Key Formulas and Concepts

Why This Matters

Actuarial formulas aren't just math problems—they're the foundation of every insurance decision you'll encounter on the exam. When you see questions about pricing, reserves, or profitability, you're really being tested on whether you understand how insurers quantify uncertainty and why specific calculations matter for financial stability. These concepts connect directly to core principles like the time value of money, risk pooling, and the fundamental insurance equation.

Here's the key insight: every formula in this guide answers one of three questions—What should we charge? (pricing), What should we save? (reserves), or How are we doing? (performance metrics). Don't just memorize the formulas—know which question each one answers and when you'd apply it. That's what separates a 3 from a 5 on the exam.

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Time Value of Money Foundations

Before insurers can price anything, they need to account for the fact that a dollar today isn't worth the same as a dollar ten years from now. These calculations form the mathematical bedrock of all actuarial work.

Present Value and Future Value

• Present Value (PV) discounts future cash flows to today's dollars—calculated as $PV = \frac{FV}{(1 + r)^n}$ where $r$ is the interest rate and $n$ is the number of periods
• Future Value (FV) projects current investments forward—essential for understanding how premium dollars grow before claims are paid
• Time value of money appears constantly in exam questions about policy pricing, settlement options, and investment income assumptions

Expected Value Calculations

• Expected value equals $E(X) = \sum [x_i \times P(x_i)]$—the probability-weighted average of all possible outcomes
• Risk quantification depends on this formula; it tells actuaries the "average" claim they should anticipate from a pool of policies
• Decision-making tool that transforms uncertainty into a single number insurers can use for pricing and planning

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Premium and Pricing Calculations

Setting the right price is the core actuarial challenge. Charge too little and the company fails; charge too much and customers leave.

Net Premium Calculations

• Net premium covers only expected claims and expenses—calculated as the present value of expected future benefits, with no profit margin included
• Formula foundation: $\text{Net Premium} = PV(\text{Expected Claims}) + PV(\text{Expenses})$—this is your baseline before loading for profit
• Financial stability depends on accurate net premium calculations; underestimate and reserves fall short

Actuarial Pricing Models

• Statistical techniques combine mortality data, expense assumptions, and investment returns into comprehensive premium calculations
• Factor integration includes risk classification, policy features, and market conditions—not just pure loss expectations
• Continuous refinement means models evolve with emerging data; expect exam questions about why pricing assumptions change over time

Credibility Theory Calculations

• Credibility factor (Z) determines how much weight to give your own data versus industry benchmarks—ranges from 0 to 1
• Small dataset problem: when you lack sufficient experience data, credibility theory tells you to lean more heavily on broader industry statistics
• Formula application: $\text{Credibility-Weighted Estimate} = Z \times (\text{Own Experience}) + (1-Z) \times (\text{Industry Data})$

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Statistical Tools for Risk Assessment

Actuaries don't guess—they model. These probability tools transform raw uncertainty into predictable patterns.

Probability Distributions

• Normal distribution models continuous variables like claim sizes—characterized by mean ($\mu$) and standard deviation ($\sigma$), with 68% of values within one standard deviation
• Poisson distribution models rare, independent events like claim frequency—defined by $\lambda$ (average rate), ideal when events are random and countable
• Model selection matters: using the wrong distribution leads to systematic pricing errors; know when each applies

Mortality Tables and Life Expectancy

• Mortality tables show $q_x$, the probability of death at each age—the foundation of all life insurance pricing
• Life expectancy calculations estimate average remaining years, influencing policy design and reserve requirements
• Trend analysis is critical; improving mortality means insurers collect premiums longer but also pay death benefits later—both affect pricing

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Reserves and Financial Stability

Collecting premiums means nothing if you can't pay claims. Reserve calculations ensure insurers remain solvent when policyholders need them most.

Reserves Estimation

• Loss reserves represent funds set aside for future claims—calculated using historical loss development patterns and actuarial projections
• Regulatory requirement: insurers must maintain adequate reserves or face intervention; this isn't optional financial planning
• Methods vary: chain-ladder, Bornhuetter-Ferguson, and expected loss methods each have strengths depending on data availability and line of business

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Performance and Profitability Metrics

Once policies are sold and claims are paid, insurers need to know: Did we get it right?

Loss Ratio Calculations

• Loss ratio equals $\frac{\text{Incurred Losses}}{\text{Earned Premiums}}$—the single most important profitability indicator in insurance
• Benchmark interpretation: ratios above 100% mean losses exceed premiums; sustainable companies typically target 60-80% depending on expense loads
• Diagnostic tool that reveals whether underwriting is too aggressive, pricing is inadequate, or claims management needs improvement

Risk-Adjusted Return on Capital (RAROC)

• RAROC formula: $\frac{\text{Risk-Adjusted Return}}{\text{Economic Capital}}$—measures profitability relative to the risk taken
• Capital allocation decisions depend on RAROC; business units with higher RAROC get more resources
• Strategic alignment ensures companies don't chase volume at the expense of risk-appropriate returns

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Quick Reference Table

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Self-Check Questions

1. Which two calculations both reduce complex scenarios to single numbers but adjust for different factors—and what does each adjust for?

2. If an insurer has limited claims experience for a new product line, which formula helps them blend their data with industry benchmarks, and what does the credibility factor (Z) represent?

3. Compare and contrast loss ratio and RAROC: What question does each metric answer, and why might a product with an acceptable loss ratio still have an unacceptable RAROC?

4. You're pricing a life insurance policy. In what order would you apply mortality tables, present value calculations, and expected value—and why does sequence matter?

5. An FRQ asks you to explain why an insurer's reserves proved inadequate after a catastrophic year. Which concepts from this guide would you reference, and how do they connect to the insurer's financial stability?