Free SOA Exam ALTAM Survival Models for Contingent Cash Flows Practice Questions (2026)

Sample Questions

Question 1 Easy

The force of mortality

μ_{x + t}

and the survival function

_{t} p_{x}

are related by which of the following identities?

Solution

C is correct.

The fundamental relationship between the force of mortality and the survival function follows from the definition of the force of mortality as:

μ_{x + t} = - \frac{d}{d t} ln_{t} p_{x}

Integrating from 0 to

t

and using

_{0} p_{x} = 1

_{t} p_{x} = exp (- \int_{0}^{t} μ_{x + s} d s)

This is the general formula valid for any non-negative integrable force of mortality.

Question 2 Medium

For a Weibull survival model with survival function

S (t) = e^{- λ t^{γ}}

, where

λ = 0.002

and

γ = 2

, derive the force of mortality

μ (t)

and evaluate it at

t = 10

Solution

C is correct.

The force of mortality is

μ (t) = - \frac{d}{d t} ln S (t)

. With

S (t) = e^{- λ t^{γ}}

, we get

ln S (t) = - λ t^{γ}

, so

μ (t) = λγ t^{γ - 1}

. With

λ = 0.002

and

γ = 2

μ (t) = 0.002 \times 2 \times t = 0.004 t

. At

t = 10

μ (10) = 0.004 \times 10 = 0.04

Question 3 Hard

For a Makeham mortality law with

μ_{x} = A + B c^{x}

, the 10-year survival probability for a life aged 30 is given by

_{10} p_{30} = exp (- 10 A - \frac{B c ^{30} ( c ^{10} - 1 )}{l n c})

. With

A = 0.0005

B = 0.00005

c = 1.10

, compute

_{10} p_{30}

Solution

D is correct.

The exact 10-year survival probability under Makeham's law is:

_{10} p_{30} = exp (- 10 A - \frac{B c ^{30} ( c ^{10} - 1 )}{ln c})

Computing each component: -

10 A = 10 \times 0.0005 = 0.005

c^{30} = 1. 1^{30}

. Note

1. 1^{10} \approx 2.5937

1. 1^{20} \approx 6.7275

1. 1^{30} \approx 17.4494

c^{10} - 1 = 2.5937 - 1 = 1.5937

ln c = ln 1.1 \approx 0.09531

- Gompertz term:

\frac{0.00005 \times 17.4494 \times 1.5937}{0.09531} = \frac{0.001390}{0.09531} \approx 0.01459

- Total exponent:

- (0.005 + 0.01459) = - 0.01959

_{10} p_{30} = e^{- 0.01959} \approx 0.9806

, closest to 0.9827 among options.

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