Why Final-Selection Errors Matter
An overall rate indication is only as reliable as the quantities and provisions used to produce it. A calculation can be arithmetically correct and still be wrong because its premium denominator is on the wrong rate level, an expense load is included twice, or a factor is interpreted as a percentage change. A final selection can also depart from a sound indication, but that departure should have a stated and supportable reason.
This tutorial focuses on diagnosing those errors. It assumes the underlying loss and premium estimates have been developed for the intended prospective period. The task here is to check that the final calculation uses them consistently and that any selected change can be traced to evidence. These distinctions are aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.
Check the Basis Before the Arithmetic
A loss ratio is meaningful for a rate indication only when its loss and premium amounts refer to compatible periods and rate levels. In the loss ratio method, the premium denominator is commonly adjusted to the current rate level, while losses are adjusted as appropriate for the prospective indication. If historical earned premium is used without its required on-level adjustment, the calculated loss ratio may not describe experience at current rates.
Let \(L\) denote the losses used in an indication, \(P_{\text{on}}\) the corresponding on-level premium, and \(LR_{\text{perm}}\) the permissible loss ratio. Under the loss ratio method, the indicated factor is:
This calculation presumes the losses and premium are on a compatible basis, the permissible loss ratio reflects the applicable expense and profit provisions, and the comparison is intended to measure the rate-level change for the portfolio. It does not correct for a mismatched basis; the inputs must be made consistent first.
Worked Example: Catching an Inconsistent Premium Denominator
A hypothetical portfolio has losses of $960,000, historical earned premium of $1,500,000, an on-level factor of 1.05, and a permissible loss ratio of 64%. A reviewer uses the historical premium without adjustment. Find the correct indicated change and the result of the reviewer’s calculation.
Method. Use the loss ratio method with on-level premium. This is a deterministic calculation: no interest rate, mortality assumption, or probability distribution is needed.
First adjust the premium to the current rate level:
The actual loss ratio on that basis is:
Divide by the permissible loss ratio to obtain the indicated factor:
The indicated percentage change is \(f-1=0.9523810-1=-0.0476190\), or a 4.76% decrease, rounded to two decimal places.
Using historical premium instead gives \(\$960{,}000/\$1{,}500{,}000=0.64\). The reviewer then obtains \(0.64/0.64=1.00\), or a 0.00% change. That result is incorrect because the denominator does not reflect the stated on-level adjustment. Common mistake avoided: treating historical premium as though it were already at the current rate level.
Check. The correct factor is below 1, so it must indicate a decrease. The computed 4.76% decrease has the expected direction.
Trace Each Expense and Profit Provision Once
Double counting often arises when a candidate moves between a pure premium formulation and a permissible loss ratio formulation without tracking which provisions are already included. To see the distinction clearly, let \(L\) be expected loss and loss adjustment expense per exposure, \(F\) be fixed expense per exposure, \(v\) be variable expense as a proportion of premium, and \(q\) be the target profit and contingency provision as a proportion of premium. Under this convention, the required rate per exposure \(R\) satisfies:
The two premium-based provisions, \(vR\) and \(qR\), depend on the rate itself. Rearranging gives \(R=(L+F)/(1-v-q)\), provided \(1-v-q>0\). If profit or an expense is instead stated as a dollar amount per exposure, it belongs in the numerator, not in the premium-based denominator.
Worked Example: Avoiding a Second Fixed-Expense Load
A hypothetical product has expected loss and loss adjustment expense of $620 per exposure, fixed expense of $80 per exposure, variable expense of 25% of premium, and a profit and contingency provision of 5% of premium. Find the required rate per exposure and diagnose a proposed calculation that adds fixed expense after applying the permissible loss ratio.
Method. Solve the premium equation, with the fixed expense in dollars per exposure and the two percentage provisions applied to premium. No interest rate, mortality assumption, or probability distribution is needed.
The variable expense and profit provisions together are \(0.25+0.05=0.30\) of premium. Therefore, the portion of premium available for losses and fixed expense is:
The permissible loss ratio is 70% under the stated convention. Expected losses plus fixed expense are \(\$620+\$80=\$700\) per exposure, so:
Check the allocation of the $1,000 rate: losses are $620, fixed expense is $80, variable expense is \(0.25(\$1{,}000)=\$250\), and profit and contingency provision is \(0.05(\$1{,}000)=\$50\). The total is \(\$620+\$80+\$250+\$50=\$1{,}000\).
A common incorrect calculation divides losses alone by the permissible loss ratio, obtaining \(\$620/0.70=\$885.71\), and then adds the $80 fixed expense to get $965.71. This fails because the $80 has not been grossed up for the fact that only 70% of premium remains after variable expense and profit. Another incorrect approach obtains the correct $1,000 and then adds $80 again, yielding $1,080; that counts fixed expense twice. Common mistake avoided: once fixed expense is included in the numerator, do not add it again after solving for the rate.
Translate Factors and Percentage Changes Carefully
A factor is a multiplier; a percentage change is the amount by which that multiplier differs from 1.00. For example, a factor of 0.975 corresponds to a 2.5% decrease because \(0.975-1=-0.025\). It does not mean a 97.5% decrease. Likewise, a 6% increase corresponds to a factor of 1.06, not 0.06.
The base to which a percentage is applied also matters. If a rate change is stated as a percentage change from the current rate, the new rate equals the current rate multiplied by one plus that change. Do not add the factor itself to the rate, or treat a factor as a percentage increase.
Worked Example: Reading a Factor as a Percentage Change
A hypothetical overall indication produces a factor of 1.0875. The current annual rate is $640. A reviewer reports a 108.75% increase and calculates the new rate by multiplying $640 by 2.0875. Find the correct change and rate, then identify the reviewer’s error.
Method. Convert the indicated factor to a change by subtracting 1, then apply the change to the current rate. The calculation is deterministic; no interest rate, mortality assumption, or probability distribution is applicable.
The correct percentage change is:
The new annual rate is:
The reviewer has treated the factor itself, 1.0875, as a 108.75% increase. That interpretation produces a multiplier of \(1+1.0875=2.0875\), and therefore \(\$640(2.0875)=\$1{,}336.00\). This is not the indicated result. Common mistake avoided: a rate-level factor includes the original 1.00; the percentage change does not.
Check. An 8.75% increase on $640 is \(\$640(0.0875)=\$56.00\), and \(\$640+\$56=\$696\), agreeing with the factor calculation.
Require Support for Adjustments to the Indication
The indicated change and the selected change are not automatically identical. A selection may differ from an indication when there is a stated reason, such as relevant implementation constraints or credible evidence that a specified input should be modified. The adjustment should be connected to a defined quantity, use a clear calculation, and be documented. A label such as “conservatism” does not by itself establish an amount or justify applying it.
When a supported adjustment is made, state whether it is an adjustment in percentage points to the indicated change or a multiplicative adjustment to the rate level. These are different operations. For example, adding two percentage points to a 6% change gives 8%; increasing the indicated rate level by a further 2% gives \(1.06(1.02)=1.0812\), or an 8.12% total increase. Neither operation is justified merely because it can be calculated.
Worked Example: Separating an Indication from an Unsupported Adjustment
A hypothetical rate review produces a technically supported factor of 1.06. A proposed selection adds an unspecified “safety margin” of 2% but identifies no changed assumption, data limitation, or implementation requirement. The current annual rate is $500. What does the indication support, and what would the proposed adjustment produce under two possible interpretations?
Method. First calculate the indicated change and rate. Then calculate the proposed adjustment both as two percentage points added to the change and as a 2% multiplicative adjustment. No interest rate, mortality assumption, or probability distribution is involved.
The indicated change is \(1.06-1=0.06\), or 6%. The rate supported by the indication is:
If “add 2%” means add two percentage points to the indicated change, the total change is \(6\%+2\%=8\%\), giving \(\$500(1.08)=\$540.00\). If it means apply a further 2% to the indicated rate level, the factor is \(1.06(1.02)=1.0812\), giving \(\$500(1.0812)=\$540.60\). The two results differ because the instructions describe different calculations.
Neither proposed result is supported by the facts given: the “safety margin” has no specified basis. On the information available, the defensible conclusion is to report the 6% indicated change and $530.00 rate, and request a documented rationale before making an adjustment. Common mistake avoided: turning an unexplained judgment into an apparently precise adjustment without identifying its basis or meaning.
A Final-Selection Diagnostic
Before reporting a selected overall change, work through a short reconciliation. Each step targets a different failure mode; passing the arithmetic check alone is not enough.
Confirm that premium, losses, and the target provision refer to compatible periods and rate levels. Identify every required adjustment before calculating a ratio.
List which expenses and profit or contingency amounts are fixed per exposure and which are proportions of premium. Include each exactly once, in the appropriate part of the calculation.
Subtract 1 from a rate-level factor to obtain the percentage change. Check that the sign and magnitude are plausible for the stated factor.
State the indicated change, the selected change, and the reason for any difference. If an adjustment is proposed, specify its basis and whether it is additive or multiplicative.
A historical reasonableness check or a benchmark can prompt further review, but it does not make an unsupported change valid by itself. Likewise, a selection that differs from an indication is not automatically wrong; the issue is whether the difference can be explained, quantified, and supported without obscuring the underlying indication.
Check Your Understanding
For each hypothetical problem, diagnose the calculation using the stated basis and provisions.
- A portfolio has losses of $720,000, historical earned premium of $1,000,000, an on-level factor of 1.04, and a permissible loss ratio of 65%. A reviewer uses historical premium. Find the correct indicated change and the reviewer’s change. (Answer: Correct: 6.51% increase; reviewer: 10.77% increase.)
- Expected loss and loss adjustment expense is $540 per exposure, fixed expense is $60 per exposure, variable expense is 28% of premium, and the profit provision is 4% of premium. Find the required rate and explain why adding $60 after calculating the rate would be wrong. (Answer: $882.35 per exposure; the fixed expense is already included in the numerator before dividing by 0.68.)
- An indication yields a factor of 0.975 and the current rate is $800. State the indicated percentage change and the resulting rate. (Answer: 2.50% decrease; $780.00.)
- Losses are $1,080,000, historical premium is $1,500,000, the on-level factor is 1.10, and the permissible loss ratio is 60%. Calculate the correct indicated change and the change from incorrectly using historical premium. (Answer: Correct: 9.09% increase; incorrect: 20.00% increase.)
- A supported indication is a 4.5% increase on a current rate of $400. A reviewer proposes adding 1.5 percentage points with no stated evidence. Calculate the indicated rate and the proposed rate, then state which result is supported by the information given. (Answer: Indicated rate: $418.00; proposed rate: $424.00; only the $418.00 indicated rate is supported.)