lazy man's question

[Andy the Actuary]

Can anyone provide an expression for a J&50% survivor annuity with 10 years certain -- the continued payment for the balance of the 10 years is either 100% or 50% depending upon whether or not the beneficiary predeceases the primary annuitant?

[Blinky the 3-eyed Fish]

J&50%S10C100%IBND50%IC!

IBND = if beneficiary not dead

IC = if croaked

[SoCalActuary]

Can anyone provide an expression for a J&50% survivor annuity with 10 years certain -- the continued payment for the balance of the 10 years is either 100% or 50% depending upon whether or not the beneficiary predeceases the primary annuitant?

I would call it a J&[fill in]S benefit guaranteed 10 years at the benefit rate payable upon the second death.

[Andy the Actuary]

I forgot to add the adjective "mathematical" to expression. I.e., is there a reference how to evaluate the little bugger?

We have:

(1) 50% 10 years certain

(2) 50% temporary a_xy:10 (i.e., so long as both are living)

(3) 50% certain if benefiicary predeceases participant withing 10 years -- this starts with a signma sign, contains some funny probabilities, and certain annuities of diminishing duration.

(4) a 10 year deferred reversionary J&50% annuity

Does this cover it?

Sorry to ask but I passed Part 4 in 1973 and have been out to lunch since then.

[SoCalActuary]

With the new requirements for cash flow projections, I just don't use the commutation functions anymore.

So I define an array of lx values for the primary & the secondary beneficiary separately.

I value at each year the probability of payment.

p(t) = l(t)/l(x) defines the probability of the primary participant payment at age t. Those probabilities are multipled by the full 100% benefit each year.

Similarly for the secondary beneficiary sp(t) = sl(t)/sl(x). They get a survivor portion (50%) x [1 - p(t)]

But in your special designed annuity, I also need to know who died first. So the probability of a payment on the secondary participant's 10 yr certain is determined by computing the probability that the primary had already died: [ 1 - p(t) ] x sp(t) within the first 10 years, as a factor in determining the probability that the secondary dies, computed for each of the remaining years. Those contingent payments get the 50% benefit rate.

Then I need to know the converse: I need the probability that the primary survived the secondary for each of the first 10 years, as a factor to determine the probability that the primary dies, applied to each of the remaining years of the guaranteed period. Those get the full 100% rate.

Have fun figuring this in your spreadsheet.

[david rigby]

SoCal has a good point, about avoiding commutation functions.

However, it may depend on the purpose for which Andy needs this: if used for an actuarial conversion factor, then good precision is desired. If used for valuing liabilities, a close approximation may suffice (assuming any affected liabilities are very small relative to the total).

[Andy the Actuary]

Very old plan has very old option which client refuses to remove even though it has never been elected. It is actually j&2/3 with 10 years guaranteed. Figuring on a j&100 the 10 year guarantee is worth about 2%, my inclination is value a J&2/3 and add 1 1/2 percent.

[jevd]

Very old plan has very old option which client refuses to remove even though it has never been elected. It is actually j&2/3 with 10 years guaranteed. Figuring on a j&100 the 10 year guarantee is worth about 2%, my inclination is value a J&2/3 and add 1 1/2 percent.

I think the Theory of Relativity would be simpler to explain.

"You can choose your friends but not your relatives"

"Insanity is hereditary. You get it from your kids."

Etc. Etc.

It's Friday and it's hot in Denver.

Go Rockies!!

[Mike Preston]

Can anyone provide an expression for a J&50% survivor annuity with 10 years certain -- the continued payment for the balance of the 10 years is either 100% or 50% depending upon whether or not the beneficiary predeceases the primary annuitant?

So, you are saying the certain isn't?

[Andy the Actuary]

Can anyone provide an expression for a J&50% survivor annuity with 10 years certain -- the continued payment for the balance of the 10 years is either 100% or 50% depending upon whether or not the beneficiary predeceases the primary annuitant?

So, you are saying the certain isn't?

Certainly, the certain is certain, but only for a certain amount, depending upon who dies first -- the primary annuitant, beneficiary, or the actuary. About this I am certain.

[Mike Preston]

Can anyone provide an expression for a J&50% survivor annuity with 10 years certain -- the continued payment for the balance of the 10 years is either 100% or 50% depending upon whether or not the beneficiary predeceases the primary annuitant?

So, you are saying the certain isn't?

Certainly, the certain is certain, but only for a certain amount, depending upon who dies first -- the primary annuitant, beneficiary, or the actuary. About this I am certain.

So, wouldn't the formula be the normal 100% certain benefit, less the reduction caused by the inopportune death?

[Andy the Actuary]

So, wouldn't the formula be the normal 100% certain benefit, less the reduction caused by the inopportune death?

Yes, sorta, but it seems as if I the carve out is still a sigma sign with with some decreasing guarantee since in (Chester Jordan terms), it only subtracts out is "x" dies before "y".

each term looks something like v^t_tp_xyq_x+tcertain(10-t)

It's been awhile but I'm possibly off in my subscripts. My point is there does not appear to be a closed expression for this monster.

[Mike Preston]

Sure there is. Give me a couple of numerical examples showing the original benefit and the continuation benefit and duration in the case of death, just to make sure it is understood what you are driving at, and I'm sure somebody will come up with it.

[Andy the Actuary]

Sure there is. Give me a couple of numerical examples showing the original benefit and the continuation benefit and duration in the case of death, just to make sure it is understood what you are driving at, and I'm sure somebody will come up with it.

Okay, son, I'm from Missouri, so show me.

(1) Payment so long as participant is living, $1,000

(2) Payments are guaranteed for 10 years

(a) If within first 10 years, participant dies with surviving beneficiary, payment of $667 is payable for balance of 10

years and thereafter for the life of beneficiary

(b) If withing first 10 years, participant dies after beneficiary dies, payment of $1,000 is payable for balance of 10

years.

(3) If death of participant occurs after 10 years

(a) If beneficiary survives, then payment of $667 is payable so long as beneficiary survives

(b) If beneficiary has predeceased participant, no further payments are due

[Mike Preston]

"Figuring on a j&100 the 10 year guarantee is worth about 2%, my inclination is value a J&2/3 and add 1 1/2 percent."

I don't know what assumptions you are using, but at 65/62/94GAR/5% 100%J&S is about 172.28249 and adding the 10 year guarantee increases it only to 172.74440, so the increase is not 2% but 0.27%. That is an order of magnitude less.

In any event, try this. The value you seek is the sum of three separate terms.

Term 1: 100% J&S of $666.67 with 10 year certain

Term 2: SLA of $333.33 (primary annuitant)

Term 3: Certain death benefit of $333.33/month for X years, payable on death of the primary annuitant within 10 years, where X is the number of years left in the initial 10 year period, but only if the beneficiary predeceases the primary annuitant.

The first two are trivial. The third one is a pretty simple 10 year calculation (or 120 months if you develop your mortality in monthly q's, as I have done - using Balducci or Constant Force of Mortality, whichever suits your fancy). You know how to do this calculation, but it is merely:

Sum over 10 years (or 120 months) of the following terms:

probability beneficiary is dead at beginning of period

times

probability primary annuitant is alive at beginning of period

times

probability primary annuitant dies during period

times

discount factor for interest only

times

present value of $333/month for remaining certain period.

Are we having fun, yet?

Term 3 is very, very low. My guess is about one order of magnitude less than the adding of the 10 year guarantee to begin with. Something like 0.02%.

In any event, I don't know why you decided that, in the end, you would value a J&2/3 plus 1.5%, but that is almost exactly what I get when I add the 10 year guarantee to the joint and 2/3 (162.03134 versus 164.44434), above.

So, if you are saying that you are going to essentially ignore the increase from $667 to $1000 which takes place only if the primary is predeceased by the secondary within the first 10 years, I think you came to the right conclusion as the cost is very, very low.

[Andy the Actuary]

Mike, thank you. I didn't mean for you to go to all this effort. What you needed to say was "Andy, you're getting older, and your eyes may be deceiving you. Ccheck your math." You're correct. My magnitude is off and I had gotten .2% rather than 2%. What this means is unless the participant and beneficiary are ancient, the guarantee is deminimis (we're dealing in theory anyway) and can be ignored. The guarantee is certainly worth less if some of the benefits are smaller. Thanks for your thoughtful reply.

Other than Balducci Vineyards, I've not heard the name Balducci in over 35 years.