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Actuarial notation cheat sheet

The symbols CM1 and the life contingencies exams are written in, on one page: what each one means and the key formula behind it. The small grey text under each name is the plain Memori syntax that produces the symbol, so this doubles as a reference for writing your own cards.

Interest theory

The four ways of quoting the same interest rate, and how they connect.

Discount factor

Present value of 1 paid in one year

v = (1)/(1+i)
v = 11+i

Effective rate of discount

Interest paid in advance; d = iv

d = (i)/(1+i)
d = i1+i

Force of interest

Continuously compounded rate

delta = ln(1+i)
δ = ln(1+i)

Nominal rate convertible p-thly

p payments per year, each at rate i⁽ᵖ⁾/p

i^{(p)} = p((1+i)^{1/p} - 1)
i(p) = p((1+i)1/p - 1)

Annuities-certain

Payments of 1 per year for n years, guaranteed. Due means paid in advance, bar means continuous.

Annuity-certain (in arrears)

a_{angle(n)} = (1 - v^n)/(i)
an = 1 - vni

Annuity-due

Payments at the start of each year

ddot(a)_{angle(n)} = (1 - v^n)/(d)
an = 1 - vnd

Accumulated annuity

Value at the end of the term

s_{angle(n)} = ((1+i)^n - 1)/(i)
sn = (1+i)n - 1i

Continuous annuity

bar(a)_{angle(n)} = (1 - v^n)/(delta)
an = 1 - vnδ

Increasing annuity

Payments 1, 2, …, n

(Ia)_{angle(n)} = (ddot(a)_{angle(n)} - nv^n)/(i)
(Ia)n = an - nvni

The life table

Survival and mortality for a life aged x. A leading subscript is a duration; a plain one is an age.

Lives at age x

Expected survivors to age x in the table

l_x
lx

Deaths at age x

d_x = l_x - l_{x+1}
dx = lx - lx+1

Mortality rate

Probability a life aged x dies within a year

q_x
qx

Survival rate

p_x = 1 - q_x
px = 1 - qx

t-year survival probability

_tp_x
tpx

n-year mortality probability

_nq_x
nqx

Deferred mortality

Survive n years, then die in the next

_{n|}q_x = _np_x * q_{x+n}
n|qx = npx × qx+n

Force of mortality

Instantaneous death rate at age x

mu_x
μx

Curtate life expectancy

Expected complete years of future life

e_x
ex

Assurances

Present value of 1 paid on death. Plain symbols pay at the end of the year of death; a bar means paid immediately.

Whole life assurance

A_x
Ax

Continuous whole life

Benefit paid at the moment of death

bar(A)_x
Ax

Term assurance

Pays only if death occurs within n years

A^1_{x:angle(n)}
A1x:n

Pure endowment

Pays 1 at time n if still alive

_nE_x = v^n _np_x
nEx = vn npx

Endowment assurance

Term assurance plus pure endowment

A_{x:angle(n)}
Ax:n

Life annuities

Payments of 1 per year while a life aged x survives.

Whole life annuity (in arrears)

a_x
ax

Whole life annuity-due

ddot(a)_x
ax

Continuous life annuity

bar(a)_x
ax

Temporary annuity-due

At most n payments

ddot(a)_{x:angle(n)}
ax:n

Temporary annuity (in arrears)

a_{x:angle(n)}
ax:n

Two relations worth memorising

Assurance-annuity relation

Turns any annuity value into an assurance

A_x = 1 - d ddot(a)_x
Ax = 1 - d ax

Net annual premium

Equivalence principle for whole life cover

P_x = (A_x)/(ddot(a)_x)
Px = Axax

Want these as flashcards instead of a list? Memori renders all of this notation natively, and the shop carries a ready-made CM1 notation and formulas set built by actuarial students. Join the beta or see the full formula guide for how to type it.