ASA cheat sheet

MSE(θ̂)=Var(θ̂)+[Bias(θ̂)]²; an estimator is unbiased iff E[θ̂]=θ.

Maximum likelihood estimator maximizes the likelihood L(θ)=∏ f(x_i|θ), equivalently the log-likelihood ℓ(θ).

Method of moments sets sample moments equal to population moments and solves for parameters.

A 100(1−α)% confidence interval for μ with known σ is x̄ ± z_{α/2}(σ/√n).

Neyman–Pearson lemma: the most powerful test rejects for large values of the likelihood ratio.

Sufficient statistic T(X) satisfies f(x|θ)=g(T(x),θ)h(x) by the factorization theorem.

For a random sample, the sample mean has variance σ²/n and is consistent for μ by the weak law.

ANOVA partitions total sum of squares into treatment and error components: SST=SSB+SSW.

Simple linear regression estimates β₁ by Cov(X,Y)/Var(X) and β₀ by ȳ−β̂₁x̄.

Price elasticity of demand ε_d=(%ΔQ_d)/(%ΔP); |ε_d|>1 is elastic, <1 inelastic.

Consumer surplus is the area below demand and above price; producer surplus is above supply and below price.

A competitive firm maximizes profit where P=MC; a monopolist sets MR=MC and prices from demand.

Short-run costs: TC=FC+VC, ATC=TC/Q, MC=dTC/dQ; MC crosses ATC at its minimum.

GDP can be measured by expenditure C+I+G+(X−M), income, or value added.

CPI measures a fixed basket's cost relative to a base year; inflation is the percentage change in a price index.

Real interest rate ≈ nominal rate − expected inflation (Fisher equation).

Central bank open-market purchases increase reserves and money supply, lowering short-term rates.

Fiscal multiplier = 1/(1−MPC) in a simple closed economy without taxes.

Accounting equation: Assets = Liabilities + Equity, and it must hold after every transaction.

Income statement: Net income = Revenues − Expenses; it flows into retained earnings on the balance sheet.

Cash flow statement sections: operating, investing, and financing activities.

Current ratio = current assets/current liabilities; quick ratio excludes inventory.

NPV = Σ CF_t/(1+r)^t; accept positive-NPV projects; IRR sets NPV=0.

WACC = (E/V)r_e + (D/V)r_d(1−T), where V=E+D.

CAPM: r_e = r_f + β(E[r_m]−r_f).

DuPont: ROE = net income/sales × sales/assets × assets/equity.

Depreciation allocates a fixed asset's cost over its useful life; it is non-cash but affects taxes.

P(A∪B)=P(A)+P(B)−P(A∩B); if A and B are disjoint, P(A∪B)=P(A)+P(B).

P(A|B)=P(A∩B)/P(B); Bayes' rule: P(A|B)=P(B|A)P(A)/P(B).

E[aX+bY]=aE[X]+bE[Y]; Var(aX)=a²Var(X) and Var(X+Y)=Var(X)+Var(Y)+2Cov(X,Y).

For independent X,Y, E[XY]=E[X]E[Y] and Var(X+Y)=Var(X)+Var(Y).

Binomial(n,p): mean np, variance np(1−p); Poisson(λ): mean λ, variance λ.

Exponential(λ): mean 1/λ, variance 1/λ²; memoryless: P(X>s+t|X>s)=P(X>t).

Normal(μ,σ²): standardize Z=(X−μ)/σ; MGF M_X(t)=exp(μt+σ²t²/2).

MGF: M_X(t)=E[e^{tX}]; derivatives at t=0 give moments, M_X^{(n)}(0)=E[X^n].

CLT: (X̄−μ)/(σ/√n) converges in distribution to N(0,1) as n→∞.

Accumulation: a(t)=(1+i)^t; present value discount factor v^t=(1+i)^{-t}.

Nominal rates: (1+i)=(1+i^{(m)}/m)^m; force of interest δ=ln(1+i).

Annuity-immediate a_angle_n=(1−v^n)/i; annuity-due ä_angle_n=(1−v^n)/d.

Perpetuity-immediate a_angle_∞=1/i; perpetuity-due =1/d.

Loan balance after k payments = outstanding balance via prospective or retrospective method.

Price of a bond = coupon a_angle_n + redemption v^n at yield i.

Macaulay duration D=Σ t·CF_t v^t / Σ CF_t v^t; modified duration D* = D/(1+i).

Redington immunization requires present value, duration, and convexity matching at the current rate.

Forward rate f_{1,2} satisfies (1+s_2)^2=(1+s_1)(1+f_{1,2}).

PAF is an e-learning module that builds foundational skills in probability, statistics, financial mathematics, and computing.

The module includes instruction in spreadsheet and programming tools such as Excel and R.

Assessments in PAF emphasize applied projects and communication of technical results.

PAF prepares candidates for Exam P, Exam FM, and later actuarial modules.

The module covers data handling, visualization, and basic modeling workflows.

Professionalism and ethical conduct are introduced in the actuarial foundations.

PAF includes self-paced online lessons and periodic knowledge checks.

The module uses real-world datasets to connect mathematics to actuarial practice.

Survival function S(x)=P(X>x); force of mortality μ(x)=−S'(x)/S(x).

Life table: l_x survivors, d_x=l_x−l_{x+1}, q_x=d_x/l_x, p_x=1−q_x.

Whole life insurance A_x=Σ v^{k+1} k|q_x; term insurance excludes pure endowment.

Annuity-due ä_x=Σ v^k k p_x; whole life annuity-immediate a_x=ä_x−1.

Net premium via equivalence principle: E[PV premiums]=E[PV benefits].

Net premium reserve V_t = E[PV future benefits − PV future net premiums | alive at t].

For short-term: aggregate loss S=Σ X_i; E[S]=E[N]E[X], Var(S)=E[N]Var(X)+Var(N)E[X]².

Collective risk model: S has compound Poisson distribution when N is Poisson and X_i iid.

Multiple decrement: (aq)_x^{(j)} = ∫ p_x μ_x^{(j)} dt; associated single decrement differs from cause-specific.

ALTAM: Markov chain models use transition probabilities p_{ij} and reserves by state; premiums and benefits depend on state.

ALTAM: multiple decrement tables distinguish independent and dependent rates; (aq)_x^{(j)} is the dependent rate.

ALTAM: universal life reserves need account value projections with credited interest and cost of insurance.

ALTAM: profit testing discounts projected cash flows at a risk discount rate to get NPV and profit margin.

ASTAM: credibility premium = Z·sample mean + (1−Z)·prior mean; Z increases with sample size.

ASTAM: Lundberg bound ψ(u)≤e^{-Ru}, where R is the adjustment coefficient for the risk process.

ASTAM: aggregate loss models combine frequency and severity; Panjer recursion computes compound distributions.

ASTAM: GLMs model claim frequency with Poisson and severity with gamma; log link is common.

ASTAM: reinsurance structures include quota share, surplus share, excess of loss, and stop loss.

GLM: g(E[Y])=Xβ; Poisson regression uses log link; gamma regression uses log or inverse link.

Logistic regression models log-odds: log(p/(1−p))=Xβ.

ARIMA(p,d,q) combines autoregression, differencing, and moving average terms.

Credibility: limited fluctuation Z=sqrt(n/n_F); Bühlmann Z=n/(n+k).

Principal components are orthogonal linear combinations maximizing variance; eigenvalues give variance explained.

Cross-validation estimates out-of-sample error; k-fold splits data into k folds.

Regularization: ridge adds λΣβ²; lasso adds λΣ|β| and can set coefficients to zero.

Decision trees split on feature thresholds to reduce impurity; random forests average many trees.

Time series stationarity requires constant mean, variance, and autocovariance; differencing can induce stationarity.

ASF integrates probability, financial mathematics, and life contingencies into actuarial models.

The module covers cash flow models, present value, and risk measures.

ASF includes computational work with spreadsheets and statistical software.

Assessments require building and interpreting actuarial models.

The module introduces life tables, survival models, and insurance benefit structures.

ASF covers interest theory including annuities, loans, and bonds.

Risk management concepts include diversification, hedging, and insurance.

The module prepares candidates for FAM and SRM.

Predictive analytics workflow: data cleaning, exploratory analysis, feature engineering, model fitting, validation, deployment.

Bias-variance tradeoff: complex models have low bias and high variance.

Random forest reduces variance by averaging de-correlated trees.

Gradient boosting fits sequential trees to residuals with a learning rate.

Regularization penalizes model complexity to reduce overfitting.

Classification metrics include accuracy, precision, recall, F1, ROC AUC.

Imputation handles missing data; categorical variables need encoding.

Feature importance can be measured by permutation or impurity decrease.

Neural networks use layers of weighted sums and nonlinear activation functions.

ATPA covers advanced predictive methods including unsupervised learning, text analytics, and deep learning.

Clustering methods group observations by similarity; k-means minimizes within-cluster variance.

PCA reduces dimensionality by projecting onto orthogonal components.

Natural language processing converts text to numeric features via tokenization and embeddings.

Model interpretability methods include SHAP values and partial dependence plots.

Time series forecasting can use ARIMA, exponential smoothing, or recurrent neural networks.

Ethical considerations include bias, fairness, privacy, and transparency in models.

Model deployment requires monitoring for drift and performance degradation.

ATPA assessment uses case studies and projects requiring end-to-end analytics.

FAP is an e-learning course organized around modules such as actuarial practice, professionalism, and ERM.

The course uses case studies to apply actuarial judgment to business problems.

FAP covers the actuarial control cycle: define problem, develop model, monitor results.

Professionalism content includes the Code of Professional Conduct and actuarial standards.

Communication skills are assessed through written and oral assignments.

Enterprise risk management topics include risk identification, measurement, and mitigation.

FAP includes health, life, retirement, and general insurance practice areas.

The course requires completing module assessments and a final case study.

APC is a professionalism course required for Associateship.

The course focuses on the Code of Professional Conduct and Actuarial Standards of Practice.

Participants analyze ethical dilemmas and professional judgment scenarios.

APC covers communication with clients, employers, and the public.

The course addresses conflicts of interest and confidentiality.

APC uses case studies and group discussions to develop professional judgment.

Completion of APC is required before obtaining the ASA designation.

The course reinforces acting with integrity, competence, and objectivity.