  There is growing interest in the design of pension annuities that insure
against idiosyncratic longevity risk while pooling and sharing systematic risk.
This is partially motivated by the desire to reduce capital and reserve
requirements while retaining the value of mortality credits; see for example
Piggott, Valdez and Detzel (2005) or Donnelly, Guillen and Nielsen (2014). In
this paper we generalize the natural retirement income tontine introduced by
Milevsky and Salisbury (2015) by combining heterogeneous cohorts into one pool.
We engineer this scheme by allocating tontine shares at either a premium or a
discount to par based on both the age of the investor and the amount they
invest. For example, a 55 year-old allocating $\$10,000$ to the tontine might
be told to pay $\$$200 per share and receive 50 shares, while a 75 year-old
allocating $\$8,000$ might pay $\$$40 per share and receive 200 shares. They
would all be mixed together into the same tontine pool and each tontine share
would have equal income rights. The current paper addresses existence and
uniqueness issues and discusses the conditions under which this scheme can be
constructed equitably -- which is distinct from fairly -- even though it isn't
optimal for any cohort. As such, this also gives us the opportunity to compare
and contrast various pooling schemes that have been proposed in the literature
and to differentiate between arrangements that are socially equitable, vs.
actuarially fair vs. economically optimal.
