Key takeaways
  • Life tables stop at whole-number ages
  • Variable annuities mix death timing with market moves
  • Bounds replace one hard guess with a safe range
  • Two views: path by path, or on average

If an insurer prices a variable annuity, the missing gap between two birthdays can matter as much as the market. Life tables only give survival probabilities at integer ages, so they leave the deaths in between unspecified. This paper shows how to turn that missing information into hard upper and lower bounds for lifetime-dependent quantities, instead of relying on a single fractional-age assumption or mortality model. The authors derive two sets of bounds in a joint actuarial-financial framework: one where every mortality trajectory is almost surely consistent with the one-year survival probabilities, and another where mortality can vary but still matches those probabilities in expectation. The result is a robust way to characterize the worst- and best-case values of contracts whose payoffs depend on both lifespan and financial assets. That makes the output useful for life insurance and for managing mortality risk when the life table does not tell the whole story.

A life table can say how many people survive to age 70. It cannot say who dies at 70 and three months. That missing slice sounds tiny. For a variable annuity, it can change the price. These contracts pay out based on both a person's lifetime and the market path. So the gap between two birthdays can matter as much as a stock swing. A rule that fills that gap can push the result up or down. The surprise is simple. You do not need a single guess for those missing months. You can turn the gap into best- and worst-case values instead. That gives price bands, not false precision. And price bands can be safer than neat answers.

Why the missing months matter

The core result is not a new mortality guess. It is a pair of bounds. One bound gives the highest possible contract value. The other gives the lowest possible value. Both stay faithful to the life table at whole-number ages. Both also respect the financial market used in the pricing task. That makes them safer than one fragile rule that spreads deaths between birthdays. The method works with hybrid functionals, which are mixed scores that depend on death time and market payoffs. Then it searches across all allowed mortality paths. One version keeps each path fully consistent with the table. The other lets the path move, but only so the average still fits. In plain terms, one version holds every path to the table. The other holds only the average path to it. That split gives two complementary views of risk. It also shows how far a contract can move without breaking the data.

How the bounds are built

Think of the common pricing trick as filling in the blanks by hand. A fractional-age rule does that. It spreads deaths inside each year in a chosen way. A mortality-rate model does something similar. It tries to sketch a full curve between birthdays. Both can work. Both also bake in strong guesses. This method avoids that trap. It asks for every mortality process that still fits the table. Then it asks which contract values survive that whole set. That search is a way to test a price under uncertainty. You can think of it as looking for the best and worst legal moves. The result is a safer envelope around the price. It does not pretend the missing months are known.

  • Life tables give survival chances only at whole-number ages.
  • Variable annuities mix death timing with market moves.
  • The new bounds replace one hard guess with a safe range.

“These distinct yet complementary approaches provide a new robust joint actuarial-financial framework for managing mortality risk in life insurance.”

From the abstract

What insurers gain from a range

That range matters for pricing. A single guess can look neat and still be wrong. The bounds show how far the contract value can move without breaking the table. That helps when a product pays on both life span and market gains. It also helps when a guarantee gets costly near a birthday cutoff. Instead of pretending the missing months are known, the method treats them as risk. That is a better fit for mortality risk, the uncertainty about when people die. It gives insurers a way to test worst and best cases before they set a price. It also helps them see when a contract depends too much on one hidden assumption.

Where this leaves pricing

The surprise does not vanish. The missing months between birthdays do not force one answer. They force a range. That makes the next pricing question sharper. How wide is the range for this contract, this market, and this table? This method gives the frame for that test. It can now sit beside life-table data instead of a guessed fractional-age rule. For products like variable annuities, that shift could change how far a price band can stretch. It turns a hidden gap into a visible stress test. That is the real payoff.