Calculating Life Expectancy

[Guest jc719]

A question from a "recreational actuary" (mere mortal engineer who remembers a little algebra, trying to understand how this stuff works)....

I've been looking for the proper way to calculate life expectancy given the life table (Lx's & qx's) & haven't found the key.

In the context of a lump sum PV from a SL annuity PMT & the interest rate (all of which I know), I can solve the annuity PV equation for n...

n = -log(1-PV*i/PMT)/log(1+i)

then add this to age @ the first payment. This would be the LE that my plan administrator used, right? (They don't want to disclose their secret sauce recipe & the phone droids don't know a life table from a coffee table.)

Then I look at the GAR94 (or AMT2008 or whatever) table, find Lx for my age (rounding down to integral age) *0.5 and find the corresponding age. Rationale is that 1/2 of those alive @ age x will be dead by then so that's the LE (median). I don't know how to account for odd months, however. Linear interpolation? Some other curve fit? How do I do that? But even with this discrepancy, I'm off by about 2 years (calculated n is 2 years less than my life table analysis so the LS is about 4% lower). What am I doing wrong?

Thanks.

[Andy the Actuary]

(1) Start with a radix (arbitrary) at age x, say there are 100,000 living at age x, so L_x=100,000

(2) Create the entire table by L_x+1=L_xx(1-q_x)

(3) Then, curtate life expectancy e_x=(L_x+1+L_x+2+L_x+3+ ...) / L_x

(4) To get the complete expectancy, add 1/2 to approximate that death occurs mid-year

This can be done quickly with an Excel spreadsheet

What's been done is you have summed the propabilty of "x" living one year, "x" living two years, "x" living 3 years etc.

[Guest Dan Moore]

I think your formula's OK to get a certain period for an annuity, but caveat emptor: insurance companies don't price life annuities by calculating the life expectancy & then calculating the PV of an annuity with that certain period.

In fact, the PV of a certain annuity with certain period = life expectancy is always higher than the PV of a life annuity using the same assumptions, assuming interest rate > 0%. Maybe 10% higher. So, don't assume you're getting a great deal.

Say you're age 65 with a life expectancy of 17.5. You could die tomorrow, or you might die at age 100 (at double your life expectancy). The savings to the insurance company for you dying tomorrow is a lot more than the extra cost they have by paying you until age 100, due to the time value of money.

To calculate the PV of a life annuity, you need to add columns to the spreadsheet Andy the Actuary recommends discounting the probability of surviving n years with interest. Or google commutation functions - in particular Dx and Nx. These are what you want.

[Mike Preston]

(1) Start with a radix (arbitrary) at age x, say there are 100,000 living at age x, so L_x=100,000

(2) Create the entire table by L_x+1=L_xx(1-q_x)

(3) Then, curtate life expectancy e_x=(L_x+1+L_x+2+L_x+3+ ...) / L_x

(4) To get the complete expectancy, add 1/2 to approximate that death occurs mid-year

This can be done quickly with an Excel spreadsheet

What's been done is you have summed the propabilty of "x" living one year, "x" living two years, "x" living 3 years etc.

You newfound subscripting capability is indeed a thing of beauty!

[Andy the Actuary]

You newfound subscripting capability is indeed a thing of beauty!

You learned me well.

a.t.a. (aka Opie Taylor)