Common Errors in Rate Indications

Why Rate Indications Are Vulnerable to Small Errors

A prospective rate indication combines amounts measured over different periods and on different bases. Losses may need to be trended, premiums may need to be adjusted to the current rate level, and expenses may be stated either per exposure or as a percentage of premium. A calculation can look plausible while mixing these quantities in a way that makes its result invalid.

The audit is not just arithmetic. Before calculating, identify what each amount represents, when it applies, and what it is divided by. This tutorial focuses on common errors in prospective indications; it is aligned with the SOA Exam ASTAM (Advanced Short-Term Actuarial Mathematics) syllabus.

Definition: A prospective rate indication is a calculation of the premium or rate level needed for a defined prospective period, given projected losses and expenses and the selected provisions for variable expenses and profit or contingency. Its components must refer to compatible periods, coverage, and exposure units.

Start With the Time Periods

Trend converts experience-period losses to an estimated loss level for a later period. If the annual trend rate is $t$, and the interval from the experience-period midpoint to the prospective-period midpoint is $n$ years, the projected losses are

L_{\mathrm{proj}}=L_{\mathrm{exp}}(1+t)^n.

The exponent is the elapsed time between the two midpoints—not automatically the number of years in either period. For example, if the experience midpoint is two years before the prospective midpoint, use $n=2$, even if each underlying period covers a different span of time. If the trend rate changes by year, apply the corresponding annual factors for each interval.

Worked Example: Trending to the Prospective Midpoint

A hypothetical insurer has experience-period losses of $500,000. The experience midpoint is two years before the prospective midpoint, and the annual loss trend assumption is 5%. The current premium for the comparable book, at current rates, is $700,000. Find the trended losses and the resulting percentage rate change if fixed expenses are $40,000 and variable expenses and profit are 20% and 5% of premium, respectively.

Method. First trend losses for the actual midpoint-to-midpoint interval. Then solve the premium equation using the stated premium-based provisions.

Step 1: Trend the losses. The elapsed interval is $n=2$ years, so

L_{\mathrm{proj}}=500{,}000(1.05)^2 =500{,}000(1.1025) =\$551{,}250.

Step 2: Solve for indicated premium. Let $P_{\mathrm{ind}}$ be the indicated premium. The 20% variable expense and 5% profit provisions together take $0.25P_{\mathrm{ind}}$. The premium must fund projected losses, fixed expenses, and those premium-based provisions:

P_{\mathrm{ind}}=551{,}250+40{,}000+0.25P_{\mathrm{ind}}, \qquad 0.75P_{\mathrm{ind}}=591{,}250.

Thus $P_{\mathrm{ind}}=591{,}250/0.75=\$788{,}333.33$, rounded to the nearest cent.

Step 3: Calculate the rate change. Compare the indicated premium with the $700,000 current-rate premium for the same book:

\frac{788{,}333.33}{700{,}000}-1 =0.126190\ldots \approx 12.62\%.

Check and trap. The premium equation checks because $0.75(788{,}333.33)\approx \$591{,}250$. Using a one-year trend merely because a rate change is often discussed annually would give $500{,}000(1.05)=\$525{,}000$, understating projected losses. The correct exponent comes from the midpoint dates.

Weight Overall Changes by Premium

An overall rate change across classes is generally premium-weighted. If class $j$ has current premium $P_j$ and indicated percentage change $r_j$, the aggregate change is

r_{\mathrm{overall}} =\frac{\sum_j P_jr_j}{\sum_jP_j}.

This formula weights each class by its share of the current premium base. It follows by adding the dollar premium changes, $P_jr_j$, across classes and dividing the total change by the total current premium. The weights must be on the same basis as the rate changes—for example, the same time period, coverage, and rate level.

Worked Example: Combining Class Rate Changes

A hypothetical portfolio has two classes. Class A has $600,000 of current premium and an indicated increase of 10%; Class B has $400,000 of current premium and an indicated decrease of 5%. Find the overall rate change.

Method. Calculate each class's dollar premium change, add those changes, and divide by total current premium.

Step 1: Class A change. $600{,}000(0.10)=\$60{,}000$.

Step 2: Class B change. A 5% decrease is $r_B=-0.05$, so $400{,}000(-0.05)=-\$20{,}000$.

Step 3: Aggregate. The total change is $60{,}000-20{,}000=\$40{,}000$, and current premium is $600{,}000+400{,}000=\$1{,}000{,}000$. Therefore,

r_{\mathrm{overall}} =\frac{40{,}000}{1{,}000{,}000} =0.04=4\%.

The new premium is $660{,}000+380{,}000=\$1{,}040{,}000$, which is also 4% above $1,000,000. Common mistake: the unweighted average $(10\%-5\%)/2=2.5\%$ gives each class equal influence, despite their different premium volumes. Premium weighting gives the correct 4%.

Keep Loss Ratios and Rate Changes Distinct

A loss ratio is projected losses divided by the relevant premium. In a loss-ratio indication, the premium denominator must be at the intended rate level and correspond to the same book and period as the projected losses. If the premium is at current rates and the target loss ratio is $LR_{\mathrm{target}}$, the indicated rate-level factor is

\text{rate-level factor} =\frac{L_{\mathrm{proj}}/P_{\mathrm{current}}} {LR_{\mathrm{target}}}.

This factor is not the loss ratio itself. A factor of 1.08 means an 8% increase; a loss ratio of 0.72 means losses are 72% of the premium denominator. The formula is appropriate only when the numerator is projected to the intended period and the denominator is expressed at the rate level being evaluated. If the premium is not on-level to current rates, it must first be adjusted to a consistent rate level.

Worked Example: Applying a Target Loss Ratio

For a hypothetical coverage, projected losses are $330,000. Premium at current rates for the matching exposure base is $500,000, and the target loss ratio is 60%. Find the indicated rate change.

Method. Calculate the projected loss ratio at current rates, then divide it by the target loss ratio to obtain the rate-level factor.

The projected loss ratio is $330{,}000/500{,}000=0.66$. Therefore,

\text{factor}=\frac{0.66}{0.60}=1.10, \qquad \text{indicated change}=1.10-1=10\%.

As a check, a 10% increase gives premium of $500{,}000(1.10)=\$550{,}000$, and $330{,}000/550{,}000=0.60$, the target loss ratio. Common mistake: using $0.60/0.66=0.9091$ reverses the ratio and suggests a decrease. The current-rate loss ratio is above target, so the rate must increase, not decrease.

Use the Correct Expense Denominator

A variable expense or profit provision stated as a percentage of premium is a share of the indicated premium, not a percentage to add directly to losses. Let $L$ be projected losses, $F$ fixed expenses, $v$ the variable expense proportion of premium, and $q$ the profit or contingency provision as a proportion of premium. When these are the only provisions, the premium equation is

P=L+F+vP+qP, \qquad P=\frac{L+F}{1-v-q}.

The denominator is the share of premium left to fund losses and fixed expenses after premium-based provisions. This derivation is simply a rearrangement: subtract $vP+qP$ from both sides to obtain $P(1-v-q)=L+F$, then divide by $1-v-q$. The formula requires $v+q<1$. It does not apply unchanged if a provision is instead a fixed dollar amount or a percentage of losses; those provisions must be represented according to their actual basis.

Worked Example: Grossing Up Per-Exposure Costs

A hypothetical small-business policy has projected losses of $420 per exposure and fixed expenses of $30 per exposure. Variable expenses are 22% of premium, and the profit provision is 3% of premium. The current rate is $600 per exposure. Find the indicated rate and percentage change.

Method. Use the premium equation per exposure, keeping the loss and fixed expense amounts on a per-exposure basis.

Step 1: Add the amounts not expressed as percentages of premium. $420+30=\$450$ per exposure.

Step 2: Identify the premium share available. The total premium-based provisions are $0.22+0.03=0.25$, leaving $1-0.25=0.75$.

R_{\mathrm{ind}} =\frac{450}{0.75} =\$600 \text{ per exposure}.

Step 3: Compare with the current rate. $(600/600)-1=0$, so the indicated change is 0%. The check is $450+0.22(600)+0.03(600)=450+132+18=\$600$.

Common mistake. Adding 25% to the $450 gives $450(1.25)=\$562.50$, which does not fund the required provisions: the 25% is defined as a share of premium, not a markup on losses and fixed expenses. At a $562.50 premium, those provisions total only $0.25(562.50)=\$140.625$, leaving $421.875 rather than the required $450.

Keep Exposure Bases Consistent

A per-exposure rate and an aggregate premium are different quantities. Before comparing them, put both on the same exposure basis. For example, a loss amount for 800 exposures cannot be compared directly with a premium for 1,000 exposures unless one is scaled to match the other. The same check applies to expenses: an expense per policy should not be combined with a count of vehicle-years unless the exposure definition and unit conversion are explicit.

A reliable alternative is to calculate the indicated amount per exposure first, then compare per-exposure rates. If experience counts, projected losses, fixed expenses, and current premiums do not use compatible exposure definitions, resolve that mismatch before interpreting the indicated change.

Label each quantity.
Record whether it is a dollar amount, a percentage of premium, a per-exposure amount, a loss ratio, or a rate change.

Align periods and rate levels.
Check the experience and prospective midpoints, and confirm that the premium denominator is on the intended rate level.

Align exposure units.
Confirm that losses, expenses, and premiums refer to the same coverage and exposure basis before aggregating or comparing.

Reconcile the result.
Substitute the indicated premium back into the funding equation, or calculate the implied loss ratio and compare it with the target.

Common Audit Traps

Using the wrong trend interval. Count the time from the experience midpoint to the prospective midpoint, not simply the number of years in a period.
Averaging class changes without weights. Use current premium weights for an overall percentage change when class changes apply to different amounts of premium.
Reversing the loss-ratio factor. With current-rate projected loss ratio $LR_{\mathrm{current}}$, divide by the target: $LR_{\mathrm{current}}/LR_{\mathrm{target}}$.
Adding premium-based provisions to losses as markups. Solve the premium equation; a percentage of premium belongs on the premium side of the equation.
Comparing unlike exposure totals. Convert aggregate amounts to a common exposure basis before calculating a rate change.

A final reasonableness check often reveals an error quickly. If projected losses exceed the target share of premium, a loss-ratio indication should increase rates. If an overall indication is reported as a percentage, its weighted dollar change divided by the corresponding current premium should reproduce that percentage. These checks do not replace a correctly specified model, but they can catch reversed ratios, missing weights, and incompatible denominators.

Check Your Understanding

For each problem, check the units and denominator as well as the arithmetic.

A hypothetical portfolio has $240,000 of losses at an experience midpoint three years before the prospective midpoint. Annual trend is 3%. What are projected losses? (Answer: $262,254.48)
Two hypothetical classes have current premiums of $250,000 and $750,000, with indicated changes of +8% and −2%, respectively. Find the overall rate change. (Answer: +0.50%)
Projected losses are $480,000 and fixed expenses are $20,000. Variable expenses are 18% of premium and profit is 2% of premium. Current premium is $600,000. Find indicated premium and the rate change. (Answer: $625,000; +4.17%)
Projected losses are $315,000, current-rate premium for the matching book is $450,000, and the target loss ratio is 63%. Find the indicated rate change. (Answer: +11.11%)
A hypothetical portfolio has 800 exposures, projected losses of $320,000, and fixed expenses of $24,000. Variable expenses are 20% of premium and profit is 4% of premium. The current rate is $500 per exposure. Find the indicated rate per exposure and the percentage change. (Answer: $565.79 per exposure; +13.16%)

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