The problem of expressing cumulative detrimental effect of radiation exposure is revisited. All conventionally used and computationally complex lifetime or time-integrated risks are based on current population and health statistical data, with unknown future secular trends, that are projected far into the future. It is shown that application of conventionally used lifetime or time-integrated attributable risks (LAR, AR) should be limited to exposures under 1Gy. More general quantities, such as excess lifetime risk (ELR) and, to a lesser extent, risk of exposure-induced death(REID), are free of dose constraints, but are even more computationally complex than LAR and AR and rely on the unknown total radiation effect on demographic and health statistical data. Appropriate assessment of time-integrated risk of a specific outcome following high-dose (more than 1Gy) exposure requires consideration of competing risks for other radiation-attributed outcomes and the resulting ELR estimate has an essentially non-linear dose response. Limitations caused by basing conventionally applied time-integrated risks on current population and health statistical data are that they are: (a) not well suited for risk estimates for atypical groups of exposed persons not readily represented by the general population; and (b) not optimal for risk projections decades into the future due to large uncertainties in developments of the future secular trends in the population-specific disease rates. Alternative disease-specific quantities, baseline and attributable survival fractions, based on reduction of survival chances are considered here and are shown to be very useful in circumventing most aspects of these limitations. Another main quantity, named as radiation-attributed decrease of survival (RADS), is recommended here to represent cumulative radiation risk conditional on survival until a certain age. RADS, historically known in statistical literature as cumulative risk, is only based on the radiation-attributed hazard and is insensitive to competing risks. Therefore, RADS is eminently suitable for risk projections in emergency situations and for estimating radiation risks for persons exposed after therapeutic or interventional medical applications of radiation or in other highly atypical groups of exposed persons, such as astronauts.