A well known broker quotes cap/floors as spot premium for ATM straddles but forward premium for the skew, given that the difference between spot premium and forward premium is that the option is not discounted at maturity for the latter, how do I account for this in a cap that consists of multiple caplets?
For example I have a caplet with spot premium defined as
\begin{equation} V_{caplet}^{spot}(t, T_j)={\tau_jP\left(t,\ T_j\right)\mathbb{E}_t^{T_j}\ \left[F_j\left(T_{j-1}\right)-K\right]}^+, \end{equation} then the equivalent forward premium would be \begin{equation} V_{caplet}^{forward}(t, T_j)={\tau_j\ \mathbb{E}_t^{T_j} \left[F_j\left(T_{j-1}\right)-K\right]}^+. \end{equation} However, if I have a 1Y cap consisting of caplets such as \begin{equation} V_{cap}^{spot}(t)=\sum_{j=1}^{N}{\tau_jP\left(t,\ T_j\right)\mathbb{E}_t^{T_j}\ \left[F_j\left(T_{j-1}\right)-K\right]}^+, \end{equation} where $N=4, T_N=1Y$ for simplicity, then how do you define the forward premium of this cap? Do I simply take the discount factors as $1$ everywhere? i.e.
\begin{equation} V_{cap}^{forward}(t)=\sum_{j=1}^{N}{\tau_j\mathbb{E}_t^{T_j} \left[F_j\left(T_{j-1}\right)-K\right]}^+. \end{equation}
