Hi, Can anyone explain me,In the derivation of EPV of immediate annuity payable continuously .Why we take w= -tPx ????? ( in chapter 2 ,topic-continuous annuities)
Integration by substitution Well, this can be explained as follows: dw --- = tpx * (mu)x+t. dt (read tpx as survival probability ..can't post its notation here..!) We need to integrate it wrt to t to get w. It becomes: w = Integral of (tpx * (mu)x+t * dt) ... (1) Insert full form of tpx: exp(-integral 0 to t: (mu)x+s ds). Then (1) becomes: w = Integral of [ (exp(-integral 0 to t: (mu)x+s * ds)) * (mu)x+t * dt] ... (2) Using integration by substitution: Let K= [integral 0 to t: (mu)x+s * ds] ...these are components of tpx. Then dk/dt = [(mu)x+t] => dk = (mu)x+t * dt [Note, after substituting the upper and lower integral limits of t and 0, the (mu)x term becomes zero because it does not have a 't' in it and hence a constant when we differentiate wrt t. ] Substitute this in (2) above: w = Integral of [ (exp(-K) ) * dk ] => - exp (-K) Substitute back K: w = - exp (-[integral 0 to t: (mu)x+s * ds]) = - tpx. therefore w = -tpx. Another idea is: Try from the other end..i.e. try to differentiate 'w' wrt to t i.e. find out dw/dt, after expanding tpx into exponential function. You will end up with tpx * (mu)x+t. if not clear, pls let me know... Thanks, Raj
