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Math Help - Chain Rule

  1. #1
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    Chain Rule

    Using the chain rule on the equation...

    Ux = -(lnlx)' I'm supposed to get the force of mortality equation

    Ux = (-l'x / lx) I've tried three or four times trying to arrive at this conclusion but can't. Can anyone help me out?
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  2. #2
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    #1 - There is no way you will be able to express much useful actuarial notation without learning some LaTeX. Even WITH LaTeX, it will be a struggle, but we can get through this one.

    #2 - Normally, l_{x} is defined only annually. Its derivative tends to be zero unless you take some time to define it between year ends. If you want to talk differences, d_{x}, that's a different story. If you have a continuous l_{x}, you'll have to state that.

    #3 - You probably mean \mu_{x}(t)\;=\;-\frac{s_{x}'(t)}{s_{x}(t)}, where s(t) is the survival function for the year of interest. It is continuous and should have a derivative, except maybe at the year ends.

    Where does that leave us?
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  3. #3
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    With force of mortality we can express Ux in terms of logarithms by the following....

    Ux = -(lnlx)'

    using the chain rule on ^^ I'm supposed to be able to see that it's the same as (-l'x / lx).

    where -l'x is the rate of change of mortality of the total population aged x.


    and lx = the number of lives which have survived to age x

    ux = the rate of change of mortality per surviving person.

    this all concerns life tables and survival models, Gompertz in particular
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  4. #4
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    You seem to be repeating yourself.

    If you are using Gompertz, then you should have said so in the first place. Gompertz is continuous throughout the lifetime of your population. This is very useful for a theoretical discussion - not so much for practical applications.

    Anyway, if s(x) = l_{x} for any integer value of x, and we define l_{x} between integer values as s(x), we are on our way.

    First "l'x" is just a really horrible notation. Let's decide not to use it. The differential notation is far less confusing.

    Second, we should decide that the force of mortality is ALWAYS positive, otherwise we get resurrections and that's kind of a mess in the Life Insurance business.

    Third, one possible version of Gompertz defines the survival probability as s(x) = ae^{-bx} over some reasonable domain and b > 0 and a > 0.

    Fourth, the instantaneous change in survival, then, is the first derivative, s'(x) = -abe^{-bx}.

    Fifth, the instantaneous rate of change (or maybe the rate of change per unit), is the derivative divided by the function. \frac{s'(x)}{s(x)} = -b

    Sixth, if we want this to be a force of mortality, we'll have to change the sign. \mu(x) = b This should strike you as particularly interesting. What do you think of a force of mortality that is constant throughout the lifetime of the population? Doesn't fit humans very well, does it?

    Seventh, and finally, just as a matter of convenience, we can rewrite it as \mu(x)\;=\;-\frac{d}{dx}ln(s(x))\;=\;-\frac{d}{dx}ln(l_{x})\;=\;-\frac{s'(x)}{s(x)}. In this form, we can play like we don't know it's constant.

    What say you? Are we getting anywhere?
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  5. #5
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    I see the relation between the two and I know how they are related. it's just my professor asked me to perform the chain rule on -d/dx ln(lx) to get to (-l'x/lx) I know I'm just repeating myself but...

    whats my g(x) and f(x) so that I can use to perform chain rule

    also s'(x) will always be negative thus neg * neg = positive for Ux, on account of were starting from general population of 100,000 and its always decreasing yr by yr
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