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Actuarial Mathematics

A complete guide to the mathematical core of the profession: how the actuary puts a price today on an uncertain future event. From compound interest to ruin theory.

Actuarial mathematics is the set of mathematical techniques used to value financial commitments that depend on uncertain events (living, dying, falling ill, suffering a loss) by combining the time value of money with the probability that those events occur. This guide walks through its foundations using standard notation and examples.

Contents

  1. The pillars
  2. Theory of interest
  3. Mortality and survival
  4. Life insurance
  5. Life annuities
  6. Premiums: net and gross
  7. Policy reserves
  8. Risk theory
  9. Credibility
  10. Pensions (PUC)

1. The five pillars of actuarial mathematics

Theory of interest

The time value of money: accumulation, discounting, annuities certain and equivalent rates.

Mortality models

The probability of living or dying at each age; mortality tables and the force of mortality.

Life contingencies

Insurance and annuities whose payment depends on the survival of the insured.

Premiums and reserves

The fair price of a policy and the liability the insurer must hold.

Risk theory

The aggregate claims distribution, solvency capital and the probability of ruin.

2. Theory of interest: the time value of money

Everything starts with the annual effective interest rate $i$. A peso today is not worth the same as a peso a year from now. The discount factor brings a future payment back to its present value, and the force of interest $\delta$ is its continuously compounded counterpart.

$$ v=\frac{1}{1+i}, \qquad d=1-v=\frac{i}{1+i}, \qquad 1+i=\left(1+\tfrac{i^{(m)}}{m}\right)^{m}=e^{\delta} $$
Discount factor, rate of discount, and the equivalence between the effective rate, the nominal rate compounded $m$ times a year and the force of interest.

These are used to value annuities certain, the basis of loans, mortgages and payment plans:

$$ a_{\overline{n}|}=\frac{1-v^{n}}{i}, \qquad \ddot a_{\overline{n}|}=\frac{1-v^{n}}{d}, \qquad s_{\overline{n}|}=\frac{(1+i)^{n}-1}{i} $$
Present value of an annuity-immediate and an annuity-due, and accumulated (future) value of an annuity.

Example

What is it worth today to receive $1 at the end of each year for 10 years, if $i=8\%$? With $v=1/1.08$: $a_{\overline{10}|}=\dfrac{1-1.08^{-10}}{0.08}\approx 6.71$. In other words, that stream of payments is worth $6.71 today.

3. Mortality and survival

A mortality table starts from a cohort $\ell_x$ of people alive at age $x$. From it we derive the probabilities of surviving and of dying, and the force of mortality $\mu_x$, which describes instantaneous mortality.

$$ {}_{t}p_x=\frac{\ell_{x+t}}{\ell_x}, \quad {}_{t}q_x=1-{}_{t}p_x, \quad {}_{t|}q_x={}_{t}p_x\,q_{x+t}, \quad \mu_x=-\frac{\ell_x'}{\ell_x} $$
Survival, mortality, deferred mortality and the force of mortality.

There are analytical laws of mortality that approximate these tables and are useful in modeling:

LawForce of mortalityUse
De Moivre$\mu_x=\dfrac{1}{\omega-x}$Simple linear model with limiting age $\omega$
Gompertz$\mu_x=B\,c^{x}$Mortality grows geometrically with age
Makeham$\mu_x=A+B\,c^{x}$Gompertz plus an accident hazard term

In Mexico, the EMSSA tables (Mexican insurance sector experience) are the benchmark for pensions and social security, and insurers use experience and regulatory tables. Modern models incorporate generational improvement: mortality at a given age falls over time thanks to medical advances, which increases longevity and pension liabilities.

4. Life insurance: actuarial present value

An insurance policy pays a benefit upon death. Its actuarial present value (APV) combines the financial discount $v$ with the probability that the payment occurs. These are the main ones:

$$ A_x=\sum_{k=0}^{\infty} v^{k+1}\,{}_{k|}q_x \quad\text{(whole life)}, \qquad A^{1}_{x:\overline{n}|}=\sum_{k=0}^{n-1} v^{k+1}\,{}_{k|}q_x \quad\text{(term)} $$
Whole life insurance (always pays at death) and $n$-year term insurance (pays only if death occurs within the term).
$$ A_{x:\overline{n}|}=A^{1}_{x:\overline{n}|}+{}_{n}E_x, \qquad {}_{n}E_x=v^{n}\,{}_{n}p_x $$
Endowment insurance = death component + pure endowment ($_nE_x$, which pays if the insured survives $n$ years).

The variance of the present value is obtained with the "rule of moments": $\operatorname{Var}(Z)={}^{2}A_x-(A_x)^2$, where ${}^{2}A_x$ is computed at the rate $i^*=(1+i)^2-1$. It is key to measuring the risk of an individual policy.

5. Life annuities

A life annuity pays for as long as the annuitant lives; it is the basis of pensions and life annuities.

$$ \ddot a_x=\sum_{k=0}^{\infty} v^{k}\,{}_{k}p_x, \qquad \ddot a_{x:\overline{n}|}=\sum_{k=0}^{n-1} v^{k}\,{}_{k}p_x $$
Whole life annuity-due (pays at the start of each year of life) and $n$-year temporary annuity-due.

Insurance and annuities are linked by a fundamental identity that connects protection and savings:

$$ \boxed{A_x=1-d\,\ddot a_x} \qquad\Longleftrightarrow\qquad \ddot a_x=\frac{1-A_x}{d} $$
The insurance-annuity relationship. All of life contingency mathematics revolves around this identity.

6. Premiums: net and gross

The equivalence principle requires that, at issue, the present value of premiums equal the present value of benefits. This yields the level net premium:

$$ P(A_x)=\frac{A_x}{\ddot a_x}=\frac{1}{\ddot a_x}-d, \qquad P_{x:\overline{n}|}=\frac{A_{x:\overline{n}|}}{\ddot a_{x:\overline{n}|}} $$
Net premium for a whole life policy and for an endowment.

The gross premium adds expenses (acquisition, administration and collection) through the extended equivalence principle:

$$ G\cdot \ddot a_x = A_x + (\text{PV of expenses}) \;\Rightarrow\; \text{loaded premium} $$
The gross premium covers benefits plus expenses. The difference from the net premium funds commissions and operations.

7. Policy reserves

Over time, the insurer must set up a reserve $_tV$: the liability that guarantees future obligations will be met. It is calculated in two equivalent ways:

$$ {}_{t}V=\underbrace{A_{x+t}-P\,\ddot a_{x+t}}_{\text{prospective}}=\underbrace{P\,\ddot s_{x:\overline{t}|}-{}_{t}k_x}_{\text{retrospective}} $$
Prospective reserve (PV of future benefits − PV of future premiums) and retrospective reserve (accumulated premiums − cost of insurance), which coincide under the same basis.

The reserve evolves from year to year according to Fackler's recursion formula, which underlies the regulation of technical reserves (required in Mexico by the CNSF, the National Insurance and Surety Commission):

$$ ({}_{t}V+P)(1+i)= {}_{t+1}V\,p_{x+t}+ b\,q_{x+t} $$
Resources at the start of the year, accumulated with interest, fund the reserve for survivors and the benefit $b$ for those who die.

8. Risk theory

In property and casualty insurance we do not model a single life but the aggregate claims amount $S$ of a portfolio: the sum of a random number $N$ of claims of size $X_i$ (collective risk model).

$$ S=\sum_{i=1}^{N}X_i, \quad E[S]=E[N]\,E[X], \quad \operatorname{Var}(S)=E[N]\operatorname{Var}(X)+\operatorname{Var}(N)\,E[X]^2 $$
Mean and variance of total cost (compound distribution formula). When $N$ is Poisson, $\operatorname{Var}(S)=\lambda\,E[X^2]$.

When $N$ belongs to the $(a,b,0)$ class (Poisson, binomial, negative binomial), the distribution of $S$ can be computed exactly with the Panjer recursion:

$$ f_S(x)=\frac{1}{1-a\,f_X(0)}\sum_{y=1}^{x}\left(a+\frac{b\,y}{x}\right) f_X(y)\,f_S(x-y) $$
Panjer recursion for the aggregate loss distribution, the basis of capital and reinsurance calculations.

Ruin theory measures the probability that an insurer's reserves are exhausted. Lundberg's inequality provides an elegant bound:

$$ \psi(u)\le e^{-Ru}, \qquad \text{where } R \text{ solves } \lambda+cR=\lambda\,M_X(R) $$
Probability of ruin with initial capital $u$; $R$ is the adjustment coefficient and $c$ the premium income rate.

9. Credibility theory

How much weight should a policyholder's own experience receive relative to that of the collective? Credibility answers with a weighted average, the basis of modern ratemaking:

$$ \text{Premium}=Z\,\bar X + (1-Z)\,\mu, \qquad Z=\frac{n}{n+k}\;\text{(Bühlmann)} $$
$Z$ is the credibility factor; $\bar X$ the individual experience and $\mu$ that of the collective. The more experience $n$, the more weight on the risk's own data.

10. Pensions: projected unit credit

The valuation of pension plans and employee benefit liabilities (in Mexico, under NIF D-3 / IAS 19) uses the projected unit credit (PUC) method, which attributes a portion of the total benefit to each year of service, based on projected salaries:

$$ \text{DBO}=\sum_{t}\; B^{*}(x,t)\;{}_{t|}q^{(\tau)}_x\;v^{t+1}, \qquad B^{*}(x,t)=B(x,t)\cdot\frac{n}{n+t} $$
Defined Benefit Obligation: the present value of the benefit attributed to service already rendered, weighted by multiple-decrement exit probabilities.

To go deeper into pensions and liability valuation, see Practice areas; for formulas developed by topic, see the Formulas page.

This guide is for general education and uses standard international notation (in the style of the SOA's Actuarial Mathematics). For formal study, see the curricula under Courses and the actuarial publications.

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