Casualty Actuarial Society Credentials (ACAS/FCAS) Probability, Financial Mathematics, and VEE Foundations Syllabus

Probability, Financial Mathematics, and VEE Foundations syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Probability, Financial Mathematics, and VEE Foundations in Casualty Actuarial Society Credentials (ACAS/FCAS), not a summary of it.

1. 1

   Exam P / Exam 1 — Probability

   4 topics

   • Fundamentals of Probability
     • Set theory, sample spaces, and events
     • Axioms of probability and basic theorems
     • Combinatorial probability: permutations and combinations
     • Conditional probability and independence
     • Bayes' Theorem and the law of total probability
   • Univariate Random Variables
     • Discrete vs. continuous random variables
     • Probability mass, density, and cumulative distribution functions
     • Expectation, variance, and higher moments
     • Moment generating functions and their uses
     • Percentiles, mode, median, and measures of skewness
   • Common Probability Distributions
     • Discrete: binomial, Poisson, geometric, negative binomial, hypergeometric
     • Continuous: uniform, exponential, gamma, normal, beta
     • Memoryless property of the exponential distribution
     • Normal approximation and the central limit theorem
   • Multivariate Distributions
     • Joint, marginal, and conditional distributions
     • Covariance, correlation, and independence of random variables
     • Conditional and double expectation (E[E[X|Y]])
     • Variance of sums and the law of total variance
     • Transformations of random variables and order statistics
2. 2

   Exam FM / Exam 2 — Financial Mathematics

   6 topics

   • Time Value of Money and Interest Theory
     • Simple, compound, and effective vs. nominal interest rates
     • Force of interest and continuous compounding
     • Discount rates and present/accumulated value
     • Equations of value and unknown time/rate problems
   • Annuities and Cash Flow Valuation
     • Level annuities-immediate and annuities-due
     • Deferred, perpetuity, and continuously payable annuities
     • Increasing, decreasing, and geometrically varying annuities
   • Loans and Amortization
     • Amortization schedules and outstanding balance methods
     • Sinking fund method and comparison to amortization
   • Bonds and Investment Analysis
     • Bond pricing, premium/discount, and yield to maturity
     • Net present value and internal rate of return
     • Spot rates, forward rates, and the term structure
   • Immunization and Interest Rate Risk
     • Macaulay and modified duration
     • Convexity and price sensitivity
     • Redington and full immunization strategies
   • Introduction to Financial Derivatives
     • Forwards, futures, and swaps
     • Put and call options and payoff diagrams
     • Hedging and basic risk management strategies
3. 3

   VEE — Validation by Educational Experience

   3 topics

   • VEE Economics
     • Microeconomics: supply, demand, and market structures
     • Macroeconomics: GDP, inflation, and monetary policy
   • VEE Accounting and Finance
     • Financial statements and corporate finance fundamentals
     • Capital structure, cost of capital, and capital budgeting
   • VEE Mathematical Statistics
     • Estimation: point estimators, bias, and consistency
     • Hypothesis testing and confidence intervals
     • Maximum likelihood and method of moments

Probability, Financial Mathematics, and VEE Foundations flashcards for Casualty Actuarial Society Credentials (ACAS/FCAS)

21 of 51 cards from the Probability, Financial Mathematics, and VEE Foundations deck — real questions with worked answers.

1. State the three Kolmogorov axioms of probability for a sample space $S$ and events $A$.

   (1) Non-negativity: $P(A) \geq 0$. (2) Normalization: $P(S) = 1$. (3) Countable additivity: for mutually exclusive events $A_1, A_2, \dots$, $P\left(\bigcup_i A_i\right) = \sum_i P(A_i)$.

2. What is the general addition rule (inclusion-exclusion for two events)?

   $$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$

3. Define conditional probability and state Bayes' Theorem.

   Conditional: $P(A \mid B) = \frac{P(A \cap B)}{P(B)}$ for $P(B) > 0$. Bayes' Theorem: $$P(A \mid B) = \frac{P(B \mid A)\,P(A)}{P(B)}$$ where $P(B) = \sum_i P(B \mid A_i)P(A_i)$ by the law of total probability.

4. When are two events $A$ and $B$ independent, and what does this imply for conditional probability?

   $A$ and $B$ are independent iff $P(A \cap B) = P(A)P(B)$. Equivalently $P(A \mid B) = P(A)$ and $P(B \mid A) = P(B)$.

5. How is the expected value of a discrete and a continuous random variable defined?

   Discrete: $E[X] = \sum_x x\,p(x)$. Continuous: $E[X] = \int_{-\infty}^{\infty} x\,f(x)\,dx$.

6. Give the two equivalent formulas for the variance of a random variable $X$.

   $$\operatorname{Var}(X) = E\big[(X - E[X])^2\big] = E[X^2] - (E[X])^2$$

7. For constants $a, b$, what are $E[aX + b]$ and $\operatorname{Var}(aX + b)$?

   $E[aX + b] = aE[X] + b$ and $\operatorname{Var}(aX + b) = a^2\operatorname{Var}(X)$ (variance is unaffected by additive constants).

8. What is a moment generating function (MGF) and how is it used to find moments?

   $M_X(t) = E[e^{tX}]$. The $n$-th moment is the $n$-th derivative evaluated at zero: $E[X^n] = M_X^{(n)}(0)$. In particular $E[X] = M_X'(0)$.

9. Relate the cumulative distribution function (CDF) to the probability density function (PDF) for a continuous random variable.

   $F(x) = \int_{-\infty}^{x} f(t)\,dt$ and, where $F$ is differentiable, $f(x) = \frac{d}{dx}F(x)$. Also $P(a < X \leq b) = F(b) - F(a)$.

10. State the probability mass function, mean, and variance of a Binomial$(n,p)$ random variable.

    $P(X = k) = \binom{n}{k}p^k(1-p)^{n-k}$ for $k = 0,\dots,n$. Mean $E[X] = np$; variance $\operatorname{Var}(X) = np(1-p)$.

11. State the PMF, mean, and variance of a Poisson$(\lambda)$ random variable.

    $P(X = k) = \frac{e^{-\lambda}\lambda^k}{k!}$ for $k = 0,1,2,\dots$. Mean and variance are both equal to $\lambda$.

12. State the PDF, mean, and variance of an Exponential random variable with rate $\lambda$.

    $f(x) = \lambda e^{-\lambda x}$ for $x \geq 0$. Mean $E[X] = \frac{1}{\lambda}$; variance $\operatorname{Var}(X) = \frac{1}{\lambda^2}$. It is memoryless: $P(X > s+t \mid X > s) = P(X > t)$.

13. State the PDF of a Normal$(\mu, \sigma^2)$ random variable and how to standardize it.

    $$f(x) = \frac{1}{\sigma\sqrt{2\pi}}\exp\!\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)$$ Standardize via $Z = \frac{X - \mu}{\sigma} \sim N(0,1)$.

14. State the mean and variance of a Uniform$(a,b)$ random variable.

    Mean $E[X] = \frac{a+b}{2}$; variance $\operatorname{Var}(X) = \frac{(b-a)^2}{12}$. The PDF is $f(x) = \frac{1}{b-a}$ for $a \leq x \leq b$.

15. State the PMF, mean, and variance of a Geometric random variable (number of trials until first success).

    $P(X = k) = (1-p)^{k-1}p$ for $k = 1,2,\dots$. Mean $E[X] = \frac{1}{p}$; variance $\operatorname{Var}(X) = \frac{1-p}{p^2}$. It is the discrete memoryless distribution.

16. How do you compute a marginal density from a joint density $f(x,y)$?

    Integrate out the other variable: $f_X(x) = \int_{-\infty}^{\infty} f(x,y)\,dy$ and $f_Y(y) = \int_{-\infty}^{\infty} f(x,y)\,dx$.

17. Define the covariance of $X$ and $Y$ and give the two equivalent formulas.

    $$\operatorname{Cov}(X,Y) = E\big[(X - E[X])(Y - E[Y])\big] = E[XY] - E[X]E[Y]$$

18. Define the correlation coefficient and state its range.

    $\rho_{X,Y} = \frac{\operatorname{Cov}(X,Y)}{\sigma_X \sigma_Y}$, with $-1 \leq \rho \leq 1$. Values near $\pm 1$ indicate strong linear association; $\rho = 0$ indicates no linear association.

19. What is the variance of a sum $X + Y$, and how does it simplify under independence?

    $\operatorname{Var}(X + Y) = \operatorname{Var}(X) + \operatorname{Var}(Y) + 2\operatorname{Cov}(X,Y)$. If $X$ and $Y$ are independent, $\operatorname{Cov}(X,Y) = 0$ so $\operatorname{Var}(X+Y) = \operatorname{Var}(X) + \operatorname{Var}(Y)$.

20. State the law of total expectation (tower property) and the law of total variance.

    Total expectation: $E[X] = E\big[E[X \mid Y]\big]$. Total variance: $$\operatorname{Var}(X) = E\big[\operatorname{Var}(X \mid Y)\big] + \operatorname{Var}\big(E[X \mid Y]\big)$$

21. State the Central Limit Theorem for the sample mean.

    For i.i.d. $X_i$ with mean $\mu$ and variance $\sigma^2$, as $n \to \infty$ the sample mean $\bar{X}$ satisfies $$\frac{\bar{X} - \mu}{\sigma/\sqrt{n}} \xrightarrow{d} N(0,1)$$

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