# Mortality and Survival Models

Study Mortality and Survival Models for SOA Exam FAM (weight 10–15%): worked examples, exam traps, a vendor chapter map, and a 5–7 day plan.

Canonical URL: https://exclam.ai/actuarial/exam-fam/topics/mortality-models/

[Actuarial](https://exclam.ai/actuarial/index.md)›[Exam FAM](https://exclam.ai/actuarial/exam-fam/index.md)›Mortality and Survival Models

Exam FAM topic · 10–15% of exam

Parametric survival models, life tables, force of mortality, and select and ultimate mortality.

[Back to Exam FAM](https://exclam.ai/actuarial/exam-fam/index.md)

## Per-objective worked-example outlines

For each learning objective on Mortality and Survival Models, here is the approach an exam item would test — the setup, the ordering of your reasoning, and the formula or identity you need to bring to the page. Approaches, not full solutions, by design. Verify against the current [soa.org](https://www.soa.org/) syllabus before your sitting.

### Compute probabilities and expectations from life tables

Setup

A life table fragment is given and you must compute t-year survival probability, deferred mortality probability, or expected curtate life.

Approach

Read the l\_x values directly: t\_p\_x = l\_{x+t} / l\_x and t\_q\_x = 1 - t\_p\_x. For deferred probability t|u\_q\_x = t\_p\_x · u\_q\_{x+t}. For curtate expectation e\_x = Σ\_{k=1}^{∞} k\_p\_x. Use the table's native age progression and don't interpolate unless you must.

Key identity

t\_p\_x = l\_{x+t} / l\_x; e\_x = Σ\_{k≥1} k\_p\_x.

### Work with force of mortality and survival functions

Setup

A survival function or force of mortality μ(x) is given and you must compute survival probabilities or moments.

Approach

Translate between forms: S(x) = exp(-∫₀^x μ(s) ds); μ(x) = -S'(x) / S(x). For t\_p\_x = S(x + t) / S(x). For complete expectation, ė\_x = ∫₀^∞ t\_p\_x dt. Sketch μ(x) — common parametric forms (Gompertz, Makeham, Weibull) all have characteristic shapes.

Key identity

μ(x) = f(x) / S(x); S(x) = exp(-∫₀^x μ(s) ds).

### Apply select and ultimate mortality tables

Setup

A select and ultimate table is given and you must compute probabilities for a select life.

Approach

Identify the select period (the columns to the left of "ultimate") and read q\_{\[x\]+s} for s less than the select period; for s beyond, use the ultimate column q\_{x+s}. Probabilities accumulate by multiplying p's across the trajectory until you join the ultimate. Recognize that select mortality is lower than ultimate at the same age.

Key identity

t\_p\_{\[x\]} = Π\_{k=0}^{t-1} p\_{\[x\]+k} until select period ends.

### Construct multiple-decrement models for life contingent risks

Setup

A model has multiple causes of decrement (e.g., death and lapse), each with its own force of decrement, and you must compute decrement probabilities or expected service.

Approach

Write μ^{(j)}(x) for the force of decrement j, and μ^{(τ)}(x) = Σ\_j μ^{(j)}(x) for the total force. Then t\_p\_x^{(τ)} = exp(-∫₀^t μ^{(τ)} ds) and t\_q\_x^{(j)} = ∫₀^t s\_p\_x^{(τ)} μ^{(j)}(x + s) ds. Distinguish absolute rates of decrement (single-decrement) from dependent rates (multi-decrement).

Key identity

t\_p\_x^{(τ)} = exp(-∫ μ^{(τ)} ds); t\_q\_x^{(j)} = ∫ s\_p\_x^{(τ)} μ^{(j)} ds.

## Common exam traps on Mortality and Survival Models

Recurring patterns where candidates lose points on Mortality and Survival Models-style items. Each entry pairs the trap with the fix.

Trap

Confusing select and ultimate ages in a select table.

Fix

Always identify the select duration s (years since selection); columns are indexed by s, rows by \[x\].

Trap

Using single-decrement rates as if they were absolute decrement rates.

Fix

Multi-decrement probabilities depend on the joint force; convert via UDD or constant force assumptions when needed.

Trap

Forgetting that complete expectation integrates t\_p\_x while curtate expectation sums k\_p\_x.

Fix

Complete: ė\_x = ∫₀^∞ t\_p\_x dt. Curtate: e\_x = Σ\_{k=1}^{∞} k\_p\_x. Approximately ė\_x ≈ e\_x + 0.5 under UDD.

Trap

Plugging q's into formulas that expect p's (or vice versa).

Fix

Write each step explicitly and double-check whether you need survival or death probability.

## Where to find Mortality and Survival Models in popular manuals

Pointers to where each major vendor covers this topic, so you can grab the right chapter without combing the full manual. We do not reproduce vendor content — just the location. Chapter and lesson numbers shift between editions; use these as a guide, not as a citation.

ASM

Mortality and survival chapters in the FAM manual (life contingencies half)

ACTEX

Life table and multiple-decrement chapters

Coaching Actuaries

Learn modules on Mortality Models; Adapt category "Mortality"

The Infinite Actuary

Life contingencies video block on mortality and life tables

## 7-day Mortality and Survival Models micro plan

A focused 7-day sub-schedule for Mortality and Survival Models specifically, at roughly 1.5–2.5 hours per day. Drop it inside your full Exam FAM plan as a single coverage module.

Day 1

Read the life table chapter; build flashcards on l\_x, q\_x, p\_x, t|u\_q\_x notation.

Day 2

Drill 15 life table problems including curtate expectation and deferred probability.

Day 3

Force of mortality problems — 10 problems with parametric μ(x).

Day 4

Select and ultimate tables — 8 problems mapping select durations to ultimate ages.

Day 5

Multiple-decrement models — 8 problems splitting τ into competing causes.

Day 6

Mixed 20-problem drill spanning life tables, parametric μ, and multi-decrement.

Day 7

Re-do flagged problems and rebuild the mortality summary sheet from memory.

## How exclam.ai helps you master Mortality and Survival Models

### Flashcards from your manual

Upload your ACTEX Exam FAM digital edition, scanned ASM pages, TIA handouts, or your own notes. exclam.ai extracts the Mortality and Survival Models sections and generates flashcards automatically, tuned to the exam traps above.

### Worked-example drilling

Each per-objective approach above maps to a quiz template. exclam.ai re-surfaces missed items until you can recall both the setup and the key identity from cold.

### FSRS spaced repetition

Because Mortality and Survival Models is 10–15% of your exam, losing it during review costs you. FSRS brings it back at the optimal moment.

## Mortality and Survival Models in the Exam FAM context

SOA Exam FAM has 7 topic areas. Mortality and Survival Models is weighted at approximately 10–15% of the exam, here is where it sits relative to the other topics.

| Topic area | Weight |
| --- | --- |
| Insurance Coverages and Retirement Products | 7.5–15% |
| Severity, Frequency, and Aggregate Models | 12.5–17.5% |
| Parametric Estimation | 2.5–7.5% |
| → Mortality and Survival Models | 10–15% |
| Life Insurance Pricing and Reserving | 27.5–42.5% |
| Short-Term Insurance Pricing and Reserving | 12.5–20% |
| Option Pricing Fundamentals | 2.5–7.5% |

## Other Exam FAM topics

[Insurance Coverages and Retirement Products 7.5–15% Short-term insurance and reinsurance coverages (syllabus Topic 1, 5–10%) plus long-term insurance coverages and retirement programs (Topic 7, 2.5–5%). Study this topic](https://exclam.ai/actuarial/exam-fam/topics/insurance-coverages-and-retirement-products/index.md)

[Severity, Frequency, and Aggregate Models 12.5–17.5% Statistical models for individual loss severity, claim frequency, and aggregate loss distributions including compound distributions. Study this topic](https://exclam.ai/actuarial/exam-fam/topics/severity-frequency-aggregate-models/index.md)

[Parametric Estimation 2.5–7.5% Maximum likelihood estimation for severity and frequency models. Study this topic](https://exclam.ai/actuarial/exam-fam/topics/parametric-estimation/index.md)

[Life Insurance Pricing and Reserving 27.5–42.5% Present value random variables for long-term coverages (syllabus Topic 9, 12.5–20%) plus premium and policy value calculation (Topic 10, 15–22.5%). Study this topic](https://exclam.ai/actuarial/exam-fam/topics/life-insurance-pricing-and-reserving/index.md)

[Short-Term Insurance Pricing and Reserving 12.5–20% Pricing and loss reserving for short-term coverages (syllabus Topic 5, 10–15%) plus introductory credibility (Topic 4, 2.5–5%). Study this topic](https://exclam.ai/actuarial/exam-fam/topics/short-term-pricing-and-reserving/index.md)

[Option Pricing Fundamentals 2.5–7.5% Introduction to option pricing including put-call parity, binomial models, and Black-Scholes pricing for European options. Study this topic](https://exclam.ai/actuarial/exam-fam/topics/option-pricing-fundamentals/index.md)