Converting a Target Return on Surplus

Why Convert a Return on Surplus?

A target return on surplus is expressed relative to the insurer’s capital, while a rate indication needs a profit provision that can be expressed relative to premium. These percentages have different denominators. Converting between them requires the premium-to-surplus ratio: the amount of premium supported by each dollar of surplus.

For example, a target return of 12% does not mean the profit provision should automatically be 12% of premium. If premium is greater than surplus, the dollar profit needed to earn the target return on surplus is a smaller percentage of premium. The conversion is direct once the target return and premium-to-surplus ratio refer to the same business, period, and accounting basis.

Definition: For a specified underwriting period, let $P$ be premium, $S$ be the surplus assigned to support that business, and $U$ be underwriting profit. In this tutorial, underwriting profit means premium less losses and expenses for the period, before investment income and taxes. The underwriting return on surplus is $U/S$, and the premium-to-surplus ratio is $P/S$. The required profit as a percentage of premium is $U/P$.

The premium measure and surplus measure must be chosen consistently with the return target. For example, if the target is based on earned underwriting results and average surplus, use earned premium and the corresponding average surplus in the conversion. Mixing written premium with an earned-period return can produce a profit percentage that does not deliver the intended return.

Deriving the Conversion

Assume $P>0$ and $S>0$, and define the target underwriting return on surplus as $r=U/S$. The premium-to-surplus ratio is $q=P/S$. The profit provision as a fraction of premium, $p$, is $U/P$. All three quantities describe the same underwriting business and period.

p=\frac{U}{P} =\frac{U/S}{P/S} =\frac{r}{q}

The middle equality justifies the formula: dividing $U/S$ by $P/S$ cancels the shared surplus denominator, leaving $U/P$. Equivalently, first find the dollar profit $U=rS$, then divide by premium $P=qS$. Thus the required profit percentage of premium is the target return on surplus divided by the premium-to-surplus ratio.

Key result: If the target underwriting return on surplus is $r$ and the premium-to-surplus ratio is $q=P/S$, then the required underwriting profit provision as a percentage of premium is $p=r/q$. In dollars, the required profit is $U=rS$, which must also equal $pP$.

Use the ratio in the direction specified by its definition. Here $q$ means premium divided by surplus, not surplus divided by premium. A ratio $q=1.50$ means that premium is 1.50 times surplus. If a problem instead gives $S/P$, first take its reciprocal to obtain $P/S$, or use $p=r(S/P)$.

Worked Conversions

Worked Example: Converting a Target Return with a Given Ratio

A hypothetical insurer targets an underwriting return on surplus of 12% for a line with projected earned premium of $240 million and average surplus of $160 million. Find the required underwriting profit in dollars and as a percentage of premium.

Method and assumptions. Use one underwriting period, no investment income or taxes, and the stated earned-premium and average-surplus bases. There is no interest or loss-distribution calculation in this conversion. The premium-to-surplus ratio is:

q=\frac{P}{S} =\frac{\$240\text{ million}}{\$160\text{ million}} =1.50.

The dollar profit needed to earn 12% on $160 million of surplus is:

U=rS=0.12(\$160\text{ million})=\$19.2\text{ million}.

Divide that dollar amount by earned premium to obtain the profit provision as a percentage of premium:

p=\frac{U}{P} =\frac{\$19.2\text{ million}}{\$240\text{ million}} =0.08=8\%.

Check. An 8% provision on $240 million is $19.2 million, and $19.2 million divided by $160 million is 12%. The result is consistent on both bases. The common mistake to avoid is multiplying 12% by 1.50, which gives 18%; the derivation shows that the return must be divided by the premium-to-surplus ratio.

Worked Example: Finding the Ratio and the Profit Requirement

A hypothetical specialty insurer expects $280 million of earned premium and assigns $160 million of average surplus to the business. Its target underwriting return on surplus is 14%. Calculate (1) the premium-to-surplus ratio, (2) the dollar profit requirement, and (3) the required profit percentage of premium.

Method and assumptions. Treat the premium and surplus as aligned measures for the same underwriting period. Ignore investment income and taxes, so the target applies to underwriting profit alone.

First calculate the ratio:

q=\frac{\$280\text{ million}}{\$160\text{ million}}=1.75.

Next calculate the required underwriting profit from the target return and surplus:

U=0.14(\$160\text{ million})=\$22.4\text{ million}.

Finally, express this amount relative to premium:

p=\frac{\$22.4\text{ million}}{\$280\text{ million}}=0.08=8\%.

The same result follows directly from the conversion formula: $p=0.14/1.75=0.08$. Check. An 8% profit provision on $280 million is $22.4 million; that profit is 14% of $160 million. The common mistake to avoid is treating 14% as the premium provision without accounting for the fact that premium is 1.75 times surplus.

Worked Example: Keeping Earned and Written Bases Separate

A hypothetical insurer’s underwriting return target is defined on earned results and average surplus. For the period, projected earned premium is $132 million, projected written premium is $165 million, and average surplus is $110 million. The target underwriting return on surplus is 9%. Find the required profit in dollars and as a percentage of earned premium. Then quantify the error from using written premium in the conversion and applying the resulting percentage to earned premium.

Method and assumptions. Since the target is on earned results, use earned premium with the stated average surplus. Use a single underwriting period with no investment income or taxes.

The appropriate premium-to-surplus ratio is:

q_{\text{earned}}=\frac{\$132\text{ million}}{\$110\text{ million}}=1.20.

The target dollar profit is:

U=0.09(\$110\text{ million})=\$9.9\text{ million}.

The required percentage of earned premium is:

p_{\text{earned}}=\frac{\$9.9\text{ million}}{\$132\text{ million}} =0.075=7.5\%.

Now calculate the result of the incorrect basis choice. Using written premium gives $q_{\text{written}}=\$165/\$110=1.50$, so the resulting percentage would be $0.09/1.50=6\%$. Applying that 6% to earned premium gives $0.06(\$132\text{ million})=\$7.92\text{ million}$. Its return on the stated surplus is $\$7.92/\$110=7.2\%$, not the 9% target.

Check. The correct $9.9 million profit is 9% of average surplus and 7.5% of earned premium. The incorrect calculation misses the target because it uses written premium to form the ratio but earned premium to apply the resulting provision. The common mistake to avoid is combining measures from different bases without checking that they still reproduce the intended dollar profit.

Using the Result in a Rate Provision

The conversion gives a profit requirement, not a new estimate of losses, expenses, or surplus. Once the percentage $p$ is determined, it can be expressed as the profit provision relative to the specified premium base. If a rate calculation uses a different premium base from the one used to derive $p$, confirm that the dollar profit remains consistent with the target return.

A useful sequence is to calculate the target dollar profit first and then convert it to a premium percentage. This order makes the capital requirement visible and provides a simple check: multiplying the selected percentage by the chosen premium should return the required dollar profit. It also makes it easier to detect a mismatch in bases before the percentage is used.

Identify the return target and its basis.
Confirm that the target is an underwriting return on surplus, and identify the period and surplus measure it uses.

Choose the matching premium measure.
Use earned or written premium consistently with the target’s intended underwriting basis.

Calculate both forms of the requirement.
Find $U=rS$, then find $p=U/P$. Alternatively, calculate $q=P/S$ and use $p=r/q$.

Check the result in both directions.
Verify that $pP=U$ and $U/S=r$, using the same premium and surplus measures throughout.

Common Exam Traps

The most frequent error is reversing the ratio. If $q=P/S$, then $P$ is the premium denominator and $S$ is the surplus denominator. Because $P$ is $q$ times $S$, the profit dollars needed for a given return on surplus are spread across $q$ times as much premium. Therefore $p$ is smaller than $r$ whenever $q>1$, as the formula $p=r/q$ shows.

A second error is to calculate a correct percentage on the wrong premium base. A profit provision is meaningful only when its denominator is stated. Do not silently substitute written for earned premium, or vice versa, if the target and surplus measure were selected for a different basis. The check $pP=rS$ exposes this problem immediately.

Finally, the formula applies to an underwriting return target as defined here. It does not by itself convert a total return target that includes investment income, or a return measured after taxes, into an underwriting profit provision. Those targets require additional components and definitions; they should not be folded into $r$ without specifying what the return includes.

Key takeaway: For an underwriting return target $r$ on surplus $S$, the required dollar profit is $rS$. If premium is $q=P/S$ times surplus, the required profit provision as a percentage of premium is $r/q$. Keep the premium and surplus bases aligned, and verify that the resulting percentage reproduces the target dollar profit.

Check Your Understanding

For each problem, use the stated underwriting basis and verify the dollar profit as well as the percentage when enough information is given.

A hypothetical portfolio has premium of $150 million and surplus of $100 million. The target underwriting return on surplus is 10%. Find the required profit provision as a percentage of premium. (Answer: 6.67%)
A hypothetical insurer has $96 million of earned premium, $80 million of average surplus, and a 11% target underwriting return on surplus. Find the dollar profit requirement and its percentage of earned premium. (Answer: $8.8 million; 9.17%)
A business unit has a premium-to-surplus ratio of 1.25 and a target underwriting return on surplus of 8.5%. Find the profit provision as a percentage of premium. If premium is $125 million, also find the dollar profit. (Answer: 6.8%; $8.5 million)
A hypothetical line targets a 13% underwriting return on average surplus of $140 million. Its matching earned premium is $210 million. Calculate the target dollar profit, the premium-to-surplus ratio, and the profit percentage of earned premium. (Answer: $18.2 million; 1.50; 8.67%)
A portfolio’s target is a 9% underwriting return on average surplus of $120 million, based on earned premium of $180 million. Written premium is $240 million. Find the correct earned-premium profit provision. Then find the percentage that would result from dividing the target profit by written premium, and explain why it is not the required percentage on earned premium. (Answer: $10.8 million profit; 6% of earned premium; 4.5% of written premium, which is on the wrong basis for the stated earned-return target)

What You'll Learn