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CM1 formula cheat sheet

The formulas CM1 is built on, in proper actuarial notation on one page: what each one says, when it applies, and the plain Memori syntax that types it. The grey text under each name doubles as a reference for writing your own cards.

Interest

The five ways of quoting the same rate, and the conversions between them.

Discount factor

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

Rate of discount

Interest paid in advance

d = (i)/(1+i) = iv = 1 - v
d = i1+i = iv = 1 - v

Force of interest

Accumulate over t by e^{δt}

delta = ln(1+i)
δ = ln(1+i)

Nominal rate convertible p-thly

Falls towards δ as p grows

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

Nominal discount rate

Rises towards δ as p grows

d^{(p)} = p[1 - v^{1/p}]
d(p) = p[1 - v1/p]

Annuities-certain

Annuity in arrears

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

Annuity-due

Advance timing swaps i for d

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

Continuously payable annuity

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

p-thly annuity in arrears

Only the denominator changes

a^{(p)}_{angle(n)} = (1 - v^n)/(i^{(p)})
a(p)n = 1 - vni(p)

Accumulated value

The same stream accumulated to time n

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

Increasing annuity

Note the annuity-DUE in the numerator

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

The life table

Survival and death probabilities

_tp_x + _tq_x = 1
tpx + tqx = 1

Deferred probability of death

Survive m years, die within the next n

_{m|n}q_x = _mp_x times _nq_{x+m}
m|nqx = mpx × nqx+m

Force of mortality

Integrate and exponentiate to recover ₜpₓ

mu_{x+t} = -(d)/(dt) ln(_tp_x)
μx+t = -ddt ln(tpx)

Pure endowment factor

The life-contingent vⁿ; defers any function

_nE_x = v^n _np_x
nEx = vn npx

Assurances

Whole life assurance

Benefit at the END of the year of death: v^{t+1}

A_x = sum_{t=0}^{infinity} v^{t+1} _tp_x q_{x+t}
Ax = ∞∑t=0 vt+1 tpx qx+t

Term assurance

Same terms, sum stops at the term's end

A^1_{x:angle(n)} = sum_{t=0}^{n-1} v^{t+1} _tp_x q_{x+t}
A1x:n = n-1∑t=0 vt+1 tpx qx+t

Endowment assurance

Mutually exclusive benefits, so the EPVs add

A_{x:angle(n)} = A^1_{x:angle(n)} + _nE_x
Ax:n = A1x:n + nEx

Immediate-payment assurance

The continuous analogue of the summation

bar(A)_x = int_{0}^{infinity} v^t _tp_x mu_{x+t} dt
Ax = ∞∫0 vt tpx μx+t dt

Variance of the present value

The leading 2 means doubled force of interest

Var[Z] = ^2A_x - (A_x)^2
Var[Z] = 2Ax - (Ax)2

One-year recursion

Built backwards from the end of the table

A_x = vq_x + vp_x A_{x+1}
Ax = vqx + vpx Ax+1

Life annuities and premium conversion

Whole life annuity-due

The t = 0 term equals 1: payment is immediate

ddot(a)_x = sum_{t=0}^{infinity} v^t _tp_x
ax = ∞∑t=0 vt tpx

Deferred annuity

Defer anything with the pure endowment factor

_{m|}ddot(a)_x = _mE_x times ddot(a)_{x+m}
m|ax = mEx × ax+m

Premium conversion

It is d, not i; endowment functions obey it too

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

Continuous premium conversion

Each payment basis pairs with its own rate

bar(A)_x = 1 - delta bar(a)_x
Ax = 1 - δ ax

Premiums, reserves and loans

Net annual premium

Equivalence principle at outset

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

Prospective reserve

Future benefits less future net premiums

_tV_x = A_{x+t} - P_x ddot(a)_{x+t}
tVx = Ax+t - Px ax+t

Annuity-ratio reserve

Needs only an annuity table

_tV_x = 1 - (ddot(a)_{x+t})/(ddot(a)_x)
tVx = 1 - ax+tax

Reserve recursion

Rolls reserves forward year by year

(_tV_x + P_x)(1+i) = q_{x+t} + p_{x+t} times _{t+1}V_x
(tVx + Px)(1+i) = qx+t + px+t × t+1Vx

Loan outstanding (prospective)

PV of the remaining payments at the loan rate

L_t = X a_{angle(n-t)}
Lt = X an-t

Every one of these has a tool. Compute them live in the annuity calculator, life contingencies calculator, loan calculator or joint life tool, and the symbol-by-symbol version is the notation cheat sheet. Memori is a flashcard app built by actuarial students, with a ready-made CM1 set in the shop. Join the beta.