Implementation

Once rate relativities and overall rate indications are finalized, they must be implemented into the insurer’s rating algorithm. This involves establishing expense fees, calculating off-balance factors, deriving new base rates, and handling rate capping.

---

Rating Algorithm Structure & Expense Fees

Insurance premium is typically structured with a multiplicative portion (representing risk relativities) and an additive portion (representing flat expenses).

$$\text{Premium} = \left( \text{Base Rate} \times \prod \text{Relativities} \right) + \text{Expense Fee}$$

Additive Expense Fees

The additive portion covers fixed expenses. Since expense fees are subject to variable underwriting expenses ($V$) and target profit provisions ($Q$), they must be loaded:

$$\text{Fixed Expense Fee per Exposure} = \frac{\text{Fixed Expense per Exposure}}{1 - V - Q}$$ $$\text{Fixed Expense Fee per Policy} = \text{Fixed Expense Fee per Exposure} \times \frac{\text{Exposures}}{\text{Policy}}$$

---

Off-Balance Factors

When rate relativities are updated, the change in the average relativity will alter the total premium collected, even if the base rate remains unchanged. To nullify this effect and ensure the overall rate change targets the selected indication, an Off-Balance Factor (OBF) is calculated.

$$\text{OBF} = \frac{\text{Premiums at Current Relativities}}{\text{Premiums at Proposed Relativities}} = \frac{\text{OLEP}}{\text{PLEP}}$$

Where:

• $\text{OLEP}$: On-Level Earned Premium under current relativities.
• $\text{PLEP}$: Proposed Level Earned Premium under proposed relativities.

Four Methods to Calculate OBF

1. Direct Premium Ratio: $$\text{OBF} = \frac{\text{OLEP}}{\text{PLEP}}$$
2. Ratio of Exposure-Weighted Average Rating Factors (EWARF): $$\text{OBF} = \frac{\text{Current EWARF}}{\text{Proposed EWARF}} = \frac{\frac{\sum \text{Current Relativity}_i \times \text{Exposures}_i}{\sum \text{Exposures}_i}}{\frac{\sum \text{Proposed Relativity}_i \times \text{Exposures}_i}{\sum \text{Exposures}_i}}$$
3. Relativity Change Inverse Weighting: $$\text{OBF} = \left( \sum \frac{\text{Relativity Change Factor}_i \times \text{OLEP}_i}{\text{OLEP}} \right)^{-1}$$
4. Ratio of Premium-Weighted Average Rating Factors (XWARF): Useful when exposures are not directly available, using current base premium distribution instead.

---

Deriving New Base Rates

To achieve the overall target rate change, the actuary must calculate a new base rate. Three methods are commonly used depending on data availability:

1. Extension of Exposures Method (Detailed Data Available)

This is the most accurate method. It uses detailed policy-by-policy data to project premiums.

\text{Proposed Base Rate} = \text{Seed Base Rate} \times \frac{\text{Proposed Average Premium} - \text{Proposed Additive Fee}}{\text{Average Premium at Seed Base Rate & Proposed Relativities} - \text{Proposed Additive Fee}}

Where:

• $\text{Seed Base Rate (SBR)}$: A trial base rate used to calculate the denominator.

2. Approximated Average Rate Differential Method (Exposures Available but not Joint Distribution)

Utilizes the product of marginal average rating factors across variables to approximate the joint EWARF.

$$\text{Proposed Base Rate} = \frac{\text{Proposed Average Premium} - \text{Proposed Additive Fee}}{\prod_{\text{Rating Variables}} \text{Proposed Average Rating Factor}_j}$$

3. Approximated Change in Average Rate Differential Method (No Exposure Data)

Used when only aggregate premium changes and current base rates are available.

$$\text{Proposed Base Rate} = \text{Current Base Rate} \times \text{Overall Rate Change Factor} \times \text{OBF}$$

---

Rate Capping

To protect policyholders from rate shock, insurers often limit (cap) the maximum premium increase for any single policyholder (e.g., maximum $+15\%$).

When a cap is implemented, the lost premium must be redistributed to uncapped policies to achieve the overall target rate indication.

Capping Non-Base Levels (Base Rate Change Factor $X$)

If non-base relativities are capped, the base rate change factor $X$ is solved for:

1. Set the premium increase for capped categories to the maximum limit: $$\text{Premium}_{\text{capped, proposed}} = \text{Premium}_{\text{capped, current}} \times (1 + \text{Cap \%})$$
2. Set up the equation to balance total proposed premium to the target: $$\sum \text{Premium}_{\text{uncapped, current}} \times \text{Relativity Change Factor}_i \times X + \sum \text{Premium}_{\text{capped, proposed}} = \text{Total Target Premium}$$
3. Solve for $X$ (the uncapped base rate adjustment factor).

Capping the Base Level (Relativity Adjustment Factor $X$)

If the base class itself is capped (e.g., base rate increase is limited to $+15\%$, but overall indication requires $+20\%$):

1. Apply the capped rate change directly to the base class.
2. Set up the equation to solve for a relativity adjustment factor $X$ applied to all non-base classes: $$\text{Premium}_{\text{base, current}} \times (1 + \text{Cap \%}) + \sum \text{Premium}_{\text{non-base, current}} \times (1 + \text{Cap \%}) \times \text{Relativity Change}_i \times X = \text{Total Target Premium}$$
3. Solve for $X$.
4. The final relativity for level $i$ is: $$\text{Proposed Relativity}_i = \frac{\text{Current Relativity}_i \times (1 + \text{Cap \%}) \times X}{\text{Capped Base Rate Change}}$$