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#1




Lesson 20 proof of the alternative formula for EPV of continuous annuity
In lesson 20, we have direct way for EPV of continuous whole life annuity:
a_x=∫ a_t*t_p_x*μ_x+t dt, integral from 0 to infinite The textbook says to plug a_t=(1v^t)/ δ into the equation above and apply integration by parts, hence the new equation: a_x=∫ (1v^t)/δ*fx(t) dt since t_p_x*μ_x+t = fx(t) = 1/δ * [∫fx(t) dt  ∫ v^t*fx(t) dt] v^t is the discount factor, which equals to e^(δt)，the first integral ∫fx(t) dt equals to 1 since I could split the integral into [0,t] and [t,∞], which gives me Fx(t) and Sx(t) that sum up to 1, so the equation becomes: a_x=1/δ * [1 ∫ v^t*fx(t) dt] now integration by parts, u = e^(δt) v= 1 as shown above du=δ*e^(δt)dt dv=fx(t) dt so ∫ v^t*fx(t) dt = e^(δt)  ∫δ*e^(δt)dt I realize that the solution is not the same as the alternative formula: a_x = ∫ v^t*t_p_x dt Any help on this one? 
#2




Quote:
e^(δt)  ∫δ*e^(δt)dt does not depend on mortality so something must be wrong there. 
#3




Quote:
u = e^(δt) v= 1 du=δ*e^(δt)dt dv=fx(t) dt I set dv = fx(t) dt and when you go back from dv to v, you integrate dv from 0 to infinite to get 1. The formula for integration by parts is: u*v  ∫v du there is no mortality anywhere in equation, that's how the mortality just disappears. I agree that things get a little weird here but I couldn't find out what goes wrong. 
#4




You are confusing definite integrals and antiderivatives.
When you do integration by parts, you select u and dv. You then compute du and v. v is an antiderivative for dv. You know that the definite integral of dv from 0 to infinity is one. That does not mean that v is one. An example of such a dv(t) is exp(t).
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