It is possible to calculate with lagged O/N rate, shifted coupon period etc.
Yes, the terms I often hear to describe these are "lookback" and "observation shift". Referring to a shifted observation of the O/N rate and a shifted coupon period respectively (ri and ni in the below equations). These methods primarily aid in market participants being able to calculate accrual and coupon payments in advance of settlement. A 2 day lookback is common in SOFR cash markets and a 5 day lookback is common in SONIA and ESTR cash markets. Observation shifts are most common in SOFR markets.
As mentioned there is no single convention for O/N rate calculations and the FRNs have introduced a lot more variation than was standard in the derivative market. However, for compounding overnight rates, the conventions can be thought of as small adjustments to the ISDA OIS compounding formula.
Here are how some of the key conventions impact the calculation of the annualized coupon rate for a period. Accrual is then just scaling the annualized rate by the applicable year fraction.
Notation:
\begin{align*}
& d_b = the\,number\,of\,business\,days\,in\,the\,interest\,period \\
& d_c = the\,number\,of\,calendar\,days\,in\,the\,interest\,(or\,observation)\,period \\
& r_i = the\,interest\,rate\,applicable\,on\,business\,day\,i \\
& n_i = the\,number\,of\,calendar\,days\,for\,which\,rate\,r_i\,applies \\
& N = days\,in\,the\,year\,based\,on\,market\,convention \\
& k = number\,of\,lookback\,days\,applied\,to\,the\,security\\
& l = number\,of\,lockout\,days\,applied\,to\,the\,security
\end{align*}
- Average
\begin{equation}
averageRate = \left[\sum_{i=1}^{d_{b}} \left(\frac{r_{i} \times n_{i}}{N} \right) \right] \times \frac{N}{d_c} + margin
\end{equation}
I deleted the 1 that was up here
- Compounded
\begin{equation}
compoundedRate = \left[\prod_{i=1}^{d_{b}} \left(1+\frac{r_{i} \times n_{i}}{N} \right) -1 \right] \times \frac{N}{d_c} + margin
\end{equation}
- Compound rate and margin
\begin{equation}
compoundedRate = \left[\prod_{i=1}^{d_{b}} \left(1+\frac{(r_{i}+margin) \times n_{i}}{N} \right) -1 \right] \times \frac{N}{d_c}
\end{equation}
- Compound with lookback
\begin{equation}
compoundedRate = \left[\prod_{i=1}^{d_{b}} \left(1+\frac{r_{i-k} \times n_{i}}{N} \right) -1 \right] \times \frac{N}{d_c} + margin
\end{equation}
- Observation shift
\begin{equation}
compoundedRate = \left[\prod_{i=1}^{d_{b}} \left(1+\frac{r_{i-k} \times n_{i-k}}{N} \right) -1 \right] \times \frac{N}{d_c} + margin
\end{equation}
Note: Days of accrued can also be adjusted for the observation shift. In my mind this is very similar in impact to a payment delay, just applied in a convoluted way, but alas, we see it in the wild.
- Lockout
\begin{equation}
compoundedRate = \left[\prod_{i=1}^{d_{b}-l-1} \left(1+\frac{(r_{i}) \times n_{i}}{N} \right)\prod_{i=d_b-l}^{d_{b}} \left(1+\frac{(r_{d_b-l}) \times n_{i}}{N} \right) -1 \right] \times \frac{N}{d_c}+margin
\end{equation}
From the Fed: "Templates for using SOFR"
ISDA: Memorandum on RFR methods
