  The number of people who receive a stable income for life from a closed
pooled annuity fund is studied. Income stability is defined as keeping the
income within a specified tolerance of the initial income in a fixed proportion
of future scenarios. The focus is on quantifying the effect of the number of
members, which drives the level of idiosyncratic longevity risk in the fund, on
the income stability. To do this, investment returns are held constant and
systematic longevity risk is omitted. An analytical expression that closely
approximates the number of fund members who receive a stable income is derived
and is seen to be independent of the mortality model. An application of the
result is to calculate the length of time for which the pooled annuity fund can
provide the desired level of income stability
