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 et al. (2005) or Donnelly et al. (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.

This article examines the impact of organizational structure on risk taking across different lines of property insurance (fire, marine, vehicle and specialized property insurance) in Sweden from 1913 to 1939. Based on the theoretical arguments whereby the mutual organizational form has a competitive advantage in underwriting homogeneous but unknown risk distribution, while the stock organizational form is more likely to underwrite more volatile and heterogeneous risk categories, we conclude that organizational form has a significant impact on risk taking. Our empirical analysis shows that the risk taking, measured as incurred claims to anticipated losses, was on average lower among mutual insurers. When comparing across lines of insurance, the analysis shows that the mutual form was more successful in keeping down risks in fire and marine, while less so in vehicle and specialized property insurance. Stock companies mitigated the higher risk by ceding more premiums to reinsurers and by diversifying more across different lines of insurance.