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.
