Relationship Between Assurance and Annuity Values and Recursions

Understand Relationships and Recursions Between Assurances and Annuities

Key rules to remember

Assurance–annuity identity (annual)

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

d = i ÷ (1 + i) = i·v. Holds for whole life. Not valid with a_x (annuity-immediate) unless you convert first.

Continuous identity

Ā_x = 1 − δ·ā_x, so ā_x = (1 − Ā_x) ÷ δ

δ = ln(1 + i). Both the assurance and the annuity are continuous.

Endowment version

A_{x:n|} = 1 − d·ä_{x:n|}

Uses the n-year endowment assurance and the n-year temporary annuity-due. The whole life form is the special case n → ∞.

Annuity-immediate and annuity-due

ä_x = 1 + a_x

The first payment is at time 0 in the due form and at time 1 in the immediate form.

Assurance recursion

A_x = v·q_x + v·p_x·A_{x+1}

One-year-ahead argument. For term assurance use A¹_{x:n|} = v·q_x + v·p_x·A¹_{x+1:n−1|}.

Annuity-due recursion

ä_x = 1 + v·p_x·ä_{x+1}

For the annuity-immediate: a_x = v·p_x·(1 + a_{x+1}).

UDD conversion for death benefits

Ā_x = (i ÷ δ)·A_x; Ā¹_{x:n|} = (i ÷ δ)·A¹_{x:n|}

Applies to whole life and term assurances paid at death. A pure endowment is not changed because it pays only on survival.

Variance of assurance

Var(Z) = ²A_x − (A_x)²

²A_x is the same assurance valued at force of interest 2δ, which is interest rate 2i + i².

Variance of annuity-due

Var(Y) = (²A_x − (A_x)²) ÷ d²

From Y = (1 − Z)/d. Divide by d², not d. Continuous: Var(ā_T) = (²Ā_x − (Ā_x)²) ÷ δ².

How to solve Relationships and Recursions Between Assurances and Annuities questions

1. 1Write down what is given and what is asked, including the payment timing: start of year, end of year or moment of death.
2. 2Compute the rates you need: v, d = i·v and δ = ln(1 + i). Keep at least 5 decimal places.
3. 3Choose the tool. Use A = 1 − d·ä for converting between assurance and annuity at the same age. Use a recursion to move between ages. Use i/δ for converting end-of-year death benefits to moment-of-death under UDD.
4. 4Convert the given quantity into the form the formula needs, for example a_x to ä_x with ä_x = 1 + a_x.
5. 5Substitute and rearrange before calculating. Check the age subscripts and the value of p_x = 1 − q_x.
6. 6For variances, find ²A at the rate 2i + i² and use Var(Y) = (²A − A²) ÷ d² or the continuous analogue.
7. 7State any assumption used (such as UDD) and do a sense check: an annuity should be smaller than 1/d, and ā_x should be about ä_x − 0.5.

Quickest way: Anchor on A = 1 − d·ä and use one-step recursion

Common mistakes in Relationships and Recursions Between Assurances and Annuities

• Using i instead of d in A = 1 − d·ä.

  Students remember the identity but forget it is about interest in advance, and i looks more familiar.

  Fix: Compute d = i/(1 + i) first and write it down. With a_x (annuity-immediate) the link is A_x = 1 − d(1 + a_x), which still uses d.

• Calculating ²A_x at the same interest rate as A_x.

  Students confuse squaring the benefit with squaring the discount factor.

  Fix: ²A_x is A_x evaluated with v² (interest rate 2i + i²). It is not the square of A_x. Take the variance as ²A_x − (A_x)².

• Dividing the assurance variance by d instead of d² when finding Var(ä).

  Y = (1 − Z)/d is easy to mis-scale. Students forget variance squares constants.

  Fix: Var(Y) = Var(Z) ÷ d². Check units: the variance of a payment stream worth about 12 should be larger than that of the assurance, which lies between 0 and 1.

• Applying i/δ to a pure endowment or to an endowment assurance as a whole.

  The factor is learned as 'continuous = (i/δ) × discrete' without the condition.

  Fix: Only the death benefit part is adjusted. For an endowment, split it into the term assurance (multiply by i/δ) plus the pure endowment (leave unchanged).

• Using the wrong age or survival term in the recursion, for example ä_x = 1 + v·q_x·ä_{x+1}.

  Students mix up the death and survival branches of the assurance and annuity recursions.

  Fix: The annuity continues only if the life survives, so use p_x. The assurance pays on death, so the first term uses q_x. Draw the one-year tree to check.

• Mixing up a_x and ä_x in the identity.

  Both are called a whole life annuity, and the dots are easy to miss in notation.

  Fix: Underline the first payment time in the question. If it is now, use ä_x. If it is in one year, use a_x = ä_x − 1.

Worked examples