  Age-specific mortality rates are often disaggregated by different attributes,
such as sex, state, ethnic group and socioeconomic status. In making social
policies and pricing annuity at national and subnational levels, it is
important not only to forecast mortality accurately, but also to ensure that
forecasts at the subnational level add up to the forecasts at the national
level. This motivates recent developments in grouped functional time series
methods (Shang and Hyndman, 2017) to reconcile age-specific mortality
forecasts. We extend these grouped functional time series forecasting methods
to multivariate time series, and apply them to produce point forecasts of
mortality rates at older ages, from which fixed-term annuities for different
ages and maturities can be priced. Using the regional age-specific mortality
rates in Japan obtained from the Japanese Mortality Database, we investigate
the one-step-ahead to 15-step-ahead point-forecast accuracy between the
independent and grouped forecasting methods. The grouped forecasting methods
are shown not only to be useful for reconciling forecasts of age-specific
mortality rates at national and subnational levels, but they are also shown to
allow improved forecast accuracy. The improved forecast accuracy of mortality
rates is of great interest to the insurance and pension industries for
estimating annuity prices, in particular at the level of population subgroups,
defined by key factors such as sex, region, and socioeconomic grouping.
