Long Term FAM

1

Sx(0)

0

Sx(infinity)

lx+t/lx

Sx(t)=tPx= (in terms of lx)

uPx - u+tPx, u+tQx-uQx, uPx*tQu+x

u|tQx all starting with in terms of survival, mortality, and both

-d/dxS(x)/S(x), -d/dx(tPx)/(tPx), f(x)/S(x)

mux in terms of derivative and f(x) form

e^-integral(mux dx)

tPx in terms of mux

integral 0 to infinity (tPx dt)

e^ox

summation k=1 to infinity (kPx)

ex

integral 0 to infinity (2t*tPx dt)

2e^ox

summation k=1 to infinity ((2k-1)*kPx)

2ex

ex + .5

e^ox estimation in terms of ex

ex:n + nPx * ex+n

ex recursion

Px(1+(ex+1))

ex approximation

(1-t)lx + (t)lx+1

UDD Fractional Age

lx^(1-t) * (lx+1)^t

Constant Force Fractional Age

select age

q[50] < q[49]+1 < q[48]+2<…

0

mux+t lower limit is …

x

mux lower limit is …

t * Qx

UDD tQx

(Px)^t

Constant force tPx

t*Qx / 1-s*Qx

UDD tQx+s

Qx/(1-t*Qx)

UDD mux+t

integral 0 to infinity (e^-delta*t * tPxmux+t dt)

Abar(x)

integral 0 to infinity (e^-2delta*t * tPxmux+t dt)

2Abar(x)

Abar(x)

Abar^x:nangle + n|Abar(x)

integral 0 to infinity (tv^t * tPxmux+t dt)

(IbarAbar)x

Integral 0 to infinity((tv^t)² * tPxmux+t dt)

2(IbarAbar)x

integral 0 to n ((n-t)v^t * tPxmux+t dt)

(DbarAbar)x:n

n*Abar(x):n

(DbarAbar)x:n + (IbarAbar)x:n

b²[2Abar(x) - (Abar(x))²)

Var[Z] with b

v^n * nPx

nEx

v²n * nPx

2nEx

v²n *nPx *nQx

Var[Z] Pure Endowment

tEx + hEx+t

t+hEx Pure Endowment

e^-(delta+mu)n

Exponential nEx

e^-2(delta+mu)n

Exponential 2nEx

(1-e^-(delta+mu))(mu/(mu+delta))

Exponential Ax^1:n

mu/(mu+delta)

Exponential Abar(x)

Abar(x)(1-nEx)

Abarx^1:n in terms of Abar(x)

(e^-delta*n)* (w-x-n)/(w-x)

Uniform nEx

(1-v^(w-x))/delta(w-x)

Uniform Abar(x)

(1-v^n)/delta(w-x)

Uniform Abarx^1:n

Abarx:n

Abarx^1:n + nEx

summation 0 to infinity (v^(k+1) * k|Qx)

Ax formula

vQx + v²1|Qx + v³2|Qx+…

Ax written out

summation 0 to infinity (v^(k+1)2 * k|Qx)

2Ax formula

vQx + vPx * Ax+1

Ax recursion w/Ax+1

nEx * Ax+n

n|Ax in terms of pure endowment

summation 0 to infinity ((k+1)v^k+1 * k|Qx)

(IA)x formula

vQx + 2v²1|Qx + 3v³2|Qx +…

(IA)x written out

summation k=0 to n-1 ((k+1)²v^2(k+1) * k|Qx)

2(IA)x:n formula

summation 0 to n-1 ((n-k)v^(k+1)*k|Qx)

(DA)x:n

(n+1)Ax^1:n

(IA)x^1:n + (DA)x^1:n

nEx

Ax^1:n in terms of pure endowment