Applying a Rating Factor Change to Existing Rates

Why a Factor Change Requires a Rate Recalculation

A rating plan assigns a rate to a risk by starting with a base rate and multiplying by the relativities for that risk’s selected factor levels. When one relativity changes, the indicated rate changes even if the risk profile and base rate do not. The practical task is to apply the change consistently: update the affected factor, leave the other inputs alone, and quantify both the new rate and its difference from the old rate.

This tutorial concerns the arithmetic of revising existing indicated rates. It does not determine whether a proposed relativity is adequate or whether the rating structure itself is appropriate. All scenarios and values below are hypothetical. Assume the quoted rates are on the same coverage, exposure, and time-period basis before and after the change.

Definition: For a risk with profile levels $c_1,\ldots,c_p$, a multiplicative rating plan has indicated rate $R=B\prod_{j=1}^{p} r_j(c_j)$, where $B$ is the base rate and $r_j(c_j)$ is the relativity for the risk’s level of factor $j$. If only factor $k$'s relativity changes from $r_k$ to $r'_k$, while $B$ and all other selected relativities stay fixed, the revised rate is $R'=B r'_k\prod_{j\ne k}r_j(c_j)$.

The Rate-Adjustment Rule

If the old indicated rate has already been calculated, there is no need to rebuild the entire rating calculation. For positive old and new relativities, multiply the old rate by the ratio of the new relativity to the old relativity. The same ratio applies to every risk whose selected level of the changed factor is affected, although the dollar change can differ by profile.

R'=R\frac{r'_k}{r_k}

The percentage change in the indicated rate follows directly from the adjustment ratio:

\frac{R'-R}{R} =\frac{r'_k}{r_k}-1

This percentage is not generally the same as the difference $r'_k-r_k$. A relativity rising from 0.80 to 1.00 rises by 0.20 relativity points, but the resulting rate rises by $1.00/0.80-1=25\%$, not 20%. The denominator for the percentage change is the old relativity, or equivalently the old rate.

Why the Adjustment Ratio Works

The rule is a direct consequence of the multiplicative structure. Write the old rate as the product of the base rate, the old relativity for the changed factor, and the unchanged factors. The new rate has the same terms except that the changed factor now has its new relativity. Dividing the new rate by the old rate cancels the common terms.

\frac{R'}{R} = \frac{B r'_k\prod_{j\ne k}r_j(c_j)} {B r_k\prod_{j\ne k}r_j(c_j)} = \frac{r'_k}{r_k} \qquad\Longrightarrow\qquad R'=R\frac{r'_k}{r_k}

This cancellation requires the base rate and other selected factors to remain unchanged. If the base rate changes too, or if another factor is revised, its change must also be included. A direct recalculation from the full rating formula is always a useful check. If an old relativity is zero, the ratio form is undefined; use the direct formula instead. Ordinary rating relativities are positive, so the ratio form is normally available.

Worked Example: Increasing a Territory Relativity

A hypothetical insurer uses a base rate of $500. For a particular policy, the territory relativity is 1.20, the vehicle relativity is 0.90, and the use relativity is 1.10. The territory relativity is revised from 1.20 to 1.32. Calculate the old and revised indicated rates, the dollar change, and the percentage change. Assume all values are rates for the same policy period.

Method: adjustment ratio, checked by direct recalculation. The old rate is $500(1.20)(0.90)(1.10)$. Compute in order: $500(1.20)=600$, $600(0.90)=540$, and $540(1.10)=594$. The old indicated rate is $594.00.

The factor adjustment ratio is $1.32/1.20=1.10$. Therefore, the revised rate is $594(1.10)=653.40$, or $653.40. The dollar change is $653.40-594.00=$$59.40, and the percentage change is $1.10-1=0.10$, or 10%.

Check by direct recalculation: $500(1.32)=660$, $660(0.90)=594$, and $594(1.10)=653.40$. This agrees with the ratio method. Common mistake avoided: adding 0.12, the relativity-point change, to the rate as though it were a dollar amount. The new relativity is a multiplier; the rate changes by the ratio $1.32/1.20$.

Calculating the Impact for Different Profiles

When one factor changes, every affected risk has the same percentage rate change if the old and new relativities for its selected level are the same. But those risks need not have the same dollar change. Their old rates reflect their other factors, so multiplying by the same adjustment ratio produces different dollar increases or decreases.

Worked Example: A Relativity Moving Toward the Reference Level

A hypothetical rating plan has a $720 base rate. For one class, a factor relativity is reduced from 0.80 to 1.00. A particular risk also has unchanged relativities of 1.25 and 0.96. Calculate its old and revised rates and explain the percentage change. Assume the same policy period applies to both rates.

Method: hold the other profile factors fixed and use the relativity ratio. First calculate their combined contribution with the base rate: $720(1.25)=900$, then $900(0.96)=864$. Thus the rate before applying the changed factor is $864.

The old rate is $864(0.80)=691.20$, or $691.20. The new rate is $864(1.00)=864.00$, or $864.00. The increase is $864.00-691.20=$$172.80.

The adjustment ratio is $1.00/0.80=1.25$, so the percentage change is $1.25-1=0.25$, or 25%. The direct calculation checks: $691.20(1.25)=864.00$. Common mistake avoided: calling this a 20% rate increase because the relativity increased by 0.20. The rate increase is measured relative to the old relativity: $0.20/0.80=25\%$.

A useful way to compare profiles is to calculate the part of the rate that remains fixed for each risk: the base rate multiplied by all unchanged relativities. Multiplying that fixed amount by the old and new changed-factor relativities gives the two rates. The percentage change is common across the affected profiles, while the absolute dollar change scales with that fixed amount.

Worked Example: Comparing Three Risk Profiles

A hypothetical insurer has a $640 base rate and revises one factor’s relativity from 1.15 to 1.05. Three profiles have the following unchanged factors: Profile A has 0.90 and 1.10; Profile B has 1.25 and 0.80; Profile C has 1.00 and 1.30. For each profile, calculate the old rate, revised rate, and dollar change. Assume every rate is on the same policy-period basis.

Method: calculate each profile’s fixed portion, then apply both relativities. For Profile A, the fixed portion is $640(0.90)(1.10)=576(1.10)=633.60$. Its old rate is $633.60(1.15)=728.64$; its revised rate is $633.60(1.05)=665.28$. The change is $665.28-728.64=$−$63.36.

For Profile B, the fixed portion is $640(1.25)(0.80)=800(0.80)=640.00$. Its old rate is $640(1.15)=736.00$; its revised rate is $640(1.05)=672.00$. The change is $672.00-736.00=$−$64.00.

For Profile C, the fixed portion is $640(1.00)(1.30)=832.00$. Its old rate is $832(1.15)=956.80$; its revised rate is $832(1.05)=873.60$. The change is $873.60-956.80=$−$83.20.

For every profile, the adjustment ratio is $1.05/1.15=21/23$, approximately 0.9130. The percentage change is therefore $21/23-1=-2/23$, approximately −8.70%. The largest dollar decrease is Profile C’s $83.20, because it has the largest fixed portion of the rate. Check: each revised rate equals its old rate multiplied by $21/23$, subject only to displayed rounding. Common mistake avoided: assuming the same percentage change means the same dollar change for every risk.

Calculation Sequence and Common Traps

A disciplined sequence helps prevent errors when updating a table or several profile rates. Keep the old and new relativity for the same factor level paired; keep the same base rate and other profile values in both calculations; then check that the direction and size of the rate change are consistent with the ratio.

Identify the affected relativity.
Confirm the selected factor level and record its old and new values. Do not substitute a relativity for a different level.

Keep the other inputs fixed.
Use the same base rate and all the same other profile relativities unless the problem explicitly changes them too.

Apply the ratio or recalculate directly.
Multiply the old rate by new relativity divided by old relativity, or evaluate the full product with the new relativity.

Report and check the impact.
Calculate the dollar difference and percentage difference from the old rate. Verify that an increased ratio raises the rate and a decreased ratio lowers it.

Two errors deserve particular attention. First, the relativity-point difference is not the rate percentage change: use $r'_k/r_k-1$, not $r'_k-r_k$. Second, do not apply the new relativity to an already revised rate; that would count the change twice. Apply the ratio once to the old rate, or calculate the new rate from the fixed base and factors. In calculations that require cents, retain full precision through the multiplications and round only the final reported amounts.

Key takeaway: With the base rate and all other selected relativities fixed, a one-factor change scales each affected indicated rate by the ratio of its new relativity to its old relativity. This gives a common percentage change for the affected level, but profile-specific dollar changes.

Check Your Understanding

For each hypothetical problem, assume the stated rates refer to the same coverage and policy period. Keep unmentioned rating inputs fixed, and round dollar answers to cents where needed.

A $450 base rate is multiplied by unchanged relativities 0.90 and 1.25 and a factor relativity that changes from 1.20 to 1.08. Find the old rate, revised rate, dollar change, and percentage change. (Answer: old rate $607.50; revised rate $546.75; decrease $60.75, or 10.00%.)
A risk’s old rate is based on a $800 base rate, a changed relativity of 0.75, and unchanged relativities of 1.10 and 0.95. The changed relativity becomes 0.90. Find the revised rate and percentage change. (Answer: revised rate $752.40; increase $125.40, or 20.00%.)
A $600 base rate has a factor relativity changed from 1.40 to 1.26. Profile L has unchanged factors 0.80 and 1.10; Profile H has unchanged factors 1.20 and 1.05. Calculate both old and revised rates and identify which profile has the larger dollar decrease. (Answer: Profile L, $739.20 to $665.28, decrease $73.92; Profile H, $1,058.40 to $952.56, decrease $105.84; Profile H has the larger decrease. Both decrease 10.00%.)
A $1,050 base rate has unchanged factors 1.15 and 1.05. Its changed relativity moves from 0.96 to 1.08. Calculate the old rate, revised rate, dollar change, and percentage change, rounding money to cents only at the end. (Answer: old rate $1,217.16; revised rate $1,369.31; increase $152.15, or 12.50%.)
A $720 base rate has a factor relativity changing from 1.25 to 1.10. Profile A has unchanged factors 0.80 and 1.25; Profile B has 1.10 and 0.90; Profile C has 1.30 and 1.05. Calculate each dollar decrease and determine which is greatest. (Answer: Profile A decreases $108.00; Profile B decreases $106.92; Profile C decreases $147.42. Profile C has the greatest decrease; each profile decreases 12.00%.)

What You'll Learn