Expenses & Profits

Loss Adjustment Expenses (LAE)

Loss Adjustment Expenses (LAE) are divided into Allocable (Allocated Loss Adjustment Expenses - ALAE) and Unallocable (Unallocated Loss Adjustment Expenses - ULAE).

Selecting the ULAE Factor

When selecting a ULAE factor from historical calendar year ratios (e.g., ULAE/Losses), sudden shifts require actuarial judgment:

• Operational/Structural Change: If the change is expected to continue, select the most recent year’s ratio.
• Anomaly/Random Fluctuation: If the change is temporary, select a volume-weighted average of historical years to smooth out the anomaly.

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Underwriting (UW) Expenses

Underwriting expenses are categorized based on whether they behave as fixed or variable costs relative to the size of the premium.

The Fundamental Insurance Equation for Premiums

The basic relationship to determine the indicated premium is: $\text{Premium} = \dfrac{\text{Losses} + \text{LAE} + \text{Fixed Expenses}}{1 - V - Q_t}$

Where:

• $V$: Variable expense rate as a percentage of premium.
• $Q_t$: Target profit and contingencies provision as a percentage of premium.

[!NOTE] This equation shows that once variable expenses and target profit are subtracted from premium, the remainder must cover losses, LAE, and fixed expenses.

Indicated Average Premium per Exposure

$\text{Indicated Average Premium} = \dfrac{\text{Projected Pure Premium (incl. LAE)} + \text{Fixed Expenses per Exposure}}{1 - V - Q_t}$

• Fixed Expenses (FE): Assumed to be constant per exposure, regardless of premium size (e.g., rent for home office, administrative IT systems).
• Variable Expenses (VE): Assumed to vary directly with written or earned premiums (e.g., commissions, premium taxes).

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Expense Pricing Methods

Actuaries use three primary methods to project expenses in rate indications:

1. All Variable Expense Method

• Method: Treats all expenses as variable. The variable expense ratio is the sum of all historical expense-to-premium ratios.
• Pros/Cons:
  • ✅ Simple: Requires no separation of fixed and variable costs.
  • ❌ Inaccurate: If fixed expenses exist, this method undercharges risks with lower-than-average premiums (assuming fixed costs decrease proportionally) and overcharges risks with higher-than-average premiums.
  • Mitigation: These distortions can be corrected using premium discounts or expense constants (see Special Classification).
• Trending: No expense trend is applied because expenses scale automatically with premium.

2. Premium-Based Projection Method

• Method: Separates expenses into fixed and variable components using an assumed fixed percentage. $\text{Variable Expense Ratio} + \text{Fixed Expense Ratio} = \text{Total Expense Ratio}$
• Key Assumption: Fixed expenses are assumed to trend at the same rate as premiums.
• Issues:
  • Rate Changes: Historical expense ratios will be distorted if rates change. Solution: On-level historical premiums before calculating ratios.
  • Trends: If premium and expense trends diverge, the ratio will change. Solution: Trend premiums and expenses separately (though uniform trend is often assumed in exams).
  • Interstate Inequities: Allocating countrywide fixed expenses to states as a percentage of premium overcharges states with high average premiums. Solution: Calculate fixed expense ratios on a state-specific basis.

3. Exposure/Policy-Based Projection Method

• Method: Variable expenses are projected as a percentage of premium. Fixed expenses are projected as a flat dollar amount per exposure and trended over time.
• Trend Application:
  • Written Fixed Expenses: Trended from the average written date of the experience period to the average written date of the prospective period.
  • Earned Fixed Expenses: Trended from the average earned date of the experience period to the average earned date of the prospective period.
• Shortcoming: Fails to account for economies of scale (e.g., fixed cost per exposure should decrease as the book of business grows).

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Expense Allocation: Written vs. Earned Bases

To determine the appropriate denominator (Written or Earned Premium/Exposures) for expense ratios, look at when the expense is incurred:

• Incurred at Inception $\implies$ Use Written Basis:
  • Commissions & Brokerage
  • Other Acquisition Expenses
  • Taxes, Licenses, & Fees
• Incurred Throughout the Policy $\implies$ Use Earned Basis:
  • General Expenses (e.g., administrative staff salaries, rent)

Expense Location	Common State-Level Expenses	Common Country-wide Expenses
State-Level	Taxes, Licenses, & Fees	Local commissions
Country-wide	General Expenses	CEO’s Salary, Brand Marketing

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Reinsurance Costs in Ratemaking

There are two primary methods to incorporate reinsurance costs into rate indications:

1. Net of Reinsurance Method: Restate all premium and loss data to be net of reinsurance premiums and recoveries, and perform the indication on a net basis.
2. Reinsurance Load Method: Calculate the net cost of reinsurance (Reinsurance Premium $-$ Expected Reinsurance Recoveries) and treat it as a fixed expense load.

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Permissible Loss Ratio (PLR)

1. Variable Permissible Loss Ratio (VPLR): The portion of each premium dollar available to cover losses, LAE, and fixed expenses. $\text{VPLR} = 1 - V - Q_t$
2. Total Permissible Loss Ratio (PLR): The portion of each premium dollar available to cover only losses and LAE. $\text{PLR} = 1 - V - F - Q_t = 1 - \text{Total Expense Ratio} - Q_t$

Using the PLR, the indicated average premium can be calculated as: $\text{Indicated Average Premium} = \dfrac{\text{Projected Pure Premium (incl. LAE)}}{\text{PLR}}$

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Miscellaneous Concepts

Profit Provision Components

The target profit provision ($Q_t$) reflects:

• Investment Income: Higher investment income allows for a lower underwriting profit provision.
• Risk Profile: Higher risk/volatility requires a higher profit margin.
• LTV (Lifetime Value): Insurers may accept lower initial profits to acquire long-term customers.
• Reinsurance: Greater reinsurance protection reduces volatility, allowing for a lower profit provision.
• Contingency Provision: A load for unexpected severe events or systemic deviations.

Policy Year Premium Audits

Policy Year (PY) premiums are not immediately fixed at the end of the year. They develop due to:

• Premium Audits: Auditing actual exposure bases (e.g., payroll, sales) after policy expiration.
• Retrospective Rating Adjustments: Adjustments based on the actual loss experience of the policy.

*[FII]: Fundamental Insurance Equation