Events, dear boy, events!

When asked what was most likely to blow a government off-course, Harold Macmillan allegedly replied "Events, dear boy, events!".  Macmillan may not have actually uttered these words (Knowles, 2006, pages 33-34), but there's no denying that unexpected events can derail your plans.  I was recently faced with some unexpected events, albeit in a rather different context.

In the UK there is a class of business called home-reversion plans.  This is where a cash-strapped property owner sells their house to an investor, subject to the right to continue living there as a tenant until death or entry into long-term care (LTC).  The investor receives no rental income and can only sell the property when all tenants have vacated it.  The investor obviously pays a lower-than-market-value price for the property at outset, since they will have to wait an unknown length of time until they get a return on their capital outlay.

On the face of it, this is a classic double-decrement model: the investor needs to set assumptions for mortality and LTC inception rates in order to price the transaction.  A traditional \(q_x\) model would be inappropriate for each risk, as a \(q_x\) model only permits one kind of event in a year, and thus only allows two possible states at the end of the year.  In contrast, with home-reversion plans there are two different events that could happen over a year to a given tenant, giving rise to three possible states at the end of a year:

1. The tenant could have died,

2. The tenant could have entered long-term care, or

3. The tenant could have survived.

\(q_x\) models are like coin-tosses: good for modelling binary outcomes only.  The traditional actuarial \(q_x\) approach to competing risks involves estimation formulae that adjust the exposure time to compensate for the fact that the outcomes are not binary.

In contrast, a survival model effortlessly handles competing risks by simply treating those events not of interest as censoring points.  Thus, both LTC events during the year and survival to the end of the year are treated as censoring points for a mortality model; similarly, both deaths during the year and survival to the end of the year are treated as censoring points for an LTC model.

Of course, the dedicated \(q_x\) practitioner could decide to lump deaths and LTC events together to form a single decrement.  This would make a degree of sense, since the investor is financially neutral regarding the tenant's actual mode of exit, just so long as it leads to repossession.  And creating a single decrement in this way avoids the hacky adjustment formulae to compensate for there being more than two states at the end of the year.

However, business reality is seldom as neat and clean as \(q_x\) models require.  In a recent analysis of the experience data for a portfolio of home-reversion plans, I found six different event types, meaning that there were seven observational possibilities for a tenant:

1. Death,

2. Entry into long-term care,

3. Repurchase of property,

4. Divorce,

5. Transfer,

6. Voluntary surrender of lease, and

7. Survival (no event).

How is a binary \(q_x\) model to deal with all this?  The answer is that it must make some assumptions about the distribution of events over the year, and these assumptions are seldom very realistic (Richards and Macdonald, 2024, Appendix B).  In contrast, survival analysis deals with such complexity using censoring - for example, all events other than death are treated as censoring points for a mortality model.  In contrast to \(q_x\) models, survival models require no assumptions about the distribution of competing exits over the year.