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When we value interest rate derivatives on any date $t$, we can estimate our future payments using some calibrated forward curve $f_s$, where $s$ is the spot date, and discount these back to $t$ using some calibrated discounting curve $d_t$.

Some software vendors (such as Bloomberg) consider any payments between $t$ and $s$ as being historical cash flows, which basically means that we value the swap on the spot date, not today. This approach has the benefit that any spot-starting swap can be directly compared to any legacy swap, since only the payments that come after the spot date will be considered (take for instance the case of a spot-starting swap vs a swap traded 1 year ago - the latter will have a payment on the spot date, this yielding present value, dv01 and par rate calculations different among the two swaps if we do not ignore the payment on the spot date). The drawback of this approach is that we naturally are not valuing the swap today which can be a problem for MTM on trading books.

Is there any convention for handling such accrued payment issues? Do we ignore all cash flows up until and including the spot date, such that we can directly compare, for instance, par rates of legacy swaps to spot-staring swaps, or is the convention to have a preference for "correct" date valuation?

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It’s important to include payments between t and s in the valuation of the trade, for several reasons: 1) the value of the trade drives the variation margin to be applied against the trade, whether it is cleared or otc. If you did not include these cash flows in value, there would be an error in the estimates of credit exposures between counterparties and clearing houses. 2) the value of the trade is used to give accurate mark to market valuations to various types of end users such as hedge funds and mutual funds. Again if these cash flows are not included, the fund valuations will be wrong.

It’s true that some vendors not concerned with the above may choose to ignore it for simplification purposes.

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