The core mathematics of the actuarial profession: interest theory, life contingencies, loss models, credibility and ratemaking, plus how to plan the SOA and CAS exams.

Actuarial Mathematics

## Why it matters

Life insurance, annuities and pensions all pay (or stop paying) when someone dies, so their value depends on **how likely a person is to be alive** at each future date. Actuaries summarise those probabilities in a **survival model**, usually presented as a **life table**. Everything in the rest of this module (insurance prices, annuity values, reserves) is built from the few probabilities you learn to read here. This material is the start of the SOA long-term actuarial mathematics exams.

## The survival function

Let `T_x` be the future lifetime of a person aged exactly `x`, written `(x)`. Two probabilities are used constantly:

- `t_p_x` = probability that `(x)` survives at least `t` more years.
- `t_q_x` = probability that `(x)` dies within `t` years `= 1 - t_p_x`.

When `t = 1` the `t` is dropped: `p_x` and `q_x`. So `q_65` is the probability that a 65-year-old dies before 66. The **mortality rate** `q_x` rises with age for adults, which is the single most important fact about human mortality.

A useful identity: surviving 5 years means surviving each year in turn, so

```
5_p_60 = p_60 * p_61 * p_62 * p_63 * p_64
```

## The life table

A life table starts with a large imaginary group, the **radix** (say 100,000 people at age 60), and lists `l_x`, the expected number still alive at each age. The illustrative table below is made up for teaching (real tables, such as the SOA's published mortality tables or national population tables, have many more ages and are based on data):

| age x | l_x | d_x = l_x - l_(x+1) | q_x = d_x / l_x |
|---|---|---|---|
| 60 | 100,000 | 1,000 | 0.010000 |
| 61 | 99,000 | 1,080 | 0.010909 |
| 62 | 97,920 | 1,170 | 0.011949 |
| 63 | 96,750 | 1,270 | 0.013127 |
| 64 | 95,480 | 1,380 | 0.014453 |
| 65 | 94,100 | 1,500 | 0.015940 |
| 66 | 92,600 | 1,630 | 0.017603 |
| 67 | 90,970 | 1,770 | 0.019457 |
| 68 | 89,200 | 1,920 | 0.021525 |
| 69 | 87,280 | 2,080 | 0.023831 |
| 70 | 85,200 | | |

From the `l_x` column alone you can answer any survival question between ages in the table:

```
t_p_x = l_(x+t) / l_x
t_q_x = (l_x - l_(x+t)) / l_x
```

## Worked examples

**Survive 5 years from 60:** `5_p_60 = l_65 / l_60 = 94,100 / 100,000 = 0.941`.

**Die within 3 years from 65:** `3_q_65 = (94,100 - 89,200) / 94,100 = 0.052072`.

**Deferred death probability.** `u|t_q_x` is the probability that `(x)` survives `u` years and then dies in the following `t` years. For a 60-year-old dying between 62 and 65:

```
2|3_q_60 = (l_62 - l_65) / l_60 = (97,920 - 94,100) / 100,000 = 0.0382
```

Deferred probabilities with `t = 1`, `k|q_x = (l_(x+k) - l_(x+k+1)) / l_x`, are exactly what you need to price insurance in the next lesson: they are the probabilities that the benefit is paid in year `k + 1`.

**Expected future lifetime.** The **curtate expectation** `e_x` (complete years lived) is the sum of the survival probabilities: `e_x = 1_p_x + 2_p_x + 3_p_x + ...`. Over just the 10 years in our table, a 60-year-old is expected to live `0.99 + 0.9792 + ... + 0.852 = 9.285` complete years.

```python
lx = {60: 100000, 61: 99000, 62: 97920, 63: 96750, 64: 95480, 65: 94100}

def tpx(x, t):
    return lx[x + t] / lx[x]

print(tpx(60, 5))  # 0.941
```

## Models behind the table

Tables can be generated from a formula. In **Gompertz's law** the force of mortality `mu_x = B * c^x` grows exponentially with age, and **Makeham** adds a constant `A` for accidents. Survival analysis in statistics (Kaplan-Meier, hazard functions) uses the same ideas. Real work also uses **select** tables (recently underwritten lives die less in the first years) and **mortality improvement** scales (people live longer each calendar year).

## Common mistakes

- **Dividing by the wrong `l`.** Probabilities for `(x)` always divide by `l_x`, the starting age, never by a later age.
- **Confusing `t_q_x` with `t|q_x`.** The first is "dies within `t` years"; the second is "survives `t` years, then dies in the next one".
- **Adding probabilities across ages.** Survival probabilities multiply across years; they do not add.
- **Using a table outside its population.** Insured lives, annuitants and the general population have different mortality.

## Quick recap

- `t_p_x = l_(x+t)/l_x`, `t_q_x = 1 - t_p_x`, `q_x = d_x/l_x`.
- `u|t_q_x = (l_(x+u) - l_(x+u+t))/l_x`.
- Curtate expectation = sum of `k_p_x` for `k = 1, 2, ...`.

## Try it

In the **Do** tab, turn this life table into Python functions for survival, death and deferred death probabilities.


Read a life table and turn it into survival and death probabilities for any age and period.

Survival models and life tables

Chapter 3 The Life Table

The Survival Function, p and q Notation

## Why it matters

A life insurer sells a promise: pay a benefit if the insured person dies during the term. The insurer does not know *when* (or whether) it will pay, but it can compute the **expected present value** of the payment, which actuaries call the **actuarial present value** (APV). Do the same for the premiums the customer will pay, set the two equal, and you have a price. This lesson joins the interest rates of module 1 with the life table of the last lesson.

## Actuarial present value

For each possible payment time, multiply three things and add them up:

```
APV = sum over times of  (amount) * (discount factor) * (probability the payment is made)
```

That is all. The three building blocks, for a life aged `x`, interest rate `i` and `v = 1/(1+i)`:

**1. Term insurance** of 1, paid at the end of the year of death if death happens within `n` years:

```
A1_(x:n) = sum for k = 0..n-1 of  v^(k+1) * k_p_x * q_(x+k)
```

`k_p_x * q_(x+k)` is the probability of surviving `k` years and then dying in year `k + 1` (the deferred probability `k|q_x` from the last lesson). The payment comes at the end of that year, hence `v^(k+1)`.

**2. Pure endowment** of 1, paid at time `n` only if `(x)` is alive: `n_E_x = v^n * n_p_x`. An endowment insurance pays on death within `n` years *or* on survival to `n`, so its APV is `A1_(x:n) + n_E_x`.

**3. Temporary life annuity-due** of 1 per year for at most `n` years while alive (this is how premiums are paid, at the start of each year):

```
ä_(x:n) = sum for k = 0..n-1 of  v^k * k_p_x
```

## The equivalence principle

The **net premium** `P` is the level annual premium that makes the APV of premiums equal the APV of benefits:

```
P * ä_(x:n) = B * A1_(x:n)     so     P = B * A1_(x:n) / ä_(x:n)
```

"Net" means no expenses or profit are loaded yet; a **gross premium** adds them.

## Worked example: a 5-year term policy

A person aged 60 buys a 5-year term policy with benefit 100,000, paid at the end of the year of death. Premiums are paid annually in advance. Use the illustrative life table from the last lesson and `i = 5%`.

| k | k_p_60 | q_(60+k) | k_p_60 * q_(60+k) | v^(k+1) | term | v^k * k_p_60 |
|---|---|---|---|---|---|---|
| 0 | 1.0000 | 0.010000 | 0.01000 | 0.952381 | 0.0095238 | 1.000000 |
| 1 | 0.9900 | 0.010909 | 0.01080 | 0.907029 | 0.0097959 | 0.942857 |
| 2 | 0.9792 | 0.011949 | 0.01170 | 0.863838 | 0.0101069 | 0.888163 |
| 3 | 0.9675 | 0.013127 | 0.01270 | 0.822702 | 0.0104483 | 0.835763 |
| 4 | 0.9548 | 0.014453 | 0.01380 | 0.783526 | 0.0108127 | 0.785516 |
| | | | | | **0.0506876** | **4.452300** |

```
APV of benefit = 100,000 * 0.0506876 = 5,068.76
ä_(60:5)       = 4.452300
Net premium    = 5,068.76 / 4.452300 = 1,138.46 per year
```

Check the reasonableness: about 5.9% of these 60-year-olds die within 5 years (`1 - 0.941`), so expected claims are about 5,900 undiscounted per policy; discounting brings that to about 5,069, spread over roughly 4.45 expected premium payments.

```python
v = 1 / 1.05
A = sum(v ** (k + 1) * lx[60 + k] / lx[60] * (lx[60 + k] - lx[61 + k]) / lx[60 + k] for k in range(5))
a = sum(v ** k * lx[60 + k] / lx[60] for k in range(5))
print(round(100000 * A / a, 2))  # 1138.46
```

## What real pricing adds

Real premiums add expenses (commission, underwriting, administration), a profit margin, lapses (policyholders who stop paying), and select mortality for newly underwritten lives. Benefits are often paid at the moment of death rather than year-end, handled with continuous versions of these formulas (an `A` with a bar) or a fractional-age assumption. Exam ALTAM and the SOA's FAM exam cover these extensions; check the current syllabus on the SOA site.

## Common mistakes

- **Wrong timing of the discount factor.** Benefits paid at the end of the year of death use `v^(k+1)`; premiums paid at the start of each year use `v^k`.
- **Using `k_p_x` alone for a death benefit.** The benefit needs survival *and then* death: `k_p_x * q_(x+k)`.
- **Dividing by `n` instead of the annuity.** Premiums are only paid while alive, and they are discounted, so divide by `ä_(x:n)`, not by `n`.
- **Forgetting the premium stops at death.** That is why the annuity uses survival probabilities.

## Quick recap

- APV = sum of amount x discount x probability.
- Term insurance `A1 = sum v^(k+1) k_p_x q_(x+k)`; pure endowment `v^n n_p_x`; annuity-due `sum v^k k_p_x`.
- Net level premium = APV(benefits) / APV(annuity of 1).

## Try it

In the **Do** tab, code these three building blocks and price the 5-year term policy.


Combine discounting with survival probabilities to price a term insurance policy with level premiums.

Life insurance and annuity values

Present Value Calculations for Insurance Products

Whole Life and Temporary Annuities

## Why it matters

In the last lesson the 60-year-old paid a level premium of 1,138.46 for 5 years. But the cost of the cover is *not* level: the chance of dying rises every year. Early premiums are more than the cost of that year's cover, and later premiums are less. The insurer must keep the early surplus to pay the later claims. That money, set aside for future obligations, is the **reserve**. It is usually the largest item on a life insurer's balance sheet, and calculating it is one of the core jobs of a valuation actuary.

## The prospective reserve

The reserve at time `t` for a policy still in force is the value of what the insurer still owes minus the value of what it will still receive:

```
t_V = APV(future benefits) - APV(future premiums)     (at time t, for a life now aged x + t)
```

This is the **prospective** method, the same idea as the loan balance in module 1 (the present value of the payments still to come). At issue, `0_V = 0` by the equivalence principle. At the end of the term, nothing is owed and nothing will be paid, so the reserve is 0 again.

## Worked example: the 5-year term policy

Using the same life table, `i = 5%`, benefit 100,000 and premium 1,138.46, the reserves at the start of each policy year (just before the premium is paid) are:

| t | age | APV future benefits | APV future premiums | reserve t_V |
|---|---|---|---|---|
| 0 | 60 | 5,068.76 | 5,068.76 | 0.00 |
| 1 | 61 | 4,365.86 | 4,168.50 | 197.36 |
| 2 | 62 | 3,531.77 | 3,216.64 | 315.13 |
| 3 | 63 | 2,543.90 | 2,208.47 | 335.43 |
| 4 | 64 | 1,376.50 | 1,138.46 | 238.04 |
| 5 | 65 | 0.00 | 0.00 | 0.00 |

The reserve rises while the early "overpayments" accumulate and then falls as they are used up. A **whole life** or **endowment** policy, by contrast, builds a large reserve that keeps growing, because a payment is certain (or very likely) eventually. Reserves per policy are small numbers, but an insurer with a million policies holds hundreds of millions.

The **retrospective** method gives the same number from the other direction: premiums received minus benefits paid, both accumulated with interest and survivorship. Prospective and retrospective reserves agree when the same basis is used for premiums and reserves.

## Why insurers must hold reserves

- **Solvency.** Without reserves, an insurer could spend early premiums and then be unable to pay later claims. Regulators require reserves so that policyholders are protected.
- **Accurate profit.** If the whole premium were counted as income when received, profits would look high early and negative later. Setting up the reserve as a liability spreads profit over the life of the policy.
- **Pricing feedback.** Reserve and capital requirements tie up money, and that cost feeds back into premiums.

## Reserve bases and regulation

The calculation depends on the **basis**: mortality table, interest rate and expenses. Different purposes use different bases:

- **Statutory reserves** are set by insurance regulators. In the US, life insurers follow the NAIC Valuation Manual, which includes principle-based reserving (VM-20) for many life products: reserves reflect company experience and stochastic scenarios instead of only fixed formulas.
- **GAAP and IFRS 17** reserves are for financial reporting to investors.
- Outside the US, regimes such as Solvency II in Europe use a best estimate plus a risk margin.

The details change often and differ by country, so in practice you read the current rules from the regulator rather than memorising them.

## Property and casualty reserves

In property and casualty (P&C) insurance (auto, home, liability), the big reserve is for claims that have *already happened* but are not yet fully paid:

- **Case reserves**: estimates on reported claims, set by claim adjusters.
- **IBNR** (incurred but not reported, plus development on known claims): the actuary's estimate of what the case reserves will grow to, usually from **loss development triangles** with methods such as the chain ladder. You will build one in module 4.

## Common mistakes

- **Thinking a reserve is a pile of profit.** It is a liability: money already committed to future claims.
- **Assuming reserves are always positive and growing.** Term reserves are small and return to zero; some product designs can even produce negative reserves, which regulators usually floor at zero.
- **Mixing timing.** State whether the reserve is just before or just after the premium due at time `t`.
- **Ignoring the basis.** A reserve is only meaningful with its mortality, interest and expense assumptions.

## Quick recap

- Level premiums for rising costs create early surplus that must be held for later claims.
- Prospective reserve = APV(future benefits) - APV(future premiums); 0 at issue under the equivalence principle.
- Reserves protect solvency and spread profit; the basis and the rules depend on the purpose and the regulator.
- P&C reserves cover claims already incurred: case reserves plus IBNR.

This lesson has no activity: check your understanding with the quiz on the **Do** tab.


Understand why insurers hold policy reserves and how a prospective reserve is calculated.

Reserves

Prospective Reserves (Contingencies: Actuarial Mathematics)

Intro to insurance reserves

Read life tables, price life insurance and annuities with actuarial present values, and understand why insurers hold reserves.