on the bond on default.

The equation on screen shows that this is done by calculating the ratio of the bond loss to the CDS payout.

To put some numbers to this, let's say we've bought $10 million of a bond at a price of 85%, which we want to hedge against default, and we assume the recovery rate will be 40%.

The notional of the CDS to trade would be given by the ratio of 45 over 60, multiplied by the bond face value of 10 million, giving a CDS notional of seven and a half million dollars, assume there is a default and the recovery was indeed 40%.

The realized loss on the bond would be 85% minus 40% times 10 million, which equals four point a half million.

The CDS payout would be 0.6 times the notional of seven point a half million, so also four and a half million.

Note that this hedge sizing requires an assumption of the recovery rate, which is not required in the case where the bond is trading at par.

So hedging non-par bonds is necessarily an approximate hedge because the true required size of the hedge will not be known until a realized recovery rate has been determined.

Still, an approximate hedge is better than no hedge.

It is also worth just pointing out here that if we were hedging the price risk rather than the default risk, then it is the cso, one of the CDS that would matter rather than the default payout, And we would solve for the CDS notional required to match our bonds.

DV O one. Hopefully you are seeing that hedging bonds in practice is not a perfectly neat scenario, but one way assumptions and choices have to be made.

However, this should not detract from the fact that a credit default swap is a viable way of managing the credit risk on a bond and is a useful tool for speculators to use to express a view on credits.