Tontines are a type of investment in which the account balances of shareholders who die are forfeited, divided up, and transferred to shareholders who are still living. Exactly how they are divided up and transferred is a critical element of tontine design. Everyone should care about this whether they are investors, prospective investors, advisors, or regulators.

#### Dividing Forfeitures Fairly and Equitably

A fundamental principle of tontines is that their design should be fair to all investors. This means that forfeited balances be transferred to survivors in an equitable manner such that no investor (and therefore no class of investor) is unfairly disadvantaged. But how? And what does this even mean? To help answer this, let’s start with a few intuitive examples.

#### Fair Games

A game or bet is fair to all parties if each party’s expected gain is zero. Consider the classic coin flip in which a player wins (say 1 USD) if the coin lands on heads, and loses (say, 1 USD) if it lands on tails. There is a 50% chance of each outcome and thus the expected gain is:

If the expected gain were greater than zero, the player would have an advantage. Similarly, if the expected gain were less than zero, the player have a disadvantage.

Now say that the game involves rolling a six-sided die in which the player wins with a roll of 6 and loses otherwise. The game is fair when wins pay five times more than losses, because the expected gain in that case is:

#### Fair Tontines

In a tontine, investors lose their balances when they die. Our objective, then, is to design a tontine such that the expected incremental gain to survivors fairly offsets the amount lost when they die. Keep in mind that this “tontine gain” (sometimes referred to as longevity credits) is an additional amount received above and beyond any underlying investment returns.

The intuition for how this must be accomplished is remarkably similar to the simple games discussed earlier. Let’s label a member’s account balance (i.e., the amount they have at stake) s, the probability that the member dies in the current month q, and the required tontine gain c. Fairness requires that:

The left side of this equation represents the expected value of losses this month due to the risk of dying. The right side of the equation represents the required amount that the investor must expect to gain this month by surviving. Solving for c, we get:

The quantity c represents the tontine gain, and q/(1-q) represents the tontine gain rate. As with flipping a coin or rolling a die, the gain received qs/(1−q) is simply the gain rate (i.e., the winning payoff rate) times the stake.

#### Insights

Perhaps the most interesting insight we can make here is that the fair tontine gain of every member is a function only of that member’s own probability of dying q and amount invested s. This fact comes as a surprise to many who may be inclined to ask, “Hey wait a minute… I don’t want to be in a tontine with a lot of younger people, do I?” Well, not if the tontine is designed unfairly! But the q’s and s’s of the other members won’t matter if the design is fair. And that is why fair design is so important.

At this point you might be saying to yourself, “Hey, wait another minute… what about adverse selection? Isn’t it true that only those who believe they have average or better life expectancy are likely to invest?” The answer is yes, but keep in mind that tontine designers will use mortality rates that reflect this. The same is true of insurance actuaries whose job it is to price and set reserves for annuities.

If you and I are different ages or different genders, our life expectancy will differ. Since a 90-year-old has a higher risk of dying next month than a 60-year-old, the 90-year-old must therefore receive a higher proportion of the tontine gains per dollar invested. Accordingly, members will expect to receive an increasing proportion of tontine gains per dollar invested as they grow older.

Another important insight is that tontine investments are scalable in that the gains and losses are a linear function of the amount s that each member places at stake.

#### Is It Really This Simple?

Conceptually, yes. Tontine design does involve a lot of little details, but the good news is that when all the details are properly considered, fair and equitable tontine design is indeed possible.