Key assurance and annuity contracts: formula sheet

Key formulas

Term assurance, discrete (present value random variable)

Z = v^(K+1) if K < n; Z = 0 if K ≥ n. Value: A¹_x:n = Σ (k = 0 to n−1) v^(k+1) × kp_x × q_(x+k)

K is the curtate future lifetime of (x). Benefit is 1 paid at end of year of death. Sum assured S multiplies the value.

Term assurance, continuous

Z = v^T if T < n; Z = 0 if T ≥ n. Value: Ā¹_x:n = ∫ (0 to n) v^t × tp_x × μ_(x+t) dt

The bar means payment at the moment of death. T is the exact future lifetime.

Whole life assurance

Z = v^(K+1) (discrete). A_x = Σ (k = 0 to ∞) v^(k+1) × kp_x × q_(x+k). Continuous: Z = v^T, Ā_x = ∫ (0 to ∞) v^t × tp_x × μ_(x+t) dt

Payment is certain, so Z is always positive.

Pure endowment

Z = v^n if K ≥ n (equivalently T ≥ n); Z = 0 otherwise. Value: A_x:n¹ (1 below n) = v^n × np_x

Written as nE_x. There is no discrete or continuous version because payment is always at time n.

Endowment assurance, discrete

Z = v^(K+1) if K < n; Z = v^n if K ≥ n. A_x:n = A¹_x:n + A_x:n¹

Term plus pure endowment. Always pays, so it costs more than either part alone.

Endowment assurance, continuous

Z = v^T if T < n; Z = v^n if T ≥ n. Ā_x:n = Ā¹_x:n + A_x:n¹

Only the death part has a bar. The survival part is the same as before.

Link between timings under UDD

Ā_x ≈ (i ÷ δ) × A_x, and Ā¹_x:n = (i ÷ δ) × A¹_x:n

Valid only if deaths are uniformly distributed over each year of age (UDD). It does not apply to the pure endowment.

Link between continuous and discrete random variables

Z_continuous = v^T and Z_discrete = v^(K+1), where K = ⌊T⌋ and K+1 ≥ T

Since the discrete benefit is paid later, Ā_x > A_x for i > 0.

Whole life annuity-due

ä_x = Σ (t = 0 to ∞) v^t × tp_x

First payment at time 0 with certainty. Use v = 1/(1 + i).

Whole life annuity-immediate

a_x = Σ (t = 1 to ∞) v^t × tp_x = ä_x − 1

First payment at time 1, needs survival to age x + 1.

Temporary annuity-due

ä_x:n̅| = Σ (t = 0 to n − 1) v^t × tp_x

n payments at times 0, 1, …, n − 1.

Temporary annuity-immediate

a_x:n̅| = Σ (t = 1 to n) v^t × tp_x = ä_x:n̅| − 1 + v^n × np_x

n payments at times 1, …, n. The last term of the sum is at time n.

Deferred annuity-due

n|ä_x = v^n × np_x × ä_x+n = ä_x − ä_x:n̅|

Payments start at time n, only if the life survives to age x + n.

Deferred annuity-immediate

n|a_x = v^(n+1) × (n+1)p_x + v^(n+2) × (n+2)p_x + … = a_x − a_x:n̅|

First payment at time n + 1.

Pure endowment factor

nE_x = v^n × np_x = v^n × l_(x+n) ÷ l_x

Use it to move values back n years: n|ä_x = nE_x × ä_x+n.

Recursion

ä_x = 1 + v × p_x × ä_x+1

Works backwards from a known value at an older age.

Annuity-due from assurance

ä_x = (1 − A_x) ÷ d, where d = i ÷ (1 + i)

Whole life, annual payments in advance, with A_x paid at the end of the year of death.

m-thly annuity approximation

ä_x^(m) ≈ ä_x − (m − 1) ÷ (2m)

Approximation for whole life annuity-due with payments of 1 per year split into m instalments. For a temporary annuity, also adjust for the term end as the question demands.

Continuous annuity approximation

ā_x ≈ ä_x − 1/2

Approximation; exact form is ∫ v^t × tp_x dt from 0 to ∞.

Whole life assurance, end of year of death

Z = v^(K_x+1); A_x = E[Z] = Σ v^(k+1) × k|q_x, summed over k = 0, 1, 2, ...

v = 1 ÷ (1 + i). The sum runs until the end of the mortality table.

Whole life assurance, payable at death

Z = v^(T_x); Ā_x = E[Z] = ∫ v^t × tp_x × μ_(x+t) dt, from 0 to ∞

The bar means payment at the moment of death. Continuous time uses the force of mortality μ.

Term assurance

Z = v^(K_x+1) if K_x < n, else 0; A¹_(x:n) = Σ v^(k+1) × k|q_x, summed over k = 0 to n−1

The superscript 1 over x marks the death benefit. The value is zero if the life survives n years.

Pure endowment

Z = v^n if T_x > n, else 0; A_(x:n)^1 = v^n × np_x

The superscript 1 over the term n marks the survival benefit. Z takes only two values.

Endowment assurance

Z = v^(min(K_x+1, n)); A_(x:n) = A¹_(x:n) + A_(x:n)^1

The endowment is a term assurance plus a pure endowment.

Second moment of an assurance

E[Z²] = ²A_x = A_x calculated at v² (force of interest 2δ)

Use the same method with v replaced by v². Do not square A_x to get this.

Variance of an assurance

Var(Z) = ²A_x − (A_x)²

For a benefit of S, the variance is S² × Var(Z).

Whole life annuity-due variable

Y = ä_(K_x+1) = (1 − v^(K_x+1)) ÷ d; d = i ÷ (1 + i) = 1 − v

Y is a linear function of Z.

Mean of annuity-due from assurance

ä_x = E[Y] = (1 − A_x) ÷ d

The continuous version is ā_x = (1 − Ā_x) ÷ δ.

Variance of annuity-due

Var(Y) = Var(Z) ÷ d² = (²A_x − (A_x)²) ÷ d²

Continuous: Var(Ȳ) = (²Ā_x − (Ā_x)²) ÷ δ².

Pure endowment variance

Var(Z) = v^(2n) × np_x × (1 − np_x)

Z is v^n with probability np_x, otherwise 0.

Annuity-immediate link

a_x = ä_x − 1

For a whole life annuity paid in arrears.

Whole life identity (discrete)

A_x = 1 − d·ä_x, so ä_x = (1 − A_x) ÷ d

Assurance paid at end of year of death; annuity-due. Here d = i ÷ (1 + i) = 1 − v.

Endowment assurance identity

A_{x:n̅|} = 1 − d·ä_{x:n̅|}

Endowment assurance pays at end of year of death or at the end of n years if the life survives. The annuity is the n-year temporary annuity-due.

Continuous identity

Ā_x = 1 − δ·ā_x

Benefit paid at the moment of death; annuity paid continuously. δ is the force of interest.

Annuity-due and immediate

ä_x = 1 + a_x

Use this to move between annuity-due and annuity-immediate before applying the identity.

Annuity recursion

ä_x = 1 + v·p_x·ä_{x+1}

First payment is certain. The rest is paid only if the life survives one year.

Assurance recursion

A_x = v·q_x + v·p_x·A_{x+1}

Pay 1 at the end of the year if death occurs; otherwise continue with the value at age x+1.

Term split of annuity

ä_x = ä_{x:n̅|} + nE_x·ä_{x+n}, where nE_x = vⁿ·ₙp_x

Splits a whole life annuity into an n-year temporary annuity and a deferred annuity.

Term assurance from endowment

A¹_{x:n̅|} = A_{x:n̅|} − nE_x

Removes the survival benefit from the endowment assurance.

Level premium link

P_x = A_x ÷ ä_x = 1 ÷ ä_x − d

Annual net premium for a whole life assurance of 1, payable annually in advance for life.

Variance of annuity from assurance

Var(ä_{K+1}) = (²A_x − (A_x)²) ÷ d²

Holds for whole life because the annuity present value is (1 − Z) ÷ d, where Z = v^(K+1). ²A_x is A_x at the rate of interest (1+i)² − 1.

Equivalence principle (net)

P × ä = EPV of benefits

Use ä_x for whole life premiums and ä_x:n for premiums limited to n years. Gross premium adds expenses to the right side.

Single premium

Single premium = EPV of benefits (+ EPV of expenses if gross)

No premium annuity is needed.

Whole life assurance, level premium

P_x = A_x ÷ ä_x

Premiums payable annually in advance for life, benefit paid at end of year of death.

Limited-pay whole life

_nP_x = A_x ÷ ä_x:n

Premiums for n years only; cover for life.

Increasing whole life assurance (discrete)

(IA)_x = Σ (k+1) × v^(k+1) × _k p_x × q_(x+k), summed over k = 0, 1, 2, …

Pays k+1 if death occurs in year k+1, paid at end of the year.

Increasing term assurance

(IA)^1_x:n = Σ (k+1) × v^(k+1) × _k p_x × q_(x+k), k = 0 to n−1

Same sum, cut off after n years.

Decreasing term assurance

(DA)^1_x:n = Σ (n−k) × v^(k+1) × _k p_x × q_(x+k), k = 0 to n−1

Pays n−k if death occurs in year k+1.

Link between increasing and decreasing term

(IA)^1_x:n + (DA)^1_x:n = (n+1) × A^1_x:n

Useful check: benefits (k+1) and (n−k) add to n+1 in every year.

Increasing annuity-due (discrete)

(Iä)_x = Σ (k+1) × v^k × _k p_x, summed over k = 0, 1, 2, …

Payment k+1 at the start of year k+1, if alive.

Quick revision

• Assurance pays on death, an endowment on death or survival to the term, a pure endowment only on survival.
• Annuity-due pays at the start of each period; annuity-immediate pays at the end.
• Whole life assurance, end of year of death: Z = v^(K+1), with APV Ax.
• Whole life annuity-due: Y = ä(K+1)| and its APV is äx = Σ v^t × tpx, summed from t = 0.
• Ax = 1 − d × äx, where d = i ÷ (1 + i), for whole life assurance paid at the end of the year of death.
• äx = 1 + ax, since the due annuity includes the payment at time 0.
• Term assurance plus pure endowment gives endowment assurance.
• Deferred annuity: n|äx = v^n × npx × ä(x+n).
• Temporary annuity: äx:n̄| = äx − n|äx.
• Pure endowment APV: nEx = v^n × npx.
• Always check timing and the condition for payment before writing the formula.
• Sense-check values: assurance APV is below the sum assured.

Common mistakes

• Treating an endowment as paying only on survival. Fix: Remember that an endowment pays in both cases. A pure endowment pays only on survival. Endowment = term + pure endowment.
• Discounting a discrete death benefit by v^K instead of v^(K+1). Fix: If death is in year k+1, payment is at time k+1. Always write v^(K+1) for the discrete case.
• Using a_x = ä_x − 1 for temporary annuities. Fix: For temporary annuities use a_x:n̅| = ä_x:n̅| − 1 + nE_x, because the last payment moves from time n − 1 to time n.
• Starting a deferred annuity-due at time n + 1. Fix: For due, the first payment is at time n. For immediate, it is at time n + 1. Draw the timeline.
• Using v^(K_x) instead of v^(K_x+1) for a whole life assurance paid at the end of the year of death. Fix: Say it in words: if K_x = 0, death is in year 1 and payment is at time 1, so the factor is v¹. Always use K_x + 1 for end-of-year payment.
• Writing E[Z²] as (A_x)², giving a variance of zero. Fix: E[Z²] = ²A_x, which is A_x calculated with v². Never square the number A_x to get the second moment.
• Using a_x instead of ä_x in A_x = 1 − d·ä_x. Fix: Convert first with ä_x = 1 + a_x. Then apply the identity. The identity uses the annuity-due only.
• Using i instead of d in the identity. Fix: Compute d = i ÷ (1 + i) and write it at the top of your working. The identity is 1 = d·ä_x + A_x, interest paid in advance.
• Using ä_x for a limited-term premium. Fix: Match the annuity term to the premium term. Premium for n years means ä_x:n.
• Using a_x (annuity-immediate) for premiums. Fix: Premiums are paid at the start of each period, so use annuity-due ä.

Exam tips

• Write the definition of Z with its cases before any calculation. Examiners award marks for the correct random variable.
• Always state 'assuming UDD' when you use i ÷ δ. Without that statement you can lose the method mark.
• In multiple-choice questions, first decide the trigger and timing. Often two options can be removed straight away.
• For endowments, split into term plus pure endowment. It reduces errors and is quick on a calculator.
• Check your final value for reasonableness: endowment value must be at least the pure endowment value and the term value.
• Write the timeline first. Most lost marks come from off-by-one errors in payment times.
• Show the formula in standard notation before substituting numbers, so you earn method marks even if the arithmetic slips.
• Keep v and survival ratios to at least four decimal places and round only at the end.