- **Long-Term Actuarial Math**
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- - **Lesson 20 proof of the alternative formula for EPV of continuous annuity**
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Lesson 20 proof of the alternative formula for EPV of continuous annuityIn 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=(1-v^t)/ δ into the equation above and apply integration by parts, hence the new equation: a_x=∫ (1-v^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? |

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e^(-δt) - ∫-δ*e^(-δt)dt does not depend on mortality so something must be wrong there. |

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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. |

You are confusing definite integrals and anti-derivatives.
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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