Free SOA Exam ALTAM Joint Life Insurance and Annuities Practice Questions (2026)

Sample Questions

Question 1 Easy

For two independent lives with

q_{x} = 0.05

and

q_{y} = 0.04

, calculate

q_{x y}

, the probability the joint-life status fails within one year.

Solution

E is correct.

The joint-life status fails within one year if at least one of the two lives dies. Under independence:

q_{x y} = 1 - p_{x y} = 1 - p_{x} \cdot p_{y} = 1 - (1 - q_{x}) (1 - q_{y}) = 1 - (0.95) (0.96) = 1 - 0.912 = 0.088

Equivalently by inclusion-exclusion:

q_{x} + q_{y} - q_{x} q_{y} = 0.05 + 0.04 - 0.002 = 0.088

Question 2 Medium

The prospective reserve at time

t

for a joint-life whole life insurance with net annual premium

P_{x y}

_{t} V_{x y}

. Which of the following is the correct prospective formula?

Solution

C is correct.

The prospective reserve equals the expected present value of future benefits minus the expected present value of future premiums, both conditioned on both lives surviving to time

t

with attained ages

x + t

and

y + t

_{t} V_{x y} = A_{x + t : y + t} - P_{x y} a ¨_{x + t : y + t}

t = 0

A_{x y} - P_{x y} a ¨_{x y} = 0

by the equivalence principle. For

t > 0

, the attained ages change and the reserve grows.

Question 3 Hard

Given

A_{x} = 0.25

A_{y} = 0.30

, and

A_{x y} = 0.40

, a student computes the last-survivor APV as

A_{x ˉ y ˉ} = A_{x} + A_{y} - A_{x y} = 0.15

. Which statement correctly evaluates this result?

Solution

A is correct.

The last-survivor status is alive whenever at least one of

(x)

(y)

is alive, so

T (x ˉ y ˉ) = max (T (x), T (y)) \geq T (x)

and

T (x ˉ y ˉ) \geq T (y)

almost surely. Since the whole life insurance APV is an increasing function of the future lifetime,

A_{x ˉ y ˉ} \geq A_{x}

and

A_{x ˉ y ˉ} \geq A_{y}

hold under any dependence structure. With the given inputs,

A_{x ˉ y ˉ} = 0.15 < 0.25 = A_{x} < 0.30 = A_{y}

, which is impossible. Furthermore,

A_{x y} = 0.40 > min (A_{x}, A_{y}) = 0.25

is itself a violation: since

T (x y) \leq T (x)

always, we require

A_{x y} \geq A_{x}

. The inconsistency arises purely from

A_{x ˉ y ˉ} = 0.15 < A_{y}

, so the identity

A_{x ˉ y ˉ} = A_{x} + A_{y} - A_{x y}

has been applied to inputs that cannot come from any joint lifetime model.

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