### How Long Do I Have?

##### Working backwards from retirement funding information to find the life expectancy used

Many companies revise their retirement plan, if they have one, now and then. The employees approaching retirement age are generally very interested in comparing the "old" and "new" plans. Now, I'm an employee "approaching retirement age", but I'm approaching it from a long way off, so figuring out which plan would have been better for me is pretty much just entertainment. Unless I want to quit, there isn't anything I can do about it either way. But I like to see that all my numbers are correct, so I still read the pamphlets.

Trouble is, they don't give you all the numbers! A recent change to my retirement plan involves "...determining life expectancies after age 65 using a standard mortality table..." I kind of wanted to know what number they were using for my life expectancy, but I don't have any actuarial tables handy. However, the pamphlet did include my vested annual benefit starting at age 65, an assumed interest rate (6%), and the number that really matters -- the present value. So I could find how much that present value would be worth, compounded at 6% until I am 65, then pay it out as an annuity at the specified annual benefit, and see how long it lasts. Like this:

Let P = present value of vested retirement F = future value of vested retirement (at age 65) i = assumed annual interest rate A = annual retirement benefit starting at age 65 t = time (in months) until I reach 65We're ready to calculate F from P, i, and t using the standard compound interest formula, but we run into a difficulty immediately. Is this 6% the effective annual rate, a nominal rate compounded monthly, or what? Sorry, that information is not available in the pamphlet: you get what you get. But assuming it's a nominal rate compounded monthly,

t F = P( 1 + i/12 )I don't carry annuity formulas around in my head; I generally have to derive whatever I need. In this case, I want to start with F dollars at age 65 and pay out A/12 each month until I simultaneously run out of money and die. As with the interest rate, some information is vague, so we have to make some assumptions. This annual benefit A is almost certainly paid out monthly, but at the beginning, middle, or end? Guessing payment is at the end of each month, after one month my retirement fund has

F( 1 + i/12 ) - A/12dollars, after two months

( F( 1 + i/12 ) - A/12 )( 1 + i/12 ) - A/12and so on, so after n months the amount remaining is

n-1 ---- \ n A \ j F( 1 + i/12 ) - --- / ( 1 + i/12 ) 12 / ---- j=0This quantity will be zero when all my money is used up, and that should happen just when the actuarial table projects I'll die. So all that's left is to solve for n, the number of months after age 65 that the table projects me to live. To get rid of the summation, there is a formula that every high school student who has been in a decent math program should recognize:

n 2 n-1 1 - x 1 + x + x + ... + x = ------- 1 - xSo we have

n-1 ---- \ n A \ j 0 = F( 1 + i/12 ) - --- / ( 1 + i/12 ) 12 / ---- j=0 n n A 1 - ( 1 + i/12 ) = F( 1 + i/12 ) - --- ( ------------------ ) 12 1 - ( 1 + i/12 ) n n ( 1 + i/12 ) - 1 = F( 1 + i/12 ) - A ( ------------------- ) i Fi 1 ---- = 1 - --------------- A n ( 1 + i/12 ) n A ( 1 + i/12 ) = -------- A - Fi A n log( 1 + i/12 ) = log( -------- ) A - Fi A log( -------- ) A - Fi n = ----------------- log( 1 + i/12 )Typical numbers might be A=$11,000, P=$30,000, t=264 months, which result in n=189 months. My own numbers are somewhat smaller, and say I'm to kick the bucket around age 81.5. Since I am planning to live to 142, it appears I will need some income source to augment this retirement plan.

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