Joint life annuity

A joint life annuity is a series of periodic payments contingent on the lifetime of two (or more) lives, where the payment amount can vary depending on if only 𝑥 is alive, or if only 𝑦 is alive, or if both 𝑥 and 𝑦 are alive.

Types of joint life annuities

To get detailed information about a specific type of joint life annuity, click its name in the first column.

Type of Annuity	Description
Joint & survivor annuity	An annuity payable during the joint lifetime of 𝑥 and 𝑦, with a provision that upon the first death, a percentage of the original amount is payable during the remaining lifetime of the survivor, regardless of who dies first.
Contingent joint & survivor annuity	An annuity payable during the joint lifetime of 𝑥 and 𝑦, with the provision that: · if 𝑦 dies first, then the original amount is payable during the remaining lifetime of 𝑥. · if 𝑥 dies first, then a percentage of the original amount is payable during the remaining lifetime of 𝑦.
Pop-up annuity	A joint & survivor annuity, with a provision that upon the death of 𝑦 while 𝑥 is alive, the payment amount to 𝑥 "pops up" to a given amount for the remaining lifetime of 𝑥.
Reversionary annuity	An annuity payable during the lifetime of 𝑦, with payments beginning upon the death of 𝑥.
Joint annuity	An annuity payable during the joint lifetime of 𝑥 and 𝑦, with payments stopping upon the first death, regardless of who dies first.
Period certain joint & survivor annuity	A joint & survivor annuity with payments for the remainder (if any) of a guaranteed period after the death of both 𝑥 and 𝑦.

General formula

The present value of a joint & survivor annuity for two lives at ages 𝑥 and 𝑦 is:

where

benefit amount payable while both 𝑥 and 𝑦 are alive

benefit amount payable to 𝑥 after the death of 𝑦

benefit amount payable to 𝑦 after the death of 𝑥

and

present value of a life annuity payable to 𝑥

present value of a life annuity payable to 𝑦

present value of a joint life annuity payable to 𝑥 and 𝑦

Joint & survivor annuity

A joint & survivor annuity is a series of periodic payments during the joint lifetime of 𝑥 and 𝑦, with a provision that upon the first death, a percentage of the original amount is payable during the remaining lifetime of the survivor, regardless of who dies first.

The present value at ages 𝑥 and 𝑦 of a 𝑃% joint & survivor annuity of 1 per year, payable at the beginning of the year is calculated by the general formula:

with

hence the present value is calculated by this formula:

Note these special cases:

· For a 100% joint & survivor annuity, 𝑃 = 1 thus

· For a 50% joint & survivor annuity, 𝑃 = .5 thus

Contingent joint & survivor annuity

A contingent joint & survivor annuity is a series of periodic payments during the joint lifetime of 𝑥 and 𝑦, with the provision that:

· if 𝑦 dies first, then the original amount is payable during the remaining lifetime of 𝑥.

· if 𝑥 dies first, then a percentage of the original amount is payable during the remaining lifetime of 𝑦.

The present value at ages 𝑥 and 𝑦 of a 𝑃% contingent joint & survivor annuity of 1 per year, payable at the beginning of the year is calculated by the general formula:

with

hence the present value is calculated by this formula:

Pop-up annuity

A pop-up annuity is a joint & survivor annuity, with a provision that:

· if 𝑦 dies first, then the amount payable to 𝑥 "pops up" to a higher amount, typically the amount of benefit payable to 𝑥 benefit before a reduction for actuarial equivalence.

· if 𝑥 dies first, then a percentage of the original amount is payable during the remaining lifetime of 𝑦.

The present value at ages 𝑥 and 𝑦 of a 𝑃% pop-up annuity of 1 per year, payable at the beginning of the year is calculated by the general formula:

with

and

the pop-up benefit amount payable to 𝑥 upon the death of 𝑦

In theory, 𝐵 could be anything, but in practice we assume that 𝐵 equals the benefit amount that would be payable to 𝑥 in the event that 𝑥 had not elected a 𝑃% joint & survivor annuity.

hence the present value is calculated by this formula:

Furthermore, the pop-up benefit 𝐵 is calculated by this formula:

Small modifications will be made to this formula if the normal form of benefit is something other than a straight life annuity.

Reversionary joint annuity

A reversionary joint annuity is a series of periodic payments where nothing is paid during the joint lifetime of 𝑥 and 𝑦, and:

· if 𝑦 dies first, then payments made during the remaining lifetime of 𝑥.

· if 𝑥 dies first, then nothing is paid.

The present value at ages 𝑥 and y of a reversionary annuity of 1 per year, payable at the beginning of the year is calculated by the general formula:

with

hence the present value is calculated by this formula:

Joint annuity

A joint annuity is a series of periodic payments payable during the joint lifetime of 𝑥 and 𝑦, with payments stopping upon the first death, regardless of who dies first.

The present value at ages 𝑥 and 𝑦 of a joint annuity of 1 per year, payable at the beginning of the year is calculated by the general formula:

with