## A/E in A&E

We have often written about how modelling the force of mortality, μx, is superior to using the rate of mortality, qx.  This is all very well when you are building a formal model, but what about when you just want to quickly compare rates?  As it happens, the μx approach is quicker and more reliable, especially for portfolios with competing risks.

Consider a portfolio of term assurances where the policyholder can either lapse the policy or die.  For simplicity we will assume that each policyholder has only one policy, although in practice this is not the case and deduplication is required.  Suppose you want to compare the mortality rates between two portfolios which have very different lapse rates.  You cannot just divide the number of deaths by the number of lives, as the number of deaths is a function of the number of lapses — even if the underlying mortality rates are identical, the portfolio with the higher rate of lapses will automatically have fewer deaths due to fewer people being exposed to risk.

A qx analysis requires calculating so-called dependent rates of mortality, then making adjustments for the exposure question above.  Typically these adjustments assume that mortality and lapse are independent processes.  However, this is clearly not the case: so-called selective lapses occur when a healthy policyholder finds a cheaper policy elsewhere.  Less-healthy policyholders are therefore more likely to stay.  Thus, the qx analysis requires a simplifying assumption which is not correct.

What of the μx approach?  The analogue here is to calculate central rates of mortality, which are simply the crude estimators of μx.  The only difference is that the number of deaths is divided by the time lived, not the number of lives.  Happily, this is all that is required: no further adjustment is required for the lapses because this has already been done by using the time lived in place of the number of lives.  The μx approach does not require any further assumptions, and is therefore simpler than the qx approach.  Even better than simplicity is correctness: the μx approach does not require the (often false) assumption that decrements are independent.