Not quite right, I'm afraid, Bystander, but nearly there. UPR's can also be calculated using an uneven incidence of risk (ie pro-rata over the year is not always the assumption when calculating a UPR - more about this in Subject ST7).
The key similarity is that both are trying to estimate the outstanding claims (and associated expenses) in the unexpired period of cover. The key difference is that whereas the URR is what you're actually trying to calculate (allowing for actual future estimated risk), the UPR is calculated simply by taking a proportion (again, not necessarily pro-rata) of the premium actually received. So, for example, if the premium is too low then taking a proportion of it is not going to cover future claims, and you need an extra amount to bring you up to the URR. (This extra amount is often called the additional unexpired risk reserve, or AURR).
Hope this helps.
P.S. I'm a GI tutor, not a health tutor, so forgive the GI bias!!