A Wishart Mortality Model Prices Joint Survival Annuities in Closed Form

When two lives are tied to the same pension-like payout, their shared survival risk matters as much as the money itself. This paper tackles that problem for joint survival annuities, contracts that keep paying while both lives survive, and for guaranteed joint survival annuity options. The authors build a linear-rational mortality model using a Wishart process, a stochastic matrix-valued process that lets the two mortality intensities move together in a flexible, dependent way while staying positive. With that setup, they derive a closed-form expression for the joint survival annuity and also obtain the distribution, including the density and cumulative distribution, of that annuity. They also develop polynomial expansions for the state variable, which give fast and accurate approximations for the guaranteed annuity option and make the model easier to implement. The result is a unified framework for pricing and managing joint mortality risk in multi-life longevity products.

Two people can retire on one payout. That payout can stop when either life ends. So the pricing problem is not just about one lifespan. It is about two lives that may move together. An annuity is a steady payout that keeps coming over time. This paper tackles that knot with a Wishart-based mortality model. A Wishart process is a matrix-valued random process that keeps the model's mortality intensities positive. Mortality intensities are the instant death rates the model tracks. The surprise is practical. Once the deaths are linked in that way, the annuity price no longer needs rough guesswork. The setup gives exact formulas for the joint survival annuity and for an option on that annuity.

Closed form for two linked lives

The model does three big jobs at once. It prices the joint survival annuity in closed form. That means the answer comes from algebra, not simulation. It also prices the guaranteed joint survival annuity option, the protected version of that payout. Beyond the prices, the model spells out the full distribution of the annuity. That includes both the density, which shows the curve's shape, and the cumulative distribution, which shows the chance below each level. In plain terms, it tells you how likely each payout level is. The parameter set also makes the mortality intensities and their dependence explicit. That is useful because the model does not hide the link between the two lives. It shows it. The result is one framework that covers pricing and risk view together.

How the Wishart state keeps the maths tractable

The Wishart process is the engine here. It is a stochastic continuous matrix process. That means it moves at random over time, but in a controlled matrix form. The model uses that state variable, the model's moving input, to drive mortality intensities. Because the intensities stay positive, the setup avoids nonsense like negative death rates. The linear-rational frame, a setup with neat algebra, then turns those moving pieces into formulas that are easier to handle. The model also develops polynomial expansions for the state variable. A polynomial expansion is a sum of powers that can stand in for a harder expression. Here, those expansions give fast and accurate approximations for the guaranteed joint survival annuity option. They also make the model easier to set up.

Why this matters for dependent-life products

Pricing becomes less like a black box and more like a map. With closed-form prices, product design can move faster. The same setup also shows how the two mortality rates depend on each other. That matters for dependent lives, where one person's survival pattern can tilt the payout. The distribution results add another layer. They do not give only one number. They show the shape of the payout risk itself. The polynomial approximations matter too. They keep the guaranteed annuity option fast enough to use in practice. In a market where longevity risk can last for decades, speed and clarity both count.

Where the surprise lands

The twist is that a matrix model gives exact answers instead of messy guesses. That is what makes the result useful. A joint survival annuity can now be priced in one setup. Its payout shape can be read. Its guarantee can be approximated quickly. That opens the door to faster comparisons across contract designs. It also gives a cleaner way to study how shared mortality risk changes the payout. The surprise is not the complexity. It is that the complexity still yields closed-form control.