In this workout, we are told that one of our euro clients is looking to raise 10 year funding. We are then asked which of the below would offer the lowest funding margin over ESTR. We are told we can ignore day count conventions and payment frequencies For simplicity. We are presented with two options. Option one is a 10 year euro bond with a 2% coupon at par. We are told that 10 year euro swap rates versus ESTR are 1.58% to 1.6%. The second option is a 10 year US dollar bond with a yield of 3.125% at par. 10 year US dollar swap rates are 2.4% to 2.42% and 10 year Euro US dollar basis is minus 27 to minus 23 basis points. We are also told that one basis point in euro equals 1.045 basis points in US dollars.

The first step in answering this question is to swap both bonds into floating. As both bonds are trading at par, we can use yield yield swaps to do this. Starting with the first option, the 10 year euro bond with a yield of 2%, this would have to be combined with a 10 year Euro receiver. As the client will be receiving the fixed rate, the bid rate applies. So the euro bond will have a funding margin over ESTR of 42 basis points, which is calculated as the 2% that they're paying minus the 1.58% they're receiving. Now looking at the 10 year US dollar bond with a 3.125% yield, that will also be combined with the receiver swap. The client will receive a fixed rate of 2.4% and therefore the margin over SOFR will be 72.5 basis points calculated as 3.125% minus 2.4%. Now to make the US dollar bond directly comparable with the euro bond, we need to swap the US dollar bond into euro. As we've already swapped both into floating, we can directly apply the cross currency basis swap The euro client will be looking to receive SOFR and therefore will pay ESTR minus 23 basis points on the cross currency basis swap versus SOFR flat. Assuming the margin should be swapped as well, the client will actually be looking to receive SOFR plus 0.725% instead of SOFR flat, and that's what we calculated above. The 72.5 basis points will be added to the US dollar leg. But we can't just add 72.5 basis points. We've been given the information that one basis point in euro is 1.045 basis points in US dollars. Or one basis point in US dollars is 0.95694 basis points in euro. So for the cross currency basis, swap to be PV neutral, the swap in including the margin would therefore be SOFR plus 72.5 basis points versus ESTR plus 46.38 basis points. And those 46.38 basis points are calculated by taking the minus 23, plus the 72.5 basis points converted at 1.045. So in conclusion, option one raising funds directly in euro would have a funding margin of 42 basis points over ESTR and option two raising funds in US dollars and then swapping into euro would have an effective funding margin of 46.38 basis points above ESTR. Making option one raising funds directly in euro cheaper.