Basic and General Annuities: Formulas, Relationships, and Applications in Actuarial Science

aₙ

Present value factor for an n-payment annuity-immediate.

sₙ

Accumulated value factor for an n-payment annuity-immediate.

äₙ

Present value factor for an n-payment annuity-due.

s̈ₙ

Accumulated value factor for an n-payment annuity-due.

i

Effective interest rate per payment period.

v

Discount factor per payment period.

Geometric series

S = a + ar + ··· + arⁿ⁻¹ = (a − arⁿ)/(1−r)

Annuity-immediate PV

aₙ = (1−vⁿ)/i

Annuity-immediate AV

sₙ = aₙ(1+i)ⁿ = [(1+i)ⁿ−1]/i

Annuity-due PV

äₙ = aₙ(1+i) = (1−vⁿ)/d

Annuity-due AV

s̈ₙ = äₙ(1+i)ⁿ = [(1+i)ⁿ−1]/d

Perpetuity-immediate

a∞ = 1/i

Perpetuity-due

ä∞ = 1/d

Grouping identity (a)

a₂ₙ = aₙ + vⁿaₙ

Grouping identity (s)

s₃ₙ = sₙ(1+i)²ⁿ + sₙ(1+i)ⁿ + sₙ

Deferred annuity

Valued before the first payment; n payments valued m+1 periods before the first payment as an annuity-immediate deferred m periods.

Fission method

Splits larger or less frequent payments into smaller payments to create a more useful pattern.

TVM worksheet

Ordinary/end-of-period payments use non-BGN mode; beginning-of-period payments use BGN mode.

aₙ⁽ᵐ⁾

PV of payments of 1/m every m-th of a year for n years, totaling 1 per year.

(Ia)ₙ / (Da)ₙ

Increasing / decreasing arithmetic annuity-immediate factors.

(Iä)ₙ / (Dä)ₙ

Increasing / decreasing arithmetic annuity-due factors.

(Ga)ₙ

Geometrically varying annuity-immediate.

āₙ

Level continuous-annuity present value factor.

δ

Constant force of interest in the continuous formulas.

m-thly annuity

aₙ⁽ᵐ⁾ = (1−vⁿ)/i⁽ᵐ⁾ = (1/m)·aₘₙ at rate i⁽ᵐ⁾/m

Increasing arithmetic immediate

(Ia)ₙ = (äₙ − nvⁿ)/i

Decreasing arithmetic immediate

(Da)ₙ = (n − aₙ)/i

Increasing arithmetic due

(Iä)ₙ = (äₙ − nvⁿ)/d

Decreasing arithmetic due

(Dä)ₙ = (n − aₙ)/d

Increasing perpetuities

(Ia)∞ = 1/(id); (Iä)∞ = 1/d²

Geometric immediate

(Ga)ₙ|ᵢ,ᵣ = [1 − ((1+r)/(1+i))ⁿ]/(i−r)

Geometric due

(Gä)ₙ|ᵢ,ᵣ = äₙ evaluated at (i−r)/(1+r)

Geometric perpetuity

(Ga)∞ = 1/(i−r) when r

Geometric special case

If i=r: (Ga)ₙ = nv and (Gä)ₙ = n

Level continuous PV

āₙ = (1−vⁿ)/δ = (i/δ)aₙ

Continuous perpetuity

ā∞ = 1/δ

Level continuous AV

s̄ₙ = āₙ(1+i)ⁿ = (i/δ)sₙ

Varying continuous PV

PV = ∫₀ⁿ f(t) exp[−∫₀ᵗ δₛ ds] dt

Varying continuous AV

AV = ∫₀ⁿ f(t) exp[∫ₜⁿ δₛ ds] dt

Continuous increasing PV

(Iā)ₙ = (āₙ − nvⁿ)/δ; (Iā)∞ = 1/δ²

Continuous decreasing PV

(Dā)ₙ = (n − āₙ)/δ

Double-dot cancellation

Xäₙ = Yäₘ ⇔ Xaₙ = Yaₘ

a₂ₙ/aₙ trick

a₂ₙ/aₙ = 1 + vⁿ

0% test

At 0% interest, the time value of an annuity equals the sum of its payments.

Pyramid annuity

PV(pyramid-immediate)=aₙäₙ; PV(pyramid-due)=äₙäₙ