We are trying to estimate the lifetime cumulative default rates for consumer loans, using the Cox proportional hazard model. Since we have a number of different maturities (and the majority have not reached maturity yet) we are wondering whether it is acceptable to use the dependent time variable expressed as a fraction of time? I.e. if a loan has a running time of 24 months and is currently in month 12, the fraction is 12/24 = 0.5. In our dataset we have maturity times ranging from 12-180 months, and the time variable would then range from 0-1 instead of the actual lenghts.
The reason I am asking is that I want to know if this causes any problems, besides the implicit assumption that the baseline hazard follows the same pattern regardless of the length of the loan? We were also thinking of potentially correcting for this assumption by using the length of the loan as one of the covariates, would this be appropriate?
Any thoughts or comments would be appreciated, as well as references to previous literature where they have used a similar procedure. Thanks!