Profit and Contingency Provisions

Why These Provisions Need a Clear Basis

A prospective rate must provide for expected losses and expenses, but a rate indication may also include a target underwriting profit and a contingency margin. Expected investment income can offset part of the amount that must be collected from policyholders. The calculation is reliable only when each item’s basis is explicit: a percentage of premium is not interchangeable with a fixed dollar amount per exposure.

This tutorial develops a prospective premium equation for these provisions. Loss and expense estimates are treated as already developed to the prospective period and matched to the exposure and premium bases. The emphasis here is how profit, contingency, and investment income enter the equation, not how to develop the underlying losses or expenses. The methods are aligned with the SOA Exam ASTAM (Advanced Short-Term Actuarial Mathematics) syllabus.

Definition: Let $R$ be the indicated premium per exposure, $L$ the prospective expected loss and loss adjustment expense per exposure, and $F$ fixed expense per exposure. Let $e$ be variable expenses as a proportion of premium and $p$ the target profit provision as a proportion of premium. A contingency provision may be a dollar amount per exposure, denoted $C$, or a proportion of premium, denoted $c$. Investment income may likewise be stated as a dollar credit per exposure, denoted $I$, or as a proportion of premium, denoted $q$.

Choose the Sign Convention Before Calculating

Use the following convention throughout: losses, fixed expenses, and a contingency amount stated in dollars are required funding amounts. Variable expenses, target profit, and a contingency percentage of premium are amounts allocated from premium. Expected investment income is a credit that reduces the premium needed to fund the other items.

Under this convention, a positive profit provision means the rate is designed to produce that profit allowance; it is not a credit subtracted from the funding requirement. By contrast, positive investment income reduces the required premium. A contingency provision also increases the indicated premium, whether it is a dollar amount or a percentage of premium.

Item	Basis	Role in the balance
Expected losses and loss adjustment expenses	Dollars per exposure	Required funding
Fixed expenses	Dollars per exposure	Required funding
Variable expenses	Percentage of premium	Premium allocation
Target profit	Percentage of premium, unless stated otherwise	Premium allocation
Contingency provision	Dollars per exposure or percentage of premium	Required funding or premium allocation
Investment income	Dollars per exposure or percentage of premium	Credit reducing required premium

Deriving the Prospective Premium Equation

Suppose the contingency provision and investment-income credit are both stated as dollars per exposure. The premium must fund losses, fixed expenses, and the contingency amount, while also covering variable expenses and profit. Investment income offsets that need. The balance is

$$ R = L + F + C + eR + pR - I. $$

Collecting the terms proportional to $R$ gives the indicated premium:

R = \frac{L+F+C-I}{1-e-p}.

This form applies when $e$ and $p$ are proportions of premium, and $C$ and $I$ are dollars per exposure. The denominator must be positive for a finite positive indication under ordinary positive-cost assumptions. If a contingency provision is instead $c$ of premium, and investment income is $q$ of premium, the balance becomes

R = L+F+(e+p+c)R-qR, \qquad R = \frac{L+F}{1-e-p-c+q}.

A positive $q$ increases the denominator and therefore reduces the indicated premium. This does not mean investment income is an additional expense: it is a credit in the balance equation. If both dollar and percentage forms occur, include each in its stated form:

R = \frac{L+F+C-I}{1-e-p-c+q}.

These equations assume that all percentage provisions are defined on the same premium basis $R$. If a problem defines a provision on another base, such as losses or written premium on a different accounting basis, translate it to the stated basis before using the equation.

Derivation: Start with the balance of premium and funding needs. For dollar provisions, the funding needs are $L+F+C$, variable expenses and profit take $eR+pR$, and investment income supplies $I$. Thus $R+I=L+F+C+eR+pR$. Moving the premium-proportional amounts to the left gives $R(1-e-p)=L+F+C-I$, which yields the formula above when $1-e-p \ne 0$. For percentage provisions, replace $C$ by $cR$ and $I$ by $qR$, producing $R(1-e-p-c+q)=L+F$.

Worked Indications

Worked Example: Percentage Provisions and an Investment Credit

A hypothetical policy class has prospective losses and loss adjustment expenses of $420 per exposure and fixed expenses of $30 per exposure. Variable expenses are 25% of premium, the target profit provision is 5% of premium, and a contingency provision is 2% of premium. Expected investment income is 1.5% of premium. Find the indicated premium per exposure.

Method. All percentage amounts are stated on the premium basis. Thus $L=420$, $F=30$, $e=0.25$, $p=0.05$, $c=0.02$, and $q=0.015$. The investment income is a positive credit, so it enters the denominator with a plus sign:

R = \frac{420+30}{1-0.25-0.05-0.02+0.015} =\frac{450}{0.695} =647.4820\ldots

Rounded to the nearest cent, the indicated premium is $647.48 per exposure. To check, using the rounded premium gives variable expenses $0.25(647.48)=161.87$, target profit $0.05(647.48)=32.37$, contingency $0.02(647.48)=12.95$, and investment income $0.015(647.48)=9.71$, each rounded to cents. The balance is $647.48-161.87-32.37-12.95+9.71=450.00$, which equals losses plus fixed expenses.

Common mistake avoided: subtracting the profit percentage as though it were a credit would overstate the premium available for losses and expenses. Profit is an allocation from premium and is therefore subtracted from the denominator.

Worked Example: Dollar Contingency and Dollar Investment Income

For a hypothetical book, expected losses and loss adjustment expenses are $500 per exposure, fixed expenses are $40 per exposure, and the contingency provision is $12 per exposure. Expected investment income is $8 per exposure. Variable expenses are 20% of premium and the target profit provision is 4% of premium. Calculate the indicated premium.

Method. The dollar amounts enter the numerator. First combine them with their signs: $500+40+12-8=544$. The premium-proportional provisions total $0.20+0.04=0.24$, leaving $1-0.24=0.76$ of premium for the dollar requirement.

R = \frac{500+40+12-8}{1-0.20-0.04} =\frac{544}{0.76} =715.7894\ldots

Rounded to the nearest cent, the indicated premium is $715.79 per exposure. Check the balance: variable expenses are $0.20(715.79)=143.16$ and profit is $0.04(715.79)=28.63$, rounded to cents. Then $500+40+12+143.16+28.63-8=715.79$. The balance holds.

Common mistake avoided: adding the $8 investment-income credit to the numerator would make the required premium too large. The credit offsets, rather than adds to, the amount that must be funded.

Worked Example: Converting an Investment Yield to a Premium Credit

A hypothetical insurer estimates losses and loss adjustment expenses of $360 per exposure and fixed expenses of $24 per exposure. Variable expenses are 22% of premium, target profit is 4% of premium, and contingency is 3% of premium. The assumed annual effective investment yield is 5%. The projected average invested funds supporting each exposure are 40% of its premium. Estimate investment income as a proportion of premium and then find the indicated premium.

Method. The yield is applied to the stated average invested funds, not directly to the full premium. The investment-income proportion is $q=0.05(0.40)=0.02$, or 2% of premium. The premium-proportional provisions and credit give a denominator of $1-0.22-0.04-0.03+0.02=0.73$.

R = \frac{360+24}{0.73} =\frac{384}{0.73} =526.0273\ldots

Rounded to the nearest cent, the indicated premium is $526.03 per exposure. Check using the rounded premium: variable expenses are $0.22(526.03)=115.73$, profit is $0.04(526.03)=21.04$, contingency is $0.03(526.03)=15.78$, and investment income is $0.02(526.03)=10.52$. The balance is $526.03-115.73-21.04-15.78+10.52=384.00$, matching $360+24$.

Common mistake avoided: using 5% of premium as the income credit would ignore that only 40% of premium is expected to be invested on average. The stated assumptions imply a 2% credit, not 5%.

Exam Traps and Practical Checks

The most common errors are basis errors and sign errors. Before calculating, label every input as either dollars per exposure or a percentage of premium. Then write the balance equation in words or symbols before rearranging it. In particular, do not put a dollar contingency directly into a percentage denominator, or a percentage investment-income credit directly into a dollar numerator.

A second trap is treating all “provisions” as if they had the same role. A target profit provision is part of the amount allocated from premium, so it reduces the denominator. A contingency margin increases the amount to be funded, whether expressed in dollars or as a premium percentage. Investment income is a credit, so it reduces the funding requirement: it is subtracted when stated in dollars and added to the denominator when stated as a percentage of premium.

Finally, verify the result by substituting it into the original balance equation, not just by repeating the rearranged formula. This catches misplaced signs and mismatched bases. The equation also requires a meaningful positive residual share of premium: for percentage provisions, check that $1-e-p-c+q>0$. If that condition fails, the stated assumptions do not produce a finite positive premium by this model.

Check Your Understanding

For each problem, identify the basis of every provision before calculating. Round dollar premiums to the nearest cent.

A hypothetical class has losses and loss adjustment expenses of $300 per exposure and fixed expenses of $20. Variable expenses are 20% of premium, profit is 3%, contingency is 2%, and investment income is 1% of premium. Find the indicated premium. (Answer: $421.05 per exposure)
Losses are $640 per exposure and fixed expenses are $35. Contingency is $15 per exposure, investment income is $5 per exposure, variable expenses are 18% of premium, and profit is 4%. Find the indicated premium. (Answer: $878.21 per exposure)
Losses are $270 per exposure and fixed expenses are $18. Variable expenses are 15% of premium, profit is 6%, and contingency is 2.5%. The annual effective investment yield is 4%, and average invested funds are 50% of premium. Find the indicated premium. (Answer: $366.88 per exposure)
For a hypothetical policy class, losses are $450 per exposure and fixed expenses are $25. Variable expenses are 24% of premium, profit is 5%, contingency is 1%, and investment income is 3% of premium. Calculate the indication and state why the income percentage has a positive sign in the denominator. (Answer: $650.68 per exposure; investment income offsets required funding)
Losses are $510 per exposure and fixed expenses are $32. Variable expenses are 23% of premium, profit is 4%, and contingency is 1% of premium. Expected investment income is $9 per exposure. Find the indicated premium, taking care to place the dollar credit on the correct side of the equation. (Answer: $740.28 per exposure)

What You'll Learn