Contradictions in longevity expectations and longevity vol as we age

David Cantor sent me a piece today on pensions and annuities which reminded me about how longevity expectations are more or less dynamic in odd ways I often avoid. While it is true that as one ages one's prospective longevity "interval" comes in a bit in absolute terms -- one has a shorter expected remaining interval at 90 than 60 -- in relative terms the uncertainty actually gets a little bigger or has more prospective "volatility," if you will. 

Works like this:

At 60 one has 

• an age = 60
• a mean expected longevity and number of years remaining of whatever
• a conditional survival "distribution" conditioned on age -- from age to oldest terminal age
• a terminal life expectation that hasn't budged in a very long time, let's call it 120

At 90 one has 

• an older age = 90
• a mean expected longevity (older now than at 60) and number of years remaining (fewer)
• a conditional survival "distribution" conditioned on age 90 -- from age to terminal
• a terminal life expectation that hasn't budged in a very long time, let's call it 120

So as one gets older one is, in fact, more likely to keel over, yes. On the other hand at 60 the likelihood of living to 120 is relatively less(?) likely if "relative" is to the number of years expected at 60. At 90, the likelihood of living to 120 is relatively more(?) likely if relative is to the number of years expected at 90. Actually, I am pretty sure I did not say that right but what I am trying to say is that it looks like life duration is a bigger dice roll at 90 than 60 in some odd way even though we are technically closer to death. Someone correct me if there is a better way to say this.   

Using a Gompertz model[1] with mode = 90 and dispersion = 8.5 (arbitrary but more or less like an annuitant life table) for ages 60 to 90, in 5 year increments, the data look like this:

and column E -- the one where we calc the ratio of: a) years between the mean_expectation and terminal 120 and b) the denominator of C (expected years at the survived-to age) -- is the column of interest that represents some flavor of growing longevity uncertainty. It charts like this:

What does this mean? I will go with the interpretation -- probably self-interested -- of the dudes that wrote the paper that D Cantor sent me. Basically their interpretation is that as one ages the beneficence of annuitizing at least some wealth goes up and there is likely some age or stage where it starts to make quite a bit more sense than not. I'll not analyze that. Too hard and I'm too lazy. I believe it more than not. Prof Milevsky has written on related topics quite a bit. I've never seen, though, any bright lines or crystal clear rules on this kind of stuff. My guess is that late 70s to early 80s are the years where this is more heavily contemplated and useful. Me? I haven't annuitized yet. Kinda want to play it out a bit but I am fully aware that there is an age and state of my portfolio where I might have a ton of regret about not pulling the trigger. But this is probably more a psychology thing than it is quant finance. TBD 

 

Notes -------------------------------

[1] g.csp <- e^((e^((x-M)/b))*(1-e^(t/b)))