Annuity and accumulation functions: formula sheet

Key formulas

Amount function

A(t) = A(0) × a(t)

a(t) = A(t) ÷ A(0). Also a(0) = 1.

Growth between two times

Value at t2 = Value at t1 × a(t2) ÷ a(t1)

Needs a(t) defined from time 0. For t1 < t2, the accumulation factor is a(t2) ÷ a(t1).

Discount function

v(t) = 1 ÷ a(t)

Present value at time 0 of 1 due at time t.

Force of interest

δ(t) = a′(t) ÷ a(t) = d/dt [ln a(t)]

Valid where a(t) is differentiable.

Accumulation from force of interest

a(t) = exp( ∫ from 0 to t of δ(s) ds )

Uses a(0) = 1.

Accumulation between two times

exp( ∫ from t1 to t2 of δ(s) ds ) = a(t2) ÷ a(t1)

Use for a payment moved from t1 to t2.

Discount from force of interest

v(t) = exp( − ∫ from 0 to t of δ(s) ds )

Reverse of accumulation.

Compound interest

a(t) = (1 + i)^t and v(t) = (1 + i)^(−t)

Holds for a constant effective annual rate i, which is equivalent to constant δ = ln(1 + i).

Simple interest

a(t) = 1 + it

Not equal to compound interest for fractional t. Check the question.

Simple interest accumulation

A(t) = 1 + it, so accumulated value = P(1 + it)

Interest is on the original principal only. i is the annual rate and t is in years.

Compound interest accumulation

A(t) = (1 + i)^t, so accumulated value = P(1 + i)^t

i is the effective annual rate. t can be fractional.

Effective rate from accumulation

i = A(1) − A(0) = A(1) − 1

More generally, the effective rate in period n is i_n = [A(n) − A(n−1)] ÷ A(n−1).

Nominal to effective

1 + i = (1 + i(m)/m)^m

i(m) is the nominal annual rate convertible m times a year. Each period earns i(m)/m.

Effective to nominal

i(m) = m[(1 + i)^(1/m) − 1]

Use this to find the nominal rate that matches a given effective rate.

Equivalent rates for different periods

(1 + i(m)/m)^m = (1 + i(p)/p)^p

Both sides equal 1 + i. Use it to convert between two nominal rates.

Rate for a fractional period

Effective rate over 1/m year = (1 + i)^(1/m) − 1

This is the compound-equivalent rate, not i/m.

Definition of force of interest

δ(t) = A'(t) ÷ A(t) = d/dt [ln A(t)]

A(t) is the accumulation of ₹1 from time 0 to t. Holds when A is differentiable.

Accumulation from force of interest

A(s,t) = exp( ∫ from s to t of δ(r) dr )

Value at time t of ₹1 invested at time s. Use s = 0 for A(t).

Discount factor

v(s,t) = exp( −∫ from s to t of δ(r) dr )

Present value at time s of ₹1 due at time t.

Constant force of interest

δ = ln(1 + i), 1 + i = e^δ, v = e^(−δ)

Applies only when the effective rate i is the same for all periods.

Varying effective annual rates

A(0,n) = (1 + i₁)(1 + i₂)…(1 + iₙ)

iₖ is the effective rate in year k. Present value is the reciprocal.

Link to discount rate

δ = ln(1 + i) = −ln(1 − d)

Useful for converting between i, d and δ when rates are constant.

Present value, annuity-immediate

a-angle-n = v + v² + ... + vⁿ = (1 − vⁿ) ÷ i

Value one period before the first payment. v = 1 ÷ (1 + i).

Accumulated value, annuity-immediate

s-angle-n = ((1 + i)ⁿ − 1) ÷ i

Value at time n, the date of the last payment.

Present value, annuity-due

ä-angle-n = 1 + v + ... + vⁿ⁻¹ = (1 − vⁿ) ÷ d

Value at the date of the first payment. d = i ÷ (1 + i).

Accumulated value, annuity-due

s̈-angle-n = ((1 + i)ⁿ − 1) ÷ d

Value at time n, one period after the last payment.

Due versus immediate

ä-angle-n = (1 + i) a-angle-n; s̈-angle-n = (1 + i) s-angle-n

Valid for the same i and n.

Link between s and a

s-angle-n = (1 + i)ⁿ a-angle-n

Same relation holds for s̈ and ä.

Due and immediate by shifting

ä-angle-n = 1 + a-angle-(n−1); a-angle-n = ä-angle-n − 1 + vⁿ

Useful for payments counted from time 0 or time n.

Reciprocal relation

1 ÷ a-angle-n = 1 ÷ s-angle-n + i

Holds for the immediate annuity with the same i and n.

Perpetuity-immediate

a_∞ = 1/i

1 paid at the end of each year forever. Needs i > 0.

Perpetuity-due

ä_∞ = 1/d = 1/i + 1

First payment at time 0. d = i ÷ (1 + i) = iv.

Deferred annuity-immediate

m| a_n = v^m a_n = a_(m+n) − a_m

Deferred m years. First payment at time m + 1, last at time m + n.

Deferred annuity-due

m| ä_n = v^m ä_n = ä_(m+n) − ä_m

First payment at time m. Last at time m + n − 1.

m-thly annuity-immediate

a_n^(m) = (1 − v^n) ÷ i^(m) = (i ÷ i^(m)) a_n

Payments of 1/m at the end of each m-th of a year. Total 1 per year.

m-thly annuity-due

ä_n^(m) = (1 − v^n) ÷ d^(m) = (d ÷ d^(m)) ä_n

Payments of 1/m at the start of each m-th of a year.

Nominal rates

i^(m) = m[(1+i)^(1/m) − 1] ; d^(m) = m[1 − v^(1/m)]

Both are convertible m-thly. 1 + i = (1 + i^(m)/m)^m = (1 − d^(m)/m)^(−m).

Link between m-thly due and immediate

ä_n^(m) = a_n^(m) + (1 − v^n)/m

The due annuity has an extra payment of 1/m at time 0 and one fewer at time n.

m-thly perpetuities

a_∞^(m) = 1/i^(m) ; ä_∞^(m) = 1/d^(m)

Set v^n = 0 in the m-thly formulas.

Continuous annuity

ā_n = (1 − v^n) ÷ δ = (i ÷ δ) a_n ; ā_∞ = 1/δ

Payment rate 1 per year. δ = ln(1 + i).

Continuous accumulation

s̄_n = ((1+i)^n − 1) ÷ δ = (1+i)^n ā_n

Accumulated value at time n of a continuous payment stream.

Increasing annuity-immediate

(Ia)n = (ä_n − n vⁿ) ÷ i

Payments 1, 2, …, n at times 1 to n. Here ä_n = (1 − vⁿ) ÷ d and v = 1 ÷ (1+i).

Increasing annuity-due

(Iä)n = (ä_n − n vⁿ) ÷ d

Payments 1, 2, …, n at times 0 to n−1. Same numerator as (Ia)n, but divide by d. Also (Iä)n = (1+i)(Ia)n.

Decreasing annuity-immediate

(Da)n = (n − a_n) ÷ i

Payments n, n−1, …, 1 at times 1 to n. Here a_n = (1 − vⁿ) ÷ i.

Decreasing annuity-due

(Dä)n = (n − a_n) ÷ d

Payments n, n−1, …, 1 at times 0 to n−1. Also (Dä)n = (1+i)(Da)n.

Link between increasing and decreasing

(Ia)n + (Da)n = (n+1) a_n

A quick check on your answers. Also useful to get one from the other.

Perpetuity increasing by 1 each year

(Ia)∞ = 1 ÷ (i d) = (1+i) ÷ i²

Payments 1, 2, 3, … at times 1, 2, 3, … forever. Needs i > 0.

Geometric annuity-immediate

PV = P × [1 − ((1+g) ÷ (1+i))ⁿ] ÷ (i − g), for i ≠ g

First payment P at time 1, then P(1+g), P(1+g)², … up to time n. If i = g, PV = n P ÷ (1+i).

Arithmetic step pattern

Payments P, P+Q, …, P+(n−1)Q at times 1 to n: PV = (P − Q) a_n + Q (Ia)n

Split into a level part and an increasing part. Q can be negative for a falling pattern.

Continuous increasing annuity

(Īā)n = (ā_n − n vⁿ) ÷ δ

Payment rate t at time t, so the total is ∫ t vᵗ dt from 0 to n. Here ā_n = (1 − vⁿ) ÷ δ.

Quick revision

• A(t) is the value at time t of 1 invested at time 0; v(t) = 1 ÷ A(t) when A(t) is the only growth path.
• Simple interest: A(t) = 1 + it. Compound interest: A(t) = (1 + i)ᵗ.
• Force of interest: δ(t) = A′(t) ÷ A(t), and A(t) = exp(∫₀ᵗ δ(s) ds).
• With constant δ, (1 + i) = e^δ, so δ = ln(1 + i).
• a-angle-n = (1 − vⁿ) ÷ i (payments at end of each period); s-angle-n = ((1 + i)ⁿ − 1) ÷ i.
• ä-angle-n = (1 − vⁿ) ÷ d; ä-angle-n = (1 + i) × a-angle-n.
• Perpetuity-immediate = 1 ÷ i; perpetuity-due = 1 ÷ d.
• Deferred annuity: value = vᵐ × a-angle-n for payments starting after m periods (immediate type).
• m-thly annuity: payments of 1/m at each m-th of a year; use the nominal rate i⁽ᵐ⁾ and d⁽ᵐ⁾ carefully.
• Increasing annuity (Ia)-angle-n = (ä-angle-n − n·vⁿ) ÷ i.
• Decreasing annuity (Da)-angle-n = (n − a-angle-n) ÷ i.
• Always check the valuation date and whether payments are at the start or end of each period.

Common mistakes

• Forgetting a(0) = 1 when finding a constant in a(t). Fix: Use a(0) = 1 first. It fixes the constant immediately.
• Exponentiating each term of δ(t) separately, or exponentiating before integrating. Fix: Integrate the full δ(s) first. Then take exp of the whole integral.
• Using the nominal rate as if it were the effective annual rate. Fix: Always check for the subscript or the words 'convertible'. Convert to effective with 1 + i = (1 + i(m)/m)^m first, or use the per-period factor with m × t periods.
• Finding the monthly effective rate as i ÷ 12 from an effective annual rate. Fix: For an effective annual rate i, the monthly effective rate is (1 + i)^(1/12) − 1. Divide by 12 only when you have a nominal rate i(12).
• Integrating δ(t) from 0 to t when the money is invested at time s > 0. Fix: Always write the limits as s to t first. Use 0 only when the investment starts at time 0.
• Forgetting to exponentiate and giving the integral as the accumulation factor. Fix: Write A = exp(integral) as a separate line. The integral alone is ln of the factor.
• Using i in the denominator for an annuity-due. Fix: Annuity-due uses d = i ÷ (1 + i). Or compute the immediate value and multiply by (1 + i).
• Valuing an annuity-due at the wrong date. Fix: Mark the timeline. ä-angle-n is the value at the time of the first payment, time 0.
• Using v^(m) instead of v^(m−1) or the wrong number of years for the deferment. Fix: Deferred m years for an annuity-immediate means the first payment is at time m + 1. Check with your timeline. Then discount by v^m.
• Using 1/i for a perpetuity whose first payment is now. Fix: 1/i is for payments at the end of each year. If the first payment is at time 0, use 1/d, or 1/i + 1.

Exam tips

• In MCQs, the quickest check is a(0) = 1. Use it to rule out options that fail this test.
• When δ(t) is given, set out the integral clearly. Marks are given for the integral limits and the exp step.
• State your units and the time convention (years from time 0) in written answers.
• If a question asks for a value between two times t1 and t2, look for a(t2) ÷ a(t1) and show it explicitly.
• In Paper B (R or Excel), use a clear function for a(t) and check that it returns 1 at t = 0.
• In multiple-choice questions, check the compounding frequency first. Many wrong options come from using i(m) as if m = 1.
• Write the formula in standard notation, such as i(12), before substituting. Written questions reward the method even if the arithmetic slips.
• State your assumptions, for example 'compound interest at effective annual rate i', so the marker can follow your working.