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SOA ASA Exam: Predictive Analysis (PA)

[mathjax] Linear Models Classification of Variables Intention (by their role in the study) target/response/dependent/output variable risk factors/drivers Characteristics (by their nature) numeric/quantitative variables categorical/qualitative/factor variables The Model Building Process Stage 1: Define the Business Problem Objectives prediction-focused (accurate prediction) vs. interpretation-focused (relationship)   Descriptive Analytics: Focuses on insights from the past and answers the question, “What happened?” Predictive Analytics: Focuses on the future and addresses, “What might happen next?” Prescriptive Analytics: Suggests decision options; for example, “What would happen if I do this?” or “What is the best course of action?” Constraints Availability of data Implementation issues Stage 2: Data Collection Data Desgin Relevance source the data from the right population and time frame. Sampling Random Sampling Voluntary surveys may be vulnerable to respondent bias. Stratified Sampling Divide the underlying population into a number of non-overlapping strata. Oversampling and undersampling: for unbalanced data. Systematic sampling: Use a set pattern. Random Sampling: set.seed(<n>) data.full$random <- runif(nrow(data.full)) data.train <- data.full[data.full$random < 0.7, ] data.test <- data.full[data.full$random >= 0.7, ] # Present the portion of training data nrow(data.train) / nrow(data.full)   Granularity How detailed the information contained by the variable is. The more detail a variable contains, the more granular it is. Data Quality Issues Reasonableness: …

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SOA FSA Module: Enterprise Risk Management

Developing an ERM Framework ERM Framework Criteria (Effective ) Scope is enterprise-wide All risk categories included Focused on key risks Enhances decision making ability Integration across risk type Aggregated metrics Balanced risk & return management Appropriate disclosures Measures value impacts Primary stakeholder focus Challenges to ERM Analysis The implementation of a strong ERM framework must address three primary hurdles. Quantification of strategic and operational risks Imprecise formulation of risk appetite Decision making is not integrated with an ERM analysis ERM Process ERM Frameworks use a control cycle process similar to that of general actuarial work, consisting of the identification, quantification, decision making, and communication of risks across the company. Risk Identification In the initial stage of the framework process, all the risks in the universe are mapped into company specific categories based on the potential losses to the company that are relevant to its business plan. Typically this categorization breaks down into: Strategic risks Operational risks Financial risks Net Risks Risks are by default discussed on a “gross” basis, without reflecting the value of techniques used to reduce such risks. But for the purposes of finalizing the quantification, the value of any risk mitigation should be subtracted from the gross …

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SOA FSA Module: Individual Life and Annuities

[mathjax] Introduction The Introduction to ILA module will give an overview of the role of an actuary in a Life and Annuity context. The module will give a strong foundation of understanding of life insurance and annuity product features, markets and distribution. Candidates will also learn the fundamentals of product development, pricing, reinsurance, valuation, financial reporting and administration. This module lays the groundwork for the Fellowship exams in the Individual Life and Annuities track. Module Learning Objectives Understand the role of actuaries in an insurance company context. Understand life insurance and annuity product types, benefits and product features; insurance market, consumer needs and distribution channels; and types of companies in the insurance space. Assess the financial reporting environment including key stakeholders. Understand the basic product designs, design process and actuarial cycle. Understand the theory of reserving. Understand the basic insurance administration, underwriting of insurance risks and payments of claims. The Introduction to ILA module consists of six sections: Module Introduction and Role of the Actuary Products, Markets, and Company Type Insurance Company Financial Reporting Environment Product Development Process Basic Principles of Reserving Insurance Administration, Underwriting, and Claims Module Introduction and Role of the Actuary The goal of this section is …

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SOA ASA Exam: Short-Term Actuarial Mathematics (STAM/C)

[mathjax] Basic Probability Functions and Moments Probability Density Function \(f(x)=\dfrac{d}{dx}F(x)\) or \(f(x)=-\dfrac{d}{dx}S(x)\) Hazard Rate Function \(\mu (x)=h(x)=\dfrac{f(x)}{S(x)}=-\dfrac{d\ln S(x)}{dx}\) Cumulative Hazard Rate Function \(H(x)=\int_{-\infty }^{x}{h(t)dt}=-\ln S(x)\) \(S(x)=e^{-H(x)}=e^{-\int_{-\infty }^{x}{h(t)dt}}\) Moment of X nth Raw Moment of X: \(\mu’_{n}=E[x^n]\) nth Central Moment of X: \(\mu_n=E[{(x-\mu )}^n]\) Covariance: \(Cov(X,Y)=E[(X-\mu_X)(Y-\mu_Y)]=E[XY]-E[X]E[Y]\) Correlation Coefficient: \(\rho_{XY}=Cov(X,Y)/({\sigma_X}{\sigma_Y})\) Coefficient of Variance: \(\sigma /\mu \) Skewness: \(\gamma_1=\mu_3/\sigma^3\) Kurtosis: \(\gamma_2=\mu_4/\sigma^4\) Moment Generating Function: \(M_X(t)=E[e^{tX}]\) Probability Generating Function: \(P_X(t)=E[t^X]\) Differentials \(\dfrac{d}{dx}{{c}^{ax}}=a\ln (c){{c}^{ax}}\) \(\dfrac{d}{dx}{{\log }_{c}}x=\frac{1}{x\ln (c)}\) Sum of Distributions If \(X_1 \sim Gamma(\alpha_1,\theta)\), …, \(X_n \sim Gamma(\alpha_n,\theta)\), then \(X=\sum{X_i} \sim Gamma(\alpha=\alpha_1+\alpha_2+…+\alpha_n, \theta)\) If \(X_1 \sim Exp(\theta)\), …, \(X_n \sim Exp(\theta)\), then \(X=\sum{X_i} \sim Gamma(\alpha=n, \theta)\) If \(X_1 \sim Poi(\lambda_1)\), …, \(X_n \sim Poi(\lambda_n)\), then \(X=\sum{X_i} \sim Poi(\lambda=\lambda_1+\lambda_2+…+\lambda_n)\) If \(X_1 \sim Bin(m_1,q)\), …, \(X_n \sim Bin(m_n,q)\), then \(X=\sum{X_i} \sim Bin(m=m_1+m_2+…+m_n,q)\) If \(X_1 \sim NB(r_1,\beta)\), …, \(X_n \sim NB(r_n,\beta)\), then \(X=\sum{X_i} \sim NB(r=r_1+r_2+…+r_n,\beta)\) Integration by Parts \(\int{udv}=uv-\int{vdu}\) Conditional probability and expectation Bayes’ Theorem \(\Pr (A|B)=\dfrac{\Pr (B|A)\Pr (A)}{\Pr (B)}\) or \(f_X(x|y)=\dfrac{f_Y(y|x)f_X(x)}{f_Y(y)}\) Law of Total Probability If \(B_i\) is a set of exhaustive (in other words, \(\Pr (\bigcup\nolimits_{i}{B_i})=1\)) and mutually exclusive (in other words \(\Pr (B_i\bigcap{B_j})=0\) for \(i\ne j\)) events, then for any event A, \(\Pr (A)=\sum\limits_{i}{\Pr (A\bigcap{B_i})}=\sum\limits_{i}{\Pr (A|B_i)\Pr (B_i)}\) Correspondingly for continuous distributions, \(\Pr (A)=\int{\Pr (A|x)f(x)dx}\) Conditional …

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SOA ASA Exam: Probability (P)

Basic Calculus Integrations \(\dfrac{d}{dx}a^x=a^x\ln(a)\) \(\int{{{}\over{}}a^xdx}=\dfrac{a^x}{\ln(a)}\text{ for }a>0\)   Logarithmic Differentiation \(\dfrac{df(x)}{dx}=f(x)(\dfrac{d\ln f(x)}{dx})\), since \(\dfrac{d\ln f(x)}{dx}=\dfrac{df(x)/dx}{f(x)}\)   Partial Fraction Decomposition \(\int{{{x}\over{1+x}}dx}=\int{(1-{{1}\over{1+x}})dx}\)   Integration by Parts \(\int{udv}=uv-\int{vdu}\) Special Cases: \(\int_{0}^{\infty }{xe^{-ax}dx}=\dfrac{1}{a^2}\), for \(a>0\) \(\int_{0}^{\infty }{x^2e^{-ax}dx}=\dfrac{2}{a^3}\), for \(a>0\)   Sets Set Properties Associative Property \((A\cup B)\cup C=A\cup (B\cup C)\) and \((A\cap B)\cap C=A\cap (B\cap C)\) Distributive Property \(A\cup (B\cap C)=(A\cup B)\cap (A\cup C))\) and \(A\cap (B\cup C)=(A\cap B)\cup (A\cap C)\) Distributive Property for Complement \((A\cup B)’=A’\cap B’\) and \((A\cap B)’=A’\cup B’\)   Basic Probability Relationships \(Pr[A\cup B]=Pr[A]+Pr[B]-Pr[A\cup B]\) \(Pr[A\cup B\cup C]=Pr[A]+Pr[B]+Pr[C]-Pr[A\cap B]-Pr[B\cap C]-Pr[A\cap C]+Pr[A\cap B\cap C]\) If A and B are independent, then: \(Pr(A\cap B)=Pr(A)Pr(B)\) \(Pr(A\cap B’)=Pr(A)Pr(B’)=Pr(A)(1-Pr(B))\) \(Pr(A’\cap B)=Pr(A’)Pr(B)=(1-Pr(A))Pr(B)\)   Combinatorics Number of Permutations \(n!\)   Number of Distinct Permutations \(_{n}P_k=\dfrac{n!}{(n-k)!}=\dfrac{n!}{{{k}_{1}}!{{k}_{2}}!\cdots {{k}_{j}}!}\)   Number of Combinations \(_{n}C_k=\left( \begin{matrix}n\\k \\\end{matrix} \right)=\dfrac{n!}{k!(n-k)!}=\dfrac{n(n-1)\cdots (n-k+1)}{k!}\)   Conditional Probabilities \(P[A|B]=\dfrac{P[A\cap B]}{P[B]}\)   Bayes’ Theorem Law of Total Probabilities \(P[A]=\sum\limits_{i=1}^{n}{P[B_i]P[A|B_i]}\)   Bayes’ Theorem \(P[A_j|B]=\dfrac{P[A_j]P[B|A_j]}{\sum\limits_{i=1}^{n}{P[A_i]P[B|A_i]}}\)   Random Variables Probability Mass Function (PMF) \(p(x)=Pr(X=x)\)   Probability Density Function (PDF) \(f(x)=\dfrac{dF(x)}{dx}\) or logarithmic differentiation: Given \(F(x)=a\) Taking log: \(\ln F(x) = ln(a)\) Differentiate: \(\dfrac{d\ln F(x)}{dx}=\dfrac{d\ln a}{dx}=\dfrac{d F(x) / dx)}{F(x)}\) Replace the \(F(x)\) in the differentiated formula: \(\dfrac{d \boxed{F(x)}}{dx}=\boxed{F(x)}\times \dfrac{d\ln a}{dx}\) i.e. \(\dfrac{d \boxed{F(x)}}{dx}=\boxed{F(x)}(\dfrac{d\ln \boxed{F(x)}}{dx})\) Similarly, \(\dfrac{d \boxed{f(x)}}{dx}=\boxed{f(x)}(\dfrac{d\ln \boxed{f(x)}}{dx})\) The pdf \(f(x)\) must …

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SOA ASA Exam: Long-Term Actuarial Mathematics (MLC/LTAM)

[mathjax] Review Basic Concepts – Integrals: \(\int_{0}^{\infty }{{{t}^{n}}{{e}^{-ct}}dt}=\dfrac{n!}{{{c}^{n+1}}}\) \(\int_{0}^{u}{{{t}^{n}}{{e}^{-ct}}dt}=\dfrac{1-(1+cu){{e}^{-cu}}}{{{c}^{2}}}\) \({{(\bar{I}\bar{a})}_{u}}=\dfrac{{{{\bar{a}}}_{\overline{u}}}-u{{v}^{u}}}{\delta }\) – Geometric Series: \(\sum\nolimits_{k=0}^{n-1}{a{{r}^{k}}}=a\dfrac{1-{{r}^{n}}}{1-r}\) \({{i}^{(m)}}=m({{(1+i)}^{{}^{1}/{}_{m}}}-1)\) \({{d}^{(m)}}=m(1-{{(1+i)}^{-{}^{1}/{}_{m}}})\) – Survival Function: \({{S}_{x}}(0)=1\) \(\underset{t\to \infty }{\mathop{\lim }}\,{{S}_{x}}(t)=0\) \({{S}_{x}}(t)\) must be a non-increasing function of t Review from MFE – Rate of Discount: \(d=\dfrac{i}{1+i}\) – Discounting Rate: \(v=\dfrac{1}{1+i}=1-d\) – Continuously Compounded Interest Rate: \(\delta =\ln (1+i)\)  – Simple Interest Rate i: \({i}_{t}=1+it\) , \({{v}_{t}}=\dfrac{1}{1+it}\) – PV of an n-Year Immediate Certain Annuity: \({{a}_{\left. {\overline {n}} \right| }}=\dfrac{1-{{v}^{n}}}{i}\) – PV of an n-Year Immediate Certain Annuity-Due: \({{\ddot{a}}_{\left. \overline{n} \right|}}=\dfrac{1-{{v}^{n}}}{d}\)  – PV of an n-Year Continuous Certain Annuity-Due: \({{\bar{a}}_{\left. \overline{n} \right|}}=\dfrac{1-{{v}^{n}}}{\delta}\) – Cumulative Value of an Annuity, the Value at the End of n Years\({{S}_{\left. {\overline {n}}\! \right| }}={{(1+i)}^{n}}{{a}_{\left. {\overline {n}}\! \right| }}\) Survival Function \(\Pr ({{T}_{x}}>t+u)=\Pr ({{T}_{x}}>t)\Pr ({{T}_{x+t}}>u)\) \({{S}_{x}}(t+u)={{S}_{x}}(t){{S}_{x+t}}(u)\) Actuarial Notation \({}_{t+u}{{p}_{x}}={}_{t}{{p}_{x}}{}_{u}{{p}_{x+t}}\) \({}_{\text{t}|u}{{q}_{x}}={}_{t}{{\text{p}}_{x}}-{}_{t+u}{{p}_{x}}={}_{t+u}{{q}_{x}}-{}_{t}{{q}_{x}}\) Life Table \({{d}_{x}}={{l}_{x}}-{{l}_{x+1}}\) \({}_{t}{{p}_{x}}=\dfrac{{{l}_{x+t}}}{{{l}_{x}}}\), \({{q}_{x}}=\dfrac{{{d}_{x}}}{{{l}_{x}}}\) \({}_{\text{t}|u}{{q}_{x}}=\dfrac{{{l}_{x+t}}-{{l}_{x+t+u}}}{{{l}_{x}}}\) Survival Function: Moments Complete Lifetime Complete Life Expectancy \({{\overset{\scriptscriptstyle\smile}{e}}_{x}}=\int_{0}^{\infty }{_{t}{{p}_{x}}dt}\), \(E[{{T}_{x}}^{2}]=2\int_{0}^{\infty }{{{t}_{t}}{{p}_{x}}dt}\) n-Yr. Temporary Life Expectancy \({{\overset{\scriptscriptstyle\smile}{e}}_{x:\left. {\overline {n}}\! \right| }}=E[\min ({{T}_{x}},n)]=\int_{0}^{n}{_{t}{{p}_{x}}dt}\) \(E[{{(\min ({{T}_{x}},n))}^{2}}]=2\int_{0}^{n}{t{}_{t}{{p}_{x}}dt}\) Special Mortality Laws Constant Force of Mortality \({{\overset{\scriptscriptstyle\smile}{e}}_{x}}={{\mu }^{-1}}\), \(Var({{T}_{x}})={{\mu }^{-2}}\) \({{\overset{\scriptscriptstyle\smile}{e}}_{x:\left. {\overline {n}}\! \right| }}=\dfrac{1-{{e}^{-\mu n}}}{\mu }\) Uniform & Beta (Derive) For any case in which mortality is uniformly distributed throughout the temporary period, \({{\overset{\smile }{\mathop{e}}\,}_{x:\left. {\overline {n}}\! \right| }}={{n}_{n}}{{p}_{x}}+{{(n/2)}_{n}}{{q}_{x}}\) \({{\overset{\smile }{\mathop{e}}\,}_{x:\left. {\overline {n}}\! \right| }}={{n}_{n}}{{p}_{x}}+{{(n/2)}_{n}}{{q}_{x}}=n(\dfrac{w-x-n}{w-x})+\dfrac{n}{2}(\dfrac{n}{w-x})\) Curtate Lifetime …

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