CM1 formula sheet

S.MONK · CM1 formula sheet · 44 formulae · smonk.in/formula-sheet/cm1

Simple and compound interest, and the principle that rules out one of them

A(0, t) = 1 + i t

A(t₁, t₃) = A(t₁, t₂) × A(t₂, t₃)

Present value, the discount factor, and why d is not i

i = interest ÷ amount at the START   |   d = interest ÷ amount at the END

d = i ÷ (1 + i)   and   i = d ÷ (1 − d)

Nominal rates payable p times a year, and the loan that costs more than it says

(1 + i^(p) ÷ p)^p = 1 + i

i^(p) = p × [ (1 + i)^1/p − 1 ]

Force of interest explained, and how it relates to i, v and d

(1 + i^(p)/p)^p = 1 + i
e^δ = 1 + i, so δ = ln(1 + i)

(1 + i)^t = e^δt

When the interest rate changes: steps, and a force that varies

A(0, n) = A(0, t₁) × A(t₁, t₂) × … × A(t_k, n)

A(0, n) = exp( ∫₀ⁿ δ(s) ds )

i = A(0, n)^1/n − 1

Continuous payment streams, and why they are worth more than they look

PV = ∫₀ⁿ ρ(t) v(t) dt

á_n = ∫₀ⁿ e^−δt dt = (1 − e^−δn) ÷ δ

continuous ÷ annual = i ÷ δ

PV = ∫₀ⁿ ρ(t) exp( −∫₀^t δ(s) ds ) dt

Annuity functions in CM1: immediate, due, deferred and temporary

a_n = v + v² + … + vⁿ = (1 − vⁿ) / i

ä_n = (1 + i) a_n = (1 − vⁿ) / d

_m|a_n = v^m × a_n = a_m+n − a_m

a_x = ∑_t≥1 v^t · _tp_x    and    ä_x = 1 + a_x

Perpetuities and increasing annuities: payments that never stop, or never stay level

a_∞ = 1 ÷ i

ä_∞ = (1 + i) ÷ i = 1 + 1 ÷ i

(Ia)_n = ( ä_n − n vⁿ ) ÷ i

PV = first payment ÷ (i − g)

Money rates, real rates and inflation: the exact link, not the subtraction

i' = (i − f) ÷ (1 + f)

Real yields from an inflation index, and index-linked payments

money payment at time t = real payment × Q(t) ÷ Q(0)

1 + money yield = (1 + real yield) × (1 + inflation rate)

Loan schedules: the prospective and retrospective methods

L = X · a_n, so X = L / a_n

L_k = X · a_n−k

L_k = L(1 + i)^k − X · s_k, where s_k = ((1 + i)^k − 1) / i

interest in instalment k+1 = i × L_k
capital in instalment k+1 = X − i × L_k

Life table functions explained: l(x), d(x), q(x) and p(x)

d_x = l_x − l_x+1

q_x = d_x / l_x

p_x = 1 − q_x = l_x+1 / l_x

_tp_x = l_x+t / l_x

_tp_x = p_x · p_x+1 · … · p_x+t−1 = (l_x+1/l_x)(l_x+2/l_x+1)…(l_x+t/l_x+t−1)

_t|q_x = _tp_x · q_x+t = (l_x+t − l_x+t+1) / l_x = d_x+t / l_x

e_x = ₁p_x + ₂p_x + ₃p_x + …

Net premiums and the equivalence principle, worked from first principles

EPV of the premiums = EPV of the benefits

EPV benefits = S · ∑_t=0ⁿ⁻¹ v^t+1 · d_x+t / l_x

EPV premiums = P · ä_x:n = P · ∑_t=0ⁿ⁻¹ v^t · l_x+t / l_x

P = S · A¹_x:n / ä_x:n

Reserves in CM1: prospective, retrospective, and why they agree

_tV = EPV of future benefits − EPV of future premiums

_tV = [ accumulated premiums − accumulated claims ] / (number of survivors, per policy issued)

Thiele's differential equation, explained term by term

d/dt (_tV) = P + δ_t · _tV − μ_x+t · (S − _tV)

Free from S.MONK School of Actuaries · smonk.in · every formula here is explained, with a worked example, on the page it links to.