If we are at time t, consider an vanilla interest rate swap starting at some future time T_0, with maturity T_N, based on single curve non-arbitrate relationship, we should have:[$]Swap_rate\sum_{i​=1}^N\delta P(0,T_i)=P(0,T_0)-P(0,T_N)[$]When we do the bootstrapping, what instrument do we usually use to determine the discount factor from present to the effective starting date T_0, from which we can then bootstrap the subsequent discount factors with market swap rates?

this is why its called "bootstrapping" - you use all the instruments in the curve in the order of their maturities.

But for P(0, T_0), shall we pick an interest rate swap with maturity T_0 and the effective starting date 0? Is that what you mean?

you don't usually use a forward starting swap to build a yield curve. but if you did, then the discount factor will be the one you obtain from the yield curve. let's say that for some reason you are using a 10 year swap that is forwarding starting in 6months then you P(0,T0) is the 6 month discount factor from the yield curve which you should be able to obtain from the curve.

I see. Which one is more liquid in terms of cross currency basis swaps, forward starting or spot? As I'm trying to construct the spread adjusted discount curve vs US dollar curve. QuoteOriginally posted by: daveangelyou don't usually use a forward starting swap to build a yield curve. but if you did, then the discount factor will be the one you obtain from the yield curve. let's say that for some reason you are using a 10 year swap that is forwarding starting in 6months then you P(0,T0) is the 6 month discount factor from the yield curve which you should be able to obtain from the curve.