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Thread: Complete Expectation of Life assuming UDD

  1. #1
    Actuary.com - Newbie Poster
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    Complete Expectation of Life assuming UDD

    Under UDD it is assumed exactly that:

    Complete expectation of Life = Kurate Expectation of Life + 1/2

    Does this apply to the limited expectation of life (related complte(ex:n) =ex:n + 1/2)?

    Does anyone know a formula for the limited life expectancy formulas under UDD? Thank you.

  2. #2
    Actuary.com - Level III Poster MathForMarines's Avatar
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    I don't recall ever having seen one, but I have also done every life con question I could get my hands on, and I never was in a situation in which I would need a formula like that.

    Quote Originally Posted by willborg35 View Post
    Under UDD it is assumed exactly that:

    Complete expectation of Life = Kurate Expectation of Life + 1/2

    Does this apply to the limited expectation of life (related complte(ex:n) =ex:n + 1/2)?

    Does anyone know a formula for the limited life expectancy formulas under UDD? Thank you.

  3. #3
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    Quote Originally Posted by willborg35 View Post
    Under UDD it is assumed exactly that:

    Complete expectation of Life = Kurate Expectation of Life + 1/2

    Does this apply to the limited expectation of life (related complte(ex:n) =ex:n + 1/2)?
    Does anyone know a formula for the limited life expectancy formulas under UDD? Thank you.
    There is an approximation: you add (1/2)*(n_q_x) instead of 1/2.

    ctperng

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    Quote Originally Posted by willborg35 View Post
    Under UDD it is assumed exactly that:

    Complete expectation of Life = Kurate Expectation of Life + 1/2

    Does this apply to the limited expectation of life (related complte(ex:n) =ex:n + 1/2)?

    Does anyone know a formula for the limited life expectancy formulas under UDD? Thank you.
    This is easy to figure out

    Complete: e_{x:n} = integral(t_p_x dt from 0 to n) = integral(1 - t/(w-x) dt from 0 to n) = n - n^2/(2(w-x)) = n[1 - n/[2(w-x)]]

    Curtate (not Kurate): e_{x:n} = sum(t_p_x from t=1 to n - 1) = sum(1 - k/(w-x) from 1 to n-1) = (n-1) - n(n-1)/[2(w-x)] = (n-1)[1 - n/[2(w-x)]] = n[1 - n/[2(w-x)]] - [1 - n/[2(w-x)]]

    So as you can see Curtate = Complete - [1 - n/[2(w-x)]]

    Obviously if the sum goes to "infinity" i.e. in this case to w - x, we let n = w-x and we get:

    Curtate = Complete- [1 - (w-x)/[2(w-x)]] = Complete - [1 - 1/2] = Complete - 1/2

    Hopefully my algebra is right.
    ________
    jugallette
    Last edited by NoMoreExams; January 20th 2011 at 06:28 PM.

  5. #5
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    Quote Originally Posted by NoMoreExams View Post
    This is easy to figure out

    Complete: e_{x:n} = integral(t_p_x dt from 0 to n) = integral(1 - t/(w-x) dt from 0 to n) = n - n^2/(2(w-x)) = n[1 - n/[2(w-x)]]

    Curtate (not Kurate): e_{x:n} = sum(t_p_x from t=1 to n - 1) = sum(1 - k/(w-x) from 1 to n-1) = (n-1) - n(n-1)/[2(w-x)] = (n-1)[1 - n/[2(w-x)]] = n[1 - n/[2(w-x)]] - [1 - n/[2(w-x)]]

    So as you can see Curtate = Complete - [1 - n/[2(w-x)]]

    Obviously if the sum goes to "infinity" i.e. in this case to w - x, we let n = w-x and we get:

    Curtate = Complete- [1 - (w-x)/[2(w-x)]] = Complete - [1 - 1/2] = Complete - 1/2

    Hopefully my algebra is right.
    I got something else, when you do the n-year curtate temporary life expectancy you get

    summation of (tPx)

    which is 1 - k/(w-x) from 1 to n (NOT n-1)

    solving the summation would give you

    n - (summation of k from 1 to n)/(w-x) this would give you

    n - n(n+1)/2(w-x) which is n - n^2/2(w-x) - n/2(w-x)

    so that is continuous e_x:n = curtate e_x:n + n/2(w-x) which would only be 1/2 when n = w-x


    which is what ctperng was saying
    Last edited by zmkramer; April 23rd 2008 at 03:04 PM.

  6. #6
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    Quote Originally Posted by zmkramer View Post
    I got something else, when you do the n-year curtate temporary life expectancy you get

    summation of (tPx)

    which is 1 - k/(w-x) from 1 to n (NOT n-1)

    solving the summation would give you

    n - (summation of k from 1 to n)/(w-x) this would give you

    n - n(n+1)/2(w-x) which is n - n^2/2(w-x) - n/2(w-x)

    so that is continuous e_x:n = curtate e_x:n + n/2(w-x) which would only be 1/2 when n = w-x


    which is what ctperng was saying

    booyah
    I agree, my summation should go to n not n - 1, however our results agree, when you plug in n = w - x, you get cont = curtate + 1/2, I wrote down curtate = cont - 1/2.

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    Quote Originally Posted by willborg35 View Post
    Under UDD it is assumed exactly that:

    Complete expectation of Life = Kurate Expectation of Life + 1/2

    Does this apply to the limited expectation of life (related complte(ex:n) =ex:n + 1/2)?

    Does anyone know a formula for the limited life expectancy formulas under UDD? Thank you.
    Actually, it's ex:n + .5(1-nPx).

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