Associate of the Society of Actuaries (ASA/FSA) Exam FAM — Fundamentals of Actuarial Mathematics Syllabus

Define Value at Risk (VaR) at security level $p$ for a loss random variable $X$.

$\text{VaR}_p(X) = \pi_p$ is the $100p$-th percentile of the loss distribution: the smallest value satisfying $F_X(\pi_p) = p$ (i.e. $\pi_p = F_X^{-1}(p)$). It is the loss exceeded with probability $1-p$.

Define Tail Value at Risk (TVaR / CTE) at level $p$ and state how it relates to VaR.

$$\text{TVaR}_p(X) = E[X \mid X > \text{VaR}_p(X)] = \frac{1}{1-p}\int_{p}^{1} \text{VaR}_u(X)\, du.$$ It equals the average of losses exceeding the VaR threshold, so $\text{TVaR}_p \geq \text{VaR}_p$ always.

What is a coherent risk measure, and which of VaR and TVaR is coherent?

A coherent risk measure satisfies four axioms: subadditivity, monotonicity, positive homogeneity, and translation invariance. TVaR is coherent; VaR is not coherent because it generally violates subadditivity.

Write the TVaR formula for losses above the threshold in terms of VaR and the mean excess.

$$\text{TVaR}_p(X) = \text{VaR}_p(X) + e\big(\text{VaR}_p(X)\big),$$ where $e(d) = E[X - d \mid X > d]$ is the mean excess loss above the VaR threshold.

State the general likelihood function for a sample of independent observations and the log-likelihood.

For independent observations $x_1, \dots, x_n$ with density $f(x; \theta)$, the likelihood is $L(\theta) = \prod_{i=1}^{n} f(x_i; \theta)$ and the log-likelihood is $\ell(\theta) = \sum_{i=1}^{n} \ln f(x_i; \theta)$. The MLE $\hat\theta$ maximizes $\ell(\theta)$.

How do you construct the likelihood contribution for a right-censored observation at limit $u$?

A right-censored observation (e.g. capped at policy limit $u$, value known only to exceed $u$) contributes the survival probability $S(u; \theta) = 1 - F(u; \theta)$ to the likelihood, rather than the density $f$.

How do you construct the likelihood contribution for a left-truncated observation at deductible $d$?

A left-truncated observation $x$ (only losses above deductible $d$ are observed) contributes the conditional density $\dfrac{f(x; \theta)}{S(d; \theta)}$ to the likelihood, normalizing by the probability the loss exceeded $d$.

What is the MLE of $\theta$ for an exponential distribution given a complete sample $x_1, \dots, x_n$?

For the exponential density $f(x) = \frac{1}{\theta}e^{-x/\theta}$, the MLE is the sample mean: $$\hat\theta = \bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i.$$

State the asymptotic distribution of the MLE and the role of Fisher information.

The MLE is asymptotically normal: $\hat\theta \approx N\!\left(\theta, \frac{1}{I(\theta)}\right)$, where $I(\theta) = -E\!\left[\frac{\partial^{2}}{\partial\theta^{2}}\ln f(X;\theta)\right]$ is the Fisher information. The estimated variance is the inverse of the observed/expected information.

Describe the likelihood ratio test for nested models.

For nested models, the test statistic is $$\Lambda = 2\big(\ell_{\text{full}} - \ell_{\text{reduced}}\big),$$ which is asymptotically $\chi^{2}$ with degrees of freedom equal to the difference in the number of free parameters. Large $\Lambda$ rejects the simpler model.

Define the Schwarz Bayesian Criterion (SBC/BIC) used for model selection and the decision rule.

$$\text{SBC} = \ell(\hat\theta) - \frac{r}{2}\ln n,$$ where $r$ is the number of estimated parameters and $n$ the sample size. Choose the model with the highest SBC; it penalizes added parameters more heavily than the AIC.

State the Akaike Information Criterion (AIC) and how it compares to SBC as a penalty.

$$\text{AIC} = \ell(\hat\theta) - r,$$ with $r$ the number of parameters. Choose the largest AIC. SBC penalizes by $\frac{r}{2}\ln n$, which exceeds the AIC penalty of $r$ whenever $\ln n > 2$ (i.e. $n > 7$), so SBC favors simpler models for large samples.

What does the Kolmogorov–Smirnov (K–S) test measure for model selection?

The K–S statistic is the maximum absolute difference between the empirical CDF $F_n(x)$ and the fitted CDF $F^{*}(x)$: $$D = \max_{x} \big| F_n(x) - F^{*}(x) \big|.$$ Smaller $D$ indicates a better fit; it is most sensitive to discrepancies near the center of the distribution.

What is the limited fluctuation (classical) credibility full-credibility standard for claim counts?

The standard for full credibility on the number of claims (Poisson assumption) is $$\lambda_F = \left(\frac{z_{(1+p)/2}}{k}\right)^{2},$$ the expected number of claims needed so that observed counts are within $\pm k$ of the mean with probability $p$.

In limited fluctuation credibility, what is the partial credibility factor $Z$ and the credibility-weighted estimate?

The square-root rule gives $Z = \sqrt{\frac{n}{n_F}}$ (capped at 1), where $n$ is the observed exposure/claims and $n_F$ the full-credibility standard. The estimate is $$\hat\mu = Z\,\bar{X} + (1 - Z)\,M,$$ with $M$ the prior/manual mean.

In Bühlmann (greatest accuracy) credibility, define the Expected Process Variance (EPV) and Variance of Hypothetical Means (VHM).

$\text{EPV} = E_\Theta\!\left[\operatorname{Var}(X \mid \Theta)\right]$ (average within-risk variance) and $\text{VHM} = \operatorname{Var}_\Theta\!\left(E[X \mid \Theta]\right)$ (variance of the hypothetical means across risks). These drive the credibility factor.

Write the Bühlmann credibility factor $Z$ and the credibility premium.

With $k = \frac{\text{EPV}}{\text{VHM}}$ and $n$ observations, $$Z = \frac{n}{n + k}.$$ The Bühlmann credibility estimate is $\hat\mu = Z\bar{X} + (1-Z)\mu$, where $\mu$ is the overall (collective) mean.

How does Bühlmann–Straub credibility differ from basic Bühlmann?

Bühlmann–Straub allows unequal exposures $m_i$ per period. The credibility factor uses total exposure $m = \sum m_i$: $$Z = \frac{m}{m + k}, \quad k = \frac{\text{EPV}}{\text{VHM}},$$ and uses an exposure-weighted average of observed claim experience.

Define the survival function $S_x(t)$ and the actuarial notation ${}_tp_x$ and ${}_tq_x$.

For a life aged $x$, ${}_tp_x = S_x(t) = \Pr[T_x > t]$ is the probability of surviving $t$ more years, and ${}_tq_x = 1 - {}_tp_x = \Pr[T_x \leq t]$ is the probability of dying within $t$ years. Note ${}_tp_x + {}_tq_x = 1$.

Express ${}_tp_x$ in terms of the force of mortality $\mu_{x+t}$.

$${}_tp_x = \exp\!\left(-\int_{0}^{t} \mu_{x+s}\, ds\right),$$ where $\mu_{x+s} = \frac{-S_x'(s)}{S_x(s)}$ is the force of mortality (hazard rate) at age $x+s$.

State the complete and curtate expectation of life formulas.

Complete expectation: $\mathring{e}_x = \int_{0}^{\infty} {}_tp_x\, dt$. Curtate expectation: $e_x = \sum_{k=1}^{\infty} {}_kp_x$. Approximately $\mathring{e}_x \approx e_x + \frac{1}{2}$ under uniform distribution of deaths.

Give the actuarial present value (EPV) of a fully discrete whole life insurance $A_x$.

$$A_x = \sum_{k=0}^{\infty} v^{k+1} \, {}_kp_x \, q_{x+k},$$ where $v = \frac{1}{1+i}$ is the annual discount factor. It pays $1$ at the end of the year of death.

Write the actuarial present value of a fully continuous whole life insurance $\bar{A}_x$.

$$\bar{A}_x = \int_{0}^{\infty} v^{t}\, {}_tp_x\, \mu_{x+t}\, dt = \int_{0}^{\infty} e^{-\delta t}\, {}_tp_x\, \mu_{x+t}\, dt,$$ where $\delta = \ln(1+i)$ is the force of interest. It pays $1$ at the moment of death.

State the relationship between the EPV of whole life insurance $A_x$ and the whole life annuity-due $\ddot{a}_x$.

$$A_x = 1 - d\,\ddot{a}_x, \quad\text{equivalently}\quad \ddot{a}_x = \frac{1 - A_x}{d},$$ where $d = \frac{i}{1+i}$ is the annual discount rate. This is the fundamental annuity–insurance identity (discrete case).

Give the variance of the present-value random variable for fully continuous whole life insurance.

$$\operatorname{Var}(Z) = {}^{2}\bar{A}_x - \left(\bar{A}_x\right)^{2},$$ where ${}^{2}\bar{A}_x$ is the insurance EPV computed at force of interest $2\delta$ (double the force of interest).

Write the EPV of a whole life annuity-due $\ddot{a}_x$ and a continuous life annuity $\bar{a}_x$.

Discrete: $\ddot{a}_x = \sum_{k=0}^{\infty} v^{k}\, {}_kp_x$ (payments at the start of each year). Continuous: $\bar{a}_x = \int_{0}^{\infty} v^{t}\, {}_tp_x\, dt = \int_{0}^{\infty} e^{-\delta t}\, {}_tp_x\, dt$.

State the equivalence principle for calculating a net premium.

The net premium $P$ is set so that the EPV of future premiums equals the EPV of future benefits at issue: $$\text{EPV(premiums)} = \text{EPV(benefits)}.$$ Equivalently, the EPV of the loss-at-issue random variable $L_0$ is zero: $E[L_0] = 0$.

Give the net annual premium for a fully discrete whole life insurance of 1, $P_x$.

$$P_x = \frac{A_x}{\ddot{a}_x} = \frac{d A_x}{1 - A_x} = \frac{1}{\ddot{a}_x} - d,$$ funding a benefit of $1$ paid at end of year of death with level premiums payable at the start of each year.

How does a gross (expense-loaded) premium differ from a net premium, and state its equivalence equation?

The gross premium $G$ funds benefits plus expenses (and possibly profit). By the equivalence principle: $$\text{EPV(gross premiums)} = \text{EPV(benefits)} + \text{EPV(expenses)}.$$ Expenses include per-policy, per-premium (percentage), and acquisition (first-year) costs, so $G \geq P$ (the net premium).

Define the prospective policy reserve and give the formula for a fully discrete whole life policy at duration $t$.

The prospective reserve is EPV(future benefits) $-$ EPV(future net premiums). For fully discrete whole life: $$\,{}_tV_x = A_{x+t} - P_x\,\ddot{a}_{x+t} = 1 - \frac{\ddot{a}_{x+t}}{\ddot{a}_x}.$$ It measures the insurer's accumulated liability per surviving policyholder.