Hi, I am a bit confused with the notation used when dealing with multiple decrement tables. For eaxample, (ad)_x^w (so subscript x and superscript w) means: number of withdrawals between exact ages x and x+1. Does the 'a' in (ad)_x^w refer to the 'able' state or is it just some notational thing indicating we are dealing with dependent probabilities? If so, would someone dying from the 'withdrawn' state be denoted by: (wd)_x^d or simply (aq)_x^d? Or something else? Another question: (ad)_x^d is the number of deaths in service between ages x and x+1. Is this equal to (aq)_x^d? Hope someone could clear up this! Thanks in advanve, Barbados.
Reply (ad)_x^w --is the number of deaths --dependent decrement which implies that this decrement is operating in the presence of other decrements like death n the like for a person aged x who can leave the population within ages x and x+1 due to withdrawal decrement. this will give the figure in numbers. on the other hand (aq)_x^w s the dependent probability spelling out the same thing..it's a notational stuff..formulawise (aq)_x^w = {(ad)_x^w}/{al} where al means number of lives alive.
Barbados, Technically speaking, you are correct, the "a" in the dependent probability notation refers to a life currently in the "able" state. See Section 2.3 of Chapter 10 for how this notation may be used in different circumstances (eg where the life is in the sick state to begin with). Using you example, the number of deaths between age x and x+1 from the withdrawn state would be (wd)_x^d. Also bear in mind that, since we're most often interested in lives leaving the "able" state (or equivalently the state of being a current policyholder), the "a" notation is used generally to denote a dependent probability (as in Chapter 13). The difference between the "d" and the "q" is exactly the same as for a standard life table - "d" is a number of "deaths", which here we interpret as the number of people leaving the population due to the specific decrement; "q" is a probability. So: (ad)_x^w - number of lives leaving the population due to decrement w (ie withdrawal) between ages x and x+1, in the presence of other decrements. (aq)_x^w - probability of a life currently aged x leaving the population due to decrement w in the coming year, in the presence of other decrements. Hope this clarifies things for you!
