Actuarial Notation Cheat Sheet

Life Table Notation

• l_x: Number of lives at age x in the life table
• d_x: Number of deaths between ages x and x+1; d_x = l_x - l_(x+1)
• q_x: Probability of death within one year for a life aged x; q_x = d_x / l_x
• p_x: Probability of surviving one year; p_x = 1 - q_x = l_(x+1) / l_x
• _t p_x: Probability of surviving t years; _t p_x = l_(x+t) / l_x
• _t q_x: Probability of dying within t years; _t q_x = 1 - _t p_x
• _(t|u) q_x: Probability of dying between times t and t+u; = _t p_x * _u q_(x+t)
• mu_x: Force of mortality at age x

Insurance Notation

• A_x: APV of whole life insurance (discrete, paid at end of year of death)
• A-bar_x: APV of whole life insurance (continuous, paid at moment of death)
• A^1_(x:n|): APV of n-year term life insurance
• A_(x:n|): APV of n-year endowment insurance
• ^2 A_x: Second moment of PV of whole life insurance (use delta' = 2 delta)

Annuity Notation

• a-double-dot_x: APV of whole life annuity-due (payments at beginning of year)
• a_x: APV of whole life annuity-immediate (payments at end of year)
• a-bar_x: APV of continuous whole life annuity
• a-double-dot_(x:n|): APV of n-year temporary life annuity-due
• _(n|) a-double-dot_x: APV of n-year deferred whole life annuity-due

Premium and Reserve Notation

• P_x: Net annual premium for whole life insurance
• P_(x:n|): Net annual premium for n-year endowment
• _t V_x: Net premium reserve at time t for whole life insurance
• _t V_(x:n|): Net premium reserve at time t for n-year endowment

Commutation Functions

• D_x = v^x l_x: Discounted number of lives
• N_x = sum D_k (k from x to omega): Used for annuity calculations
• C_x = v^(x+1) d_x: Discounted deaths
• M_x = sum C_k (k from x to omega): Used for insurance calculations
• A_x = M_x / D_x
• a-double-dot_x = N_x / D_x