Actuarial Outpost Annuity Conversion Factor
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#1
08-30-2005, 10:04 AM
 Alfies_Girl Join Date: Jun 2003 Posts: 11
Annuity Conversion Factor

I'm trying to review some option factors from a prior actuary for a new client.

Is there a nice formula for 50% J&S with "pop-up" if the beneficiary dies before the participant?
#2
08-30-2005, 02:13 PM
 Harpo Member SOA CCA AAA Join Date: Jan 2002 Posts: 32
Pop Up Formula

I usually just go back to 1st principals.

Assume:
ax = life annuity to member
ay = life annuity to beneficiary
axy = life annuity while either is alive (a 100% JS factor)
axybar = life annuity while both are alive (= ax + ay - axy)
j = JS percent (could be 50, 100 or whatever)
r = reduction factor for JS popup
\$1 = life annuity to member that will be equivalent to popup

ax = r * axybar + jr * (ay - axybar) + (ax - axybar)
\$1 for life = \$r while both are alive + jr\$ while only spouse is alive + \$1 while only member is alive

if j = .5 then ax = r*axybar + (1/2)* r * (ay - axybar) + (ax-axybar)
the ax drops out so just solve for r.

r = axybar/ [ axybar + (1/2) *(ay - axybar)]

for 100% JS popup, r = ay/axybar (kinda cool huh!)

ax is a life annuity
ay is a life annuity
axy is a 100% joint life annuity
so ... not too hard from there.

Hope that helps.
#3
09-26-2005, 07:45 PM
 Fuzzy Member SOA AAA Join Date: Oct 2001 Location: Far from the madding crowd...but closer than I used to be Studying for the "final" exam Favorite beer: Carlsberg Posts: 623

I finally got a chance to check this since my curiosity was piqued and I will probably have to explain the option in court someday: I think that Harpo had the right initial idea, but he can't spell (yes, there is money involved (principal), and he's probably referring to the rules (principles) involved), which should have warned me that something wasn't quite right: I think that he got the definitions of axy and axybar mixed up! (OK, since this forum won't support all of the pretty notation conventions ("upper 12", age subscripts, etc.) I'll have to stick with Harpo's usage and trust that no one will get too confused.)

My Life Contingencies textbook (C.W.Jordan) defines a "joint life" annuity as terminating upon the first death (i.e., it exists as long as both parties are alive), and uses the expression axy. "Joint and survivor" annuity is used for the "last survivor" function (i.e., the one that survives until the last death), and uses the expression axybar. (Yes, axybar = ax + ay - axy, and would be read as "an annuity paid as long as either is alive is equal to an annuity paid as long as one is alive plus an annuity paid as long as the other is alive minus an annuity paid as long as both are alive!") Actuarial Mathematics by Bowers, Gerber, et al, continues this convention. I hope I haven't missed a radical change in terminology and notational convention while concentrating on consulting for the past xxx years.

Compounding the problem is the terminology screw-up foisted upon us with ERISA: "joint and survivor" was used to describe the situation wherein a particular order of death was important when the order was originally immaterial. Jordan would use the term "contingent annuity" to describe the situation I wish would at least more properly be called the "qualified joint and survivor" annuity: a "50% joint and survivor" annuity technically reduces upon the first death no matter which it is; a "50% contingent annuity" reduces only if the first death is the beneficiary!

Anyway, if we're discussing an annuity wherein the "pop-up" occurs only if the beneficiary dies first, then it is a special "contingent" annuity, rather than a special "joint and survivor" annuity where the "pop-up" would occur upon the first death no matter who it was.

Properly read, Harpo's expression, r × axybar + ˝ × r × (ay - axybar) + (ax -axybar), would describe the (I think) rather strange annuity wherein a reduced amount is paid as long as either party survived with 50% of that reduced amount paid if the member survives the beneficiary and the beneficiary pays the plan the reduction if (she) survives the member! (Substitute the equivalent expression, simplify it, then read the result!)

I think that the proper expression for a 50% pop-up upon the member's death is r × axy + ˝ × r × (ay - axy) + (ax - axy) (Harpo would have been correct if he'd left off the "bar"), or (the rearragned ax + ˝ × r × ay - (1- ˝ × r) × axy.

The optional form factor then depends on the plan normal form: if it is a single life annuity, then the factor can be calculated by solving for r in the equation ax = ax + ˝ × r × ay - (1 - ˝ × r) × axy, so r = 2 × axy / (ay + axy). I'll leave it to the more ambitious actuaries to come up with the 10C&C normal form factor!

Of course, it is entirely possible that I've made a mistake in the algebra, I would welcome editorial comments and a good proofreader!

axy axybar ax ay
__________________
"Don't worry about the world coming to an end today. It's already tomorrow in Australia." (Charles Schulz)
"We created an environment where we didn't know what we were doing, but it was legal and making profits."(Bill Sharon, chief executive of Sorms)
"As soon as we solve one problem, another one appears. So let's try to keep this one going for as long as possible." (Pepper...and Salt, WSJ, 5/4/2011)
#4
09-26-2005, 09:26 PM
 Malik Shabazz Wiki Contributor SOA CCA AAA Join Date: Mar 2004 Location: God bless Chocolate City and its vanilla suburbs Posts: 12,813

Fuzzy, I'm not able to review your formulas right now, but you're right about the confusing terminology.

The annuity that US pension actuaries call a "joint and survivor annuity" is generally referred to as a "joint and contingent annuity." It's my understanding that most non-pension actuaries in the US and most non-US actuaries use the "correct" terminology, and US pension actuaries are out of step with the rest of our profession.

In 20 years of pension experience, I have only seen 1 pension plan that offered a "true" joint and survivor annuity as an optional form of payment, in which the benefit is reduced on the first death.
__________________
If I weren't out here every day battling the white man, I could spend the rest of my life reading, just satisfying my curiosity—because you can hardly mention anything I'm not curious about. — Malcolm X

#5
09-27-2005, 09:45 AM
 DaveyDo Member AAA Join Date: May 2005 Location: Cleveland Posts: 725

Quote:
 The annuity that US pension actuaries call a "joint and survivor annuity" is generally referred to as a "joint and contingent annuity." It's my understanding that most non-pension actuaries in the US and most non-US actuaries use the "correct" terminology, and US pension actuaries are out of step with the rest of our profession.
If I'm not mistaken, ERISA defined a "joint and survivor annuity" as what is actually a "joint and contingent annuity." US pension actuaries just went along with the ERISA terminology.

That's what I have been told. Has anyone else heard this? And no, I am not searching ERISA for it.
#6
09-27-2005, 10:03 AM
 lawfi5h Guest Posts: n/a

Quote:
 Originally Posted by DaveyDo If I'm not mistaken, ERISA defined a "joint and survivor annuity" as what is actually a "joint and contingent annuity." US pension actuaries just went along with the ERISA terminology. That's what I have been told. Has anyone else heard this? And no, I am not searching ERISA for it.
That was my understanding as well.
#7
09-27-2005, 10:46 AM
 Mark Cavazos Member Join Date: Sep 2004 Posts: 2,709

Quote:
 Originally Posted by DaveyDo If I'm not mistaken, ERISA defined a "joint and survivor annuity" as what is actually a "joint and contingent annuity." US pension actuaries just went along with the ERISA terminology. That's what I have been told. Has anyone else heard this? And no, I am not searching ERISA for it.
The contingent annuitant could be anyone: spouse, child, great-aunt twice removed....

I briefly looked at the Code. It defines a "qualified joint and survivor annuity" as being reduced upon the death of the participant. I do not know if there is a definition of a plain old joint and survivor annuity, i.e., without the adjective "qualified".
#8
09-27-2005, 11:09 AM
 Fuzzy Member SOA AAA Join Date: Oct 2001 Location: Far from the madding crowd...but closer than I used to be Studying for the "final" exam Favorite beer: Carlsberg Posts: 623

My point exactly, ERISA/REA uses "qualified joint and survivor annuity". When the legislators got so much of the rest of the rules technically correct, why they decided to screw up the terminology is beyond me (but understandable). The real shame is that the US pension "professionals" seem to be willing to allow the confusion to continue by allowing the common usage to drop the "qualified" clarifier.

While I'm on the soap box, how many of us are clear about using "immediate", "due", and "deferred" as proper clarifiers of "annuity"? and what about putting a frequency reference in there too? ...OK, that's enough for now...
__________________
"Don't worry about the world coming to an end today. It's already tomorrow in Australia." (Charles Schulz)
"We created an environment where we didn't know what we were doing, but it was legal and making profits."(Bill Sharon, chief executive of Sorms)
"As soon as we solve one problem, another one appears. So let's try to keep this one going for as long as possible." (Pepper...and Salt, WSJ, 5/4/2011)
#9
09-27-2005, 11:20 AM
 Traci Retired Administrator Join Date: Sep 2001 Posts: 5,507

Quote:
 Originally Posted by Fuzzy My point exactly, ERISA/REA uses "qualified joint and survivor annuity". When the legislators got so much of the rest of the rules technically correct, why they decided to screw up the terminology is beyond me (but understandable). The real shame is that the US pension "professionals" seem to be willing to allow the confusion to continue by allowing the common usage to drop the "qualified" clarifier.
I guess I continue to use it because it doesn't confuse me.

I've worked on 100's of plans and J&S has always meant reduced upon the death of the participant. If it means reduced upon the death of either, It's been called a "TRUE J&S".

One of those things that you just grow up with - so it doesn't seem strange.

Alfies_Girl,

I have a program that I use for pop-up factors -- if you want to send me your assumptions, I'd be happy to shoot you a table for comparison.
#10
09-27-2005, 08:26 PM
 Malik Shabazz Wiki Contributor SOA CCA AAA Join Date: Mar 2004 Location: God bless Chocolate City and its vanilla suburbs Posts: 12,813

Quote:
 Originally Posted by DaveyDo If I'm not mistaken, ERISA defined a "joint and survivor annuity" as what is actually a "joint and contingent annuity." US pension actuaries just went along with the ERISA terminology. That's what I have been told. Has anyone else heard this? And no, I am not searching ERISA for it.
Dumb pension actuaries, following the law instead of Chester Jordan.

Mumble mumble mumble.

__________________
If I weren't out here every day battling the white man, I could spend the rest of my life reading, just satisfying my curiosity—because you can hardly mention anything I'm not curious about. — Malcolm X