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

A property and casualty insurer does not ask "will this person die?" but "how many claims will this portfolio have, and how big will they be?" The total paid in a year, the **aggregate loss**, drives premiums, reinsurance purchases and how much capital the company must hold. Actuaries build it from two simpler pieces, **frequency** (how many claims) and **severity** (how large each claim is). This collective risk model is the backbone of short-term actuarial mathematics on the SOA and CAS exams.

## The collective risk model

```
S = X_1 + X_2 + ... + X_N
```

- `N` is the number of claims in the period (frequency). Common choices: **Poisson**, **negative binomial** (more spread than Poisson), **binomial** (fewer).
- `X_1, X_2, ...` are the claim sizes (severity), independent of each other and of `N`, with the same distribution. Common choices: exponential, gamma, lognormal, Pareto (for heavy tails).
- If `N = 0`, then `S = 0`.

## The two key formulas

```
E[S]   = E[N] * E[X]
Var[S] = E[N] * Var[X] + Var[N] * E[X]^2
```

The variance has two parts: uncertainty in claim sizes, and uncertainty in how many claims there are. When `N` is Poisson with mean `lambda`, `E[N] = Var[N] = lambda` and the formula simplifies to `Var[S] = lambda * E[X^2]`.

## Worked example 1: Poisson and exponential

A small book of policies has Poisson claim counts with mean 2 per year, and exponential claim sizes with mean 5,000 (so `Var[X] = 5000^2` and `E[X^2] = 2 * 5000^2 = 50,000,000`).

```
E[S]   = 2 * 5,000           = 10,000
Var[S] = 2 * 50,000,000      = 100,000,000
SD[S]  = 10,000
P(S = 0) = P(N = 0) = e^(-2) = 0.1353
```

The standard deviation equals the mean: one bad year can easily cost double the average. That is why small books need reinsurance or more capital.

## Worked example 2: a more variable frequency

Suppose instead `E[N] = 3`, `Var[N] = 6` (a negative binomial), `E[X] = 2,000` and `Var[X] = 1,000,000`:

```
E[S]   = 3 * 2,000 = 6,000
Var[S] = 3 * 1,000,000 + 6 * 2,000^2 = 3,000,000 + 24,000,000 = 27,000,000
SD[S]  = 5,196
```

Here most of the variance comes from frequency, a common pattern when claims are driven by shared events such as storms.

## Simulation

Formulas give the mean and variance, but pricing a reinsurance layer or a capital requirement needs the whole distribution, such as the 99.5th percentile. **Monte Carlo simulation** is the practical tool: simulate many years, and in each year draw a claim count and then that many claim sizes.

```python
import random

rng = random.Random(42)          # fixed seed: same results every run
claims = 3
year_total = sum(rng.expovariate(1 / 5000) for _ in range(claims))
```

Python's standard library has no Poisson sampler, but a short helper (Knuth's method, provided in the activity) works for small means. In larger work you would use `numpy.random.default_rng(seed).poisson(lam, size)` and vectorise the whole simulation, or exact methods such as Panjer's recursion. With 20,000 simulated years, the sample mean and variance should land within a few percent of the formulas; if they do not, the code has a bug.

## Policy features change the model

A **deductible** of `d` removes small claims and reduces every other claim by `d`, which changes both frequency (fewer claims reach the insurer) and severity. A **policy limit** caps severity. Inflation raises severity over time. Actuaries apply these adjustments before fitting or simulating.

## Common mistakes

- **Forgetting the frequency term** in the variance. `Var[S]` is not `E[N] * Var[X]` alone.
- **Assuming independence that is not there.** A hurricane creates many claims at once; claims are not independent then.
- **Unseeded simulations.** Without a fixed seed, results change every run and cannot be reviewed or reproduced.
- **Too few simulations** for tail percentiles. The 99.5th percentile needs far more runs than the mean.

## Quick recap

- Aggregate loss `S` = sum of a random number `N` of random claims `X`.
- `E[S] = E[N]E[X]`; `Var[S] = E[N]Var[X] + Var[N]E[X]^2`; Poisson: `Var[S] = lambda E[X^2]`.
- Simulate with a fixed seed to get the full distribution and check it against the formulas.

## Try it

In the **Do** tab, code the formulas and a seeded simulation, and check that they agree.


Model total annual losses as a random number of claims with random sizes, by formula and by simulation.

Frequency and severity

Determining the Mean of a Compound Distribution

Computing the aggregate claims distribution

## Why it matters

A trucking company with 40 trucks asks for a price. Its own claims last year were lower than the industry average. Should the insurer charge it less? If the company had 4,000 trucks, its experience would clearly be reliable. With 40, one lucky year might be noise. **Credibility theory** answers the question "how much weight should this group's own data get?" It is used in group health and life pricing, workers' compensation, commercial auto, and experience rating everywhere.

## The credibility formula

Every credibility method produces the same kind of estimate, a weighted average:

```
estimate = Z * (observed experience) + (1 - Z) * (manual or prior rate)
```

`Z` is the **credibility factor**, between 0 and 1. `Z = 1` (**full credibility**) means trust the group's own data completely; `Z = 0` means ignore it and charge the manual rate (the rate for the average risk). The methods differ in how they choose `Z`.

## Limited fluctuation (classical) credibility

This older US method asks: how much data do we need so that the observed value is within `k` (say 5%) of its true mean with probability `p` (say 90%)? For claim counts that follow a Poisson distribution, the **full credibility standard** in number of claims is

```
n_full = (z / k)^2,   where z is the standard normal value with P(-z < Z < z) = p
```

For `p = 90%`, `z = 1.645` and `k = 5%`:

```
n_full = (1.645 / 0.05)^2 = 1,082 claims     (the classic "1,082 rule")
```

With fewer claims, use the **square-root rule**:

```
Z = min(1, sqrt(n / n_full))
```

### Worked example 1

A group had 400 claims. Its observed pure premium (losses per exposure) is 520; the manual pure premium is 480.

```
Z = sqrt(400 / 1082.2) = 0.608
estimate = 0.608 * 520 + 0.392 * 480 = 504.32
```

The group gets a rate between its own experience and the manual, closer to its own because it has a fair amount of data. Different standards exist for severity and for pure premium; the formulas add a term for the variability of claim sizes.

## Buhlmann (greatest accuracy) credibility

The modern approach models two sources of variation:

- **EPV**, the expected value of the process variance: how much a single risk's results bounce around from year to year.
- **VHM**, the variance of the hypothetical means: how different the true average costs of different risks are.

Then

```
K = EPV / VHM          Z = n / (n + K)
```

where `n` is the number of observations (for example years of experience, or exposure units). If risks really differ a lot (large VHM), `K` is small and experience gets weight quickly. If results are very noisy (large EPV), `K` is large and you need much more data.

### Worked example 2

For a portfolio of commercial policies, EPV = 8,000 and VHM = 200, so `K = 40`. A policy with 10 years of history gets `Z = 10 / (10 + 40) = 0.2`. With 40 years it would get 0.5; `Z` approaches 1 but never reaches it. The Buhlmann estimate is the best linear approximation to the Bayesian estimate, which is why it is called "greatest accuracy" credibility. The **Buhlmann-Straub** extension handles unequal exposure in different years.

```python
def credibility_estimate(z, observed, manual):
    return z * observed + (1 - z) * manual

print(round(credibility_estimate(0.2, 1200, 1000), 2))  # 1040.0
```

## Where the manual rate comes from

The "complement of credibility" (the `1 - Z` part) should be an unbiased, relevant estimate: the industry rate, a larger related class, or last year's rate trended forward. The CAS literature on ratemaking discusses how to choose it. A poor complement weakens the whole estimate.

## Common mistakes

- **Treating `Z` as a probability.** It is a weight, not the chance that the data is right.
- **Mixing units.** The classical standard of 1,082 is in *claims*, not policies or dollars.
- **Using raw experience without adjustment.** Losses should be developed to ultimate and trended before blending.
- **Forgetting the cap.** In the square-root rule, `Z` cannot exceed 1.

## Quick recap

- Credibility estimate = `Z * observed + (1 - Z) * manual`.
- Limited fluctuation: `n_full = (z/k)^2` claims (1,082 at 90%/5%), `Z = min(1, sqrt(n/n_full))`.
- Buhlmann: `K = EPV/VHM`, `Z = n/(n + K)`.

## Try it

In the **Do** tab, code both credibility methods and blend a group's experience with a manual rate.


Decide how much to trust a group's own claims experience and blend it with a manual rate.

Credibility

Limited Fluctuation Credibility - FAM S

Discussion of the credibility factor Z and its constant K

## Why it matters

Ratemaking is how property and casualty insurers decide what to charge. Too low, and the company loses money and may not be able to pay claims; too high, and customers leave and regulators object. Pricing actuaries produce a **rate indication**: the percentage change to current rates that the data supports. Most US states require insurers to file rates and justify them, so the indication is also a regulatory document. The CAS covers ratemaking in its upper-level exams, and its principles are in the CAS "Statement of Principles Regarding Property and Casualty Insurance Ratemaking".

## The fundamental insurance equation

```
premium = losses + loss adjustment expenses (LAE) + underwriting expenses + profit
```

A rate is "adequate" when expected premium covers expected losses, the costs of settling them (LAE), other expenses and a reasonable profit.

## Preparing the data

Raw data cannot be used as is. Before computing anything, the actuary:

1. **Brings premium to current rate level** ("on-leveling"): past premium was collected at older rates, so restate it as if today's rates had applied.
2. **Develops losses to ultimate**: recent accident years still have open claims; loss development factors (module 4) estimate their final cost.
3. **Trends losses and premium** to the future period the new rates will apply to (inflation in repair and medical costs, changes in exposure).

In this lesson's activity those steps are already done, so the table holds **earned premium at current rates** and **ultimate trended losses and LAE**.

## The loss ratio method

The **experience loss ratio** is losses (including LAE) divided by premium at current rates. The **permissible loss ratio** is the share of premium available for losses and LAE once expenses and profit are taken out:

```
permissible loss ratio = 1 - variable expense ratio - profit provision      (ignoring fixed expenses)
indicated rate change  = experience loss ratio / permissible loss ratio - 1
```

### Worked example

Auto line, three accident years, all adjusted:

| year | earned premium at current rates | ultimate trended losses + LAE |
|---|---|---|
| 2023 | 1,000,000 | 640,000 |
| 2024 | 1,100,000 | 715,000 |
| 2025 | 1,200,000 | 804,000 |
| total | 3,300,000 | 2,159,000 |

```
experience loss ratio = 2,159,000 / 3,300,000 = 0.6542
variable expenses 25%, profit 5%  ->  permissible loss ratio = 0.70
indicated change = 0.6542 / 0.70 - 1 = -0.0654   (a 6.5% decrease)
```

Notice the experience loss ratio uses **totals**, not the average of the yearly ratios. Summing premium and losses first weights each year by its size. Many companies also weight years differently or use five years of data.

With **fixed expenses** (costs that do not scale with premium, such as a policy-issuance cost), the formula becomes `(loss ratio + fixed expense ratio) / (1 - variable expense ratio - profit) - 1`.

## The pure premium method

The alternative works per unit of exposure (a car-year, a house-year) rather than per dollar of premium:

```
indicated rate = (pure premium + fixed expense per exposure) / (1 - variable expense ratio - profit)
pure premium   = losses and LAE / exposures
```

It gives an indicated *rate* rather than a *change*, and it is used for new lines where there is no current rate to change. Both methods give the same answer when applied consistently.

## Doing it in SQL

Pricing data usually lives in a database. A grouped query computes the indications for every line at once:

```sql
SELECT e.line,
       ROUND(SUM(e.losses) * 1.0 / SUM(e.earned_premium), 4) AS loss_ratio
FROM experience AS e
GROUP BY e.line;
```

Multiplying by `1.0` forces real (not integer) division in SQLite.

## After the indication

The indication is a starting point. The actuary credibility-weights it (module 3, lesson 2) when data is thin, considers competition, regulatory caps and customer retention, and then distributes the overall change across rating variables (age, territory, vehicle) with methods such as generalized linear models (GLMs). Rate filings are reviewed by state regulators, who check that rates are not excessive, inadequate or unfairly discriminatory.

## Common mistakes

- **Averaging yearly loss ratios** instead of dividing total losses by total premium.
- **Using premium at historical rates.** Without on-leveling, past rate changes distort the loss ratio.
- **Forgetting LAE** or mixing it into expenses twice.
- **Integer division in SQL.** `SUM(a) / SUM(b)` on integers returns 0 in SQLite.

## Quick recap

- Premium = losses + LAE + expenses + profit.
- Adjust data first: on-level premium, develop and trend losses.
- Loss ratio method: indicated change = experience loss ratio / permissible loss ratio - 1.
- Pure premium method: rate = (pure premium + fixed expense) / (1 - V - Q).

## Try it

In the **Do** tab, write one SQL query that computes the loss ratio and the indicated rate change for each line of business.


Use the loss ratio method to compute an indicated rate change from premium and loss data.

Ratemaking basics

Pricing Insurance #2: Loss Ratio Method (CAS Exam 5)

Pricing Insurance #1: Pure Premium Method (CAS Exam 5)

Model aggregate losses from frequency and severity, blend experience with credibility, and compute an indicated rate change.