Life Contingencies Cheat Sheet

Survival Functions

• Survival probability: _t p_x = P(T_x > t) = l_(x+t) / l_x
• Mortality probability: _t q_x = 1 - _t p_x
• Force of mortality: mu_(x+t) = -S'(x+t) / S(x+t)
• Curtate future lifetime: K_x = floor(T_x)
• Complete expectation: e-circle_x = E[T_x] = integral _t p_x dt
• Curtate expectation: e_x = E[K_x] = sum _t p_x (t from 1 to omega-x)

Insurance Present Values

Type	APV Symbol	Formula
Whole life (cont.)	A-bar_x	integral v^t _t p_x mu_(x+t) dt
Whole life (disc.)	A_x	sum v^(k+1) _k p_x q_(x+k)
n-year term	A^1_(x:n|)	sum (k=0 to n-1) v^(k+1) _k p_x q_(x+k)
n-year endowment	A_(x:n|)	A^1_(x:n|) + v^n _n p_x

Annuity Present Values

• Whole life annuity-due: a-double-dot_x = sum v^k _k p_x
• Relationship: a-double-dot_x = (1 - A_x) / d
• n-year temporary: a-double-dot_(x:n|) = (1 - A_(x:n|)) / d

Reserves

• Prospective: _t V = A_(x+t) - P * a-double-dot_(x+t)
• Retrospective: _t V = (P * s-double-dot_(x:t|) - _t k_x) / _t E_x
• Recursion: (_t V + P)(1+i) = q_(x+t) + p_(x+t) * _(t+1) V