Full Price vs Clean Price

To compute the ‘full price’ of a bond, we take the present value of the expected cash flows based on the following formula: PV = CF / (1+i)^t-1+w where w = days between settlement and next coupon / days in coupon period So say the settlement date is 78 days into a 182-day period…then w = 0.4286 So the PV of the CF one year from now would be discounted based on only the unaccrued portion (1-1+w), etc… This is the ‘full price’ (according to the text) The ‘clean price’ is the full price less accrued interest, where accrued interest is 1-w. I don’t understand why we would deduct accrued interest from the full price based on the above calculation. The way I interpret the above formula, and correct me if I’m wrong, is the PV of the CF TODAY, RIGHT NOW, discounting only the days we haven’t yet received interest. If we don’t base it on the above formula, then the full price makes sense to me, i.e. take the PV value as CF / (1+i)^t. Then we could simply deduct the accrued interest, but that would effectively yield the same result as the above formula. So can someone please clarify what the ‘full price’ is, because the text says it’s the PV including accrued interest, and yet, it also says the above formula (which I understand as the PV NOT including accrued interest) as the full price. Thanks