Class 1 — Actuarial Foundations Review¶

This lesson reviews key actuarial concepts used throughout the course. The goal is to establish a common language and notation for:

probability and random variables
survival and mortality notation
life contingencies
financial mathematics

How to use this lesson

This session is a review, not a full re-teaching of preliminary actuarial material.
Focus on notation and interpretation — these will appear repeatedly in later lessons.

1. Probability and Random Variables¶

1.1 Random Variables¶

A random variable represents a numerical outcome of an uncertain future event.

Common actuarial uses:

the timing of an event
whether an event occurs within a period
the value of a payment contingent on an event

Term	Description
Random variable	Numerical representation of an uncertain outcome
Event	A specific outcome or set of outcomes
Sample space	Set of all possible outcomes

Actuarial intuition

In product modeling, random variables often show up as time-to-event (e.g., time to death) or event indicators (e.g., surrender happens vs doesn’t happen).

1.2 Indicator Variables¶

Indicator variables model whether an event occurs.

\[ I = \begin{cases} 1 & \text{if the event occurs} \\ 0 & \text{if the event does not occur} \end{cases} \]

Indicator variables are useful for:

modeling survival or termination
defining contingent payments
simplifying expected value calculations

Why actuaries love indicators

They allow you to write contingent cash flows as a single expression, then take expectations cleanly.

1.3 Expectation¶

Expectation is the primary probabilistic operator used in actuarial work.

\[ E[X] = \sum_x x \, P(X = x) \]

Key points:

expectation represents an average over all possible outcomes
many actuarial quantities are expressed as expected values
higher moments (variance, skewness) are not emphasized in this course

Focus for this course

If you can set up the random variable and compute an expectation, you can model most of what we need.

2. Survival and Mortality Concepts¶

2.1 Basic Mortality Notation¶

Standard actuarial notation is assumed.

Symbol	Meaning
\(q_x\)	Probability of death between ages \(x\) and \(x+1\)
\(p_x\)	Probability of survival between ages \(x\) and \(x+1\)
\({}_t p_x\)	Probability of survival for \(t\) years from age \(x\)

Relationship between one-year survival and death probabilities:

\[ p_x = 1 - q_x \]

Interpretation

\(q_x\) answers: “What is the probability the life dies in the next year?”
\(p_x\) answers: “What is the probability the life survives the next year?”

2.2 Survival Probabilities¶

Survival probabilities over multiple years:

\[ {}_t p_x = \prod_{k=0}^{t-1} \left( 1 - q_{x+k} \right) \]

These probabilities form the basis for valuing payments contingent on survival or death.

Mental model

Multi-year survival is the product of surviving each year along the path.

3. Life Contingencies¶

Life contingencies analyze payments that depend on human life events.

Used to:

value benefits contingent on survival or death
compare different payment structures
support actuarial valuation and reserving

Scope

This course assumes familiarity with the concepts, not derivations.
We care about applying notation consistently to real cash flows.

3.1 Decrements¶

A decrement is a cause by which a policy terminates.

Decrement	Description
Death	Termination due to death
Withdrawal	Voluntary termination
Maturity	Termination at a specified time

General assumptions:

decrements are mutually exclusive within a period
timing conventions must be defined
assumptions must be applied consistently

Common modeling pitfall

Many errors come from inconsistent timing assumptions (BOP vs EOP).
Always define when events happen within the period.

4. Financial Mathematics¶

4.1 Time Value of Money¶

The time value of money reflects that payments at different times are not directly comparable.

Discount factor:

\[ v = \frac{1}{1 + i} \]

where \(i\) is the annual effective interest rate.

4.2 Present Value¶

Present value converts future payments into an equivalent value at time 0.

For cash flows \(C_t\) paid at time \(t\):

\[ PV = \sum_{t=1}^{n} C_t v^t \]

Unless otherwise stated:

time is measured in discrete periods
payments occur at the end of each period
interest rates are deterministic

Course convention

We will be explicit when switching between annual vs monthly time steps.
In Python projects, we will mostly operate in monthly time.

5. Life Insurance¶

Life insurance provides payments contingent on death.

5.1 Cash Flow Timing¶

premiums are paid while the insured is alive
a benefit is paid upon death
timing of the benefit payment is uncertain
valuation emphasizes death probabilities and discounting

5.2 Basic Cash Flow Sketch¶

Time:     0     1     2     3     4     5
          |-----|-----|-----|-----|-----|
Premiums: P     P     P
Death:                   X
Benefit:                   |B|

6. Annuities¶

Annuities provide payments contingent on survival.

6.1 Cash Flow Timing¶

payments are made while the annuitant is alive
payments stop at death
duration of payments is uncertain
valuation emphasizes survival probabilities and discounting

6.2 Basic Cash Flow Sketch¶

Time:     0     1     2     3     4     5
          |-----|-----|-----|-----|-----|
Premiums: P
Death:                         X
Payments:      |A|   |A|   |A|   |A|

7. Life Insurance vs Annuities¶

Aspect	Life Insurance	Annuity
Primary risk addressed	Mortality (death)	Longevity (survival)
Cash flow trigger	Death	Survival
Benefit timing	Upon death	While alive
Payment uncertainty	Timing of payment	Duration of payments
Actuarial emphasis	Death probabilities	Survival probabilities

Practical takeaway

Life insurance hedges dying too soon (financial loss at death).
Annuities hedge living too long (outliving assets).

8. Key Takeaways¶

After this lesson, you should be comfortable with:

Basic probability concepts and expectation
Actuarial mortality and survival notation
Core life contingency ideas and timing conventions
Time value of money and present value
Cash flow timing for life insurance vs annuity contracts

These foundational concepts will be referenced throughout the remainder of the course.