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This is my first time delving into dual curves, or multiple yield curves. A question struck me about using OIS discounting when choosing the swap rate of a new IRS.

Without multiple yield curves I thought it went something like this

$$ V_{swap} = V_{fixed}-V_{float} ~~ \text{with $N$ as par}. $$

where at the time of inception, $V_{float} = N$ due to the fact that the discount and projection rate were the same. This way, we simply put the coupon $C$ of $V_{fixed}$ such that it equals par. Thus $C$ is a par-yield and can be used as such when looking at market data.

However, if we discount the cash flows of the instruments with an OIS rate, it mustn't hold that $V_{float}=N$ initially. If it doesn't, how can we choose a $C$ such that $V_{fixed}$ both becomes valued at the float value, but also becomes a par-yield.

To put it more explicitly, we want

$$ N(\sum_{i=1}^{n-1}Cd(T_i)+d(T_n)) = N(\sum_{j=1}^{m-1}\text{LIBOR}_jd(T_j) + d(T_m)) $$ where $d(T_i)$ are the discount factors, $c_j$ is the appropriate LIBOR rate, n is the number of fixed cash flows and m is the number of floating cash flows. Again, a coupon cannot be chosen so that it is both a par-yield and necessarily equates to the FRN initially.

Is there something I am missing here? I've searched through some articles but all they say is that "something has to be adjusted" in this scenario. But never what that exactly might be.

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    $\begingroup$ You are not missing anything. These formulas hold in an ancient world before the 2007/08 credit crunch where 1m Libor was as as good (meaning riskless) as any other Libor (3m or 6m). Post that crisis banks got anxious and charged spreads on their Libors. To factor this into the formulas an old idea of cross currency swaps was picked up that uses different curves for forecasting and discounting. (This was greatly refined over time). Think this way: keep $C$ fixed and switch to discounting with OIS. Then you will achieve par when you add a spread to LIBOR. $\endgroup$
    – Kurt G.
    Oct 12, 2022 at 8:57
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    $\begingroup$ Some good links are this and this. $\endgroup$
    – Kurt G.
    Oct 12, 2022 at 8:58

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