Chapter 10 Page 9

Discussion in 'CT4' started by rinishj28, Feb 15, 2014.

  1. rinishj28

    rinishj28 Member

    I am not able to visualise this mathematically :

    By contrast, assumption (a) implies a uniform distribution of deaths over the year of age.
    This in turn implies an increasing force of mortality to cancel the effect of the survival
    probability t px (less than 1) and maintain a constant number of deaths. (Another way to
    think about this is to consider a group of lives who die at a uniform rate over a given
    year. To maintain a constant number of deaths, the force of mortality must increase to
    counter the fact that the number of survivors is decreasing.)
     
    rinishj28, Feb 15, 2014
    #1
  2. John Potter

    John Potter ActEd Tutor Staff Member

    UDD means the same number of people are dying every month, say. They are dying out of a smaller and smaller population, so the prob of death is going up. ie the force of mortality is increasing.

    John
     
    John Potter, Feb 17, 2014
    #2
  3. CAKABOGU23

    CAKABOGU23 Active Member

    For a constant force of mortality, I thought probability of death in a particular year of age is constant. So that the probability of dying in two months say, is the same irrespective of when this two months is picked in that year of age.
    So what is the paragraph on Page 9 saying about probability of dying in the last month being lower than probability of dying in the first month???
     
    CAKABOGU23, Apr 23, 2014
    #3
  4. CAKABOGU23

    CAKABOGU23 Active Member

    Can anyone please intuitively explain how Balducci assumption implies a decreasing force of mortality.
    Thanks!
     
    CAKABOGU23, Apr 23, 2014
    #4
  5. John Potter

    John Potter ActEd Tutor Staff Member

    Good question. I've never bothered doing this! This might not be the quickest way but...

    tpx *1-t p x+t = px

    Rearrange

    tpx = px / 1-t p x+t = px / [1-(1-t)*qx] = px / (px + t*qx)

    So, log tpx = log [px / (px + t*qx)] = log px - log (px + t*qx)

    mux+t = -d/dt log tpx so differentiate above and negate:

    -d/dt log tpx = qx / (px + t*qx)

    As t goes up mu x+t goes down, decreasing force of mortality

    John
     
    John Potter, Apr 24, 2014
    #5
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  6. CAKABOGU23

    CAKABOGU23 Active Member

    Thanks John,

    But I meant intuitively. Like you explained that of Uniform Distribution of Deaths. And can you kindly help on the question on Constant Force of Mortality.:)
     
    CAKABOGU23, Apr 24, 2014
    #6
  7. John Potter

    John Potter ActEd Tutor Staff Member

    Sorry, missed the question on CFM. The next sentence answers your question really:

    "This is because the probability of a life dying in the last month of the year of age includes the probability of the life having survived up to that point, which is less than 1."

    ie you're right that the probability (60) dies before (60.1) is the SAME as
    the probability (60.9) dies before (70)

    This means that the probability (60) dies before (60.1) is HIGHER than
    the probability (60) dies between the ages of (60.9) and (70).

    It's quite hard to explain the Balducci assumption intuitively. It's saying that the probability someone in any given year of age dies before they get to the end of that year is proportional to the amount of time that they have left in that year.

    So, probability (60) dies before (61) is 1*qx
    Probability (60.5) dies before (61) is 0.5*qx etc.

    But probability (60) dies before (61)

    = probability (60) dies before (60.5) + probability (60) survives until (60.5) * Probability (60.5) dies before (61)

    A = B + C * D

    Now, A is proportional to D, and C is less than 1, which means that B > D.

    The mortality over the whole year is the "same" as the mortality over the second half the year. But given that the first half of the year and the second half of the year need to "average out" and that C<1, we must have that the mortality over the first half of the year is slightly heavier.

    I can see it in my head but it's not the clearest explanation I've ever put together! I just think it's quite hard to actually think intuitively about the Balducci assumption. My next best answer is, just look at the maths and you can see it is - we could prove it algebraically

    John
     
    Last edited: Apr 25, 2014
    John Potter, Apr 24, 2014
    #7
  8. CAKABOGU23

    CAKABOGU23 Active Member

    "This means that the probability (60) dies before (60.1) is LOWER than*the probability (60) dies between the ages of (60.9) and (70)."

    I guess you meant HIGHER.
    Thanks!
     
    CAKABOGU23, Apr 24, 2014
    #8
  9. John Potter

    John Potter ActEd Tutor Staff Member

    Yes, thank you, I've corrected it now :)
     
    John Potter, Apr 25, 2014
    #9

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