Worked Product Pricing With Capital Charges

Pricing With a Capital Requirement

A product price must cover more than the expected claims. It must also cover expenses and provide an adequate return for the capital assigned to support the risk. A useful pricing calculation therefore keeps three quantities distinct: expected underwriting profit, the charge for using capital, and the return required on capital.

This tutorial uses a one-year pricing horizon. Unless a worked example says otherwise, premiums are received at the start of the year, claims and expenses are paid during the year, and there is no investment income, tax, or change in allocated capital. All products and data below are hypothetical. The calculations are aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus.

Definition: Let $P$ be the premium, $L$ the aggregate loss over the pricing period, $E$ the expense, and $C$ the capital allocated to the product. Expected underwriting profit before any capital charge is $P-E[L]-E$. Its RAROC, when $C>0$, is this expected profit divided by $C$. If the capital charge rate is $k$, the charge is $kC$, and economic profit after that charge is $P-E[L]-E-kC$.

A RAROC target $h$ is the required ratio of expected underwriting profit before the capital charge to allocated capital. When expenses are fixed and capital is treated as independent of the premium, the target condition is

\frac{P-E[L]-E}{C}=h \qquad\Longrightarrow\qquad P=E[L]+E+hC.

This is not automatically the same as pricing to cover a capital charge. Pricing at $E[L]+E+kC$ makes economic profit zero at charge rate $k$, and gives a pre-charge RAROC of $k$. A different target $h$ gives economic profit $(h-k)C$ at the target price. The two rates coincide only when $h=k$.

Why the Required-Premium Formula Holds

The formula follows from the definition of the target, not from a separate assumption that the insurer pays out a capital charge as a claim or expense. Begin with the target condition: expected profit before the charge must equal the target rate multiplied by allocated capital. Substitute the definition of underwriting profit and solve for $P$.

Set the target return.
Expected underwriting profit must be $hC$.

Write profit in terms of premium.
$P-E[L]-E=hC$.

Isolate the premium.
Adding expected losses and expenses to both sides gives $P=E[L]+E+hC$.

This derivation assumes that the specified $C$ does not change as $P$ changes and that the RAROC numerator is the profit defined above. If expenses vary as a proportion of premium, they must remain in the equation when solving; they cannot be treated as a fixed dollar amount.

Worked Example: A Fixed-Expense Product at Its RAROC Target

A hypothetical annual product has aggregate loss $L$, in dollars, equal to $500 with probability 0.25, $700 with probability 0.50, and $900 with probability 0.25. Fixed expenses are $80, allocated capital is $500, and the target RAROC is 16%. A proposed premium is $830. Find the required premium, the proposed premium’s RAROC, and economic profit using a 10% capital charge rate.

Method and loss assumption. The stated discrete distribution is the full one-year aggregate-loss model. Its probabilities sum to $0.25+0.50+0.25=1$, and expected loss is

E[L]=(0.25)(500)+(0.50)(700)+(0.25)(900) =125+350+225=\$700.

Required premium. With fixed expenses, use $P=E[L]+E+hC$. The target profit is $0.16(500)=\$80$, so

P_{\mathrm{req}}=700+80+80=\$860.

At this premium, expected underwriting profit is $860-700-80=\$80$, and RAROC is $80/500=0.16=16\%$, as required.

Proposed premium and capital charge. At $830, expected underwriting profit is $830-700-80=\$50$. Its RAROC is $50/500=10\%$. The capital charge is $0.10(500)=\$50$, so economic profit is $50-50=\$0$. The proposed price covers the 10% capital charge but does not meet the 16% RAROC target; it is $30 below the target price.

Check. At the required price, economic profit after the same charge is $80-50=\$30$, which also equals $(0.16-0.10)(500)=\$30$.

Repricing Under Alternative Loss Assumptions

A required premium depends directly on the expected loss assumption. A pricing decision should therefore state which loss distribution it uses and show how the price changes if that assumption changes. A stress distribution is an alternative assumption for analysis; it is not automatically a new estimate of the expected loss.

Worked Example: Comparing Base and Stress Loss Assumptions

A hypothetical product is priced in thousands of dollars. Under the base one-year aggregate-loss model, $L$ is $800 thousand with probability 0.25, $1,100 thousand with probability 0.50, and $1,400 thousand with probability 0.25. Under a stress model, the corresponding outcomes are $950 thousand, $1,250 thousand, and $1,550 thousand, with the same probabilities. Expenses are $150 thousand, allocated capital is $2,000 thousand, the offered premium is $1,550 thousand, and the target RAROC is 12%. The capital charge rate is 8%. Calculate the expected profit, RAROC, and economic profit at the offered price under each loss model, and find the target premium under each model.

Method and base model. Each model is a discrete annual aggregate-loss distribution with probabilities summing to 1. Under the base model,

E[L]_{\mathrm{base}}=(0.25)(800)+(0.50)(1{,}100)+(0.25)(1{,}400) =200+550+350=1{,}100

All monetary figures in this example are in thousands of dollars. At the offered premium, base expected underwriting profit is $1{,}550-1{,}100-150=300$. Thus base RAROC is $300/2{,}000=15\%$. The capital charge is $0.08(2{,}000)=160$, so base economic profit is $300-160=140$.

Stress model. Its expected loss is

E[L]_{\mathrm{stress}}=(0.25)(950)+(0.50)(1{,}250)+(0.25)(1{,}550) =237.5+625+387.5=1{,}250.

At the same offered premium, stress expected underwriting profit is $1{,}550-1{,}250-150=150$, giving RAROC $150/2{,}000=7.5\%$. The capital charge remains 160 under the stated assumption, so stress economic profit is $150-160=-10$.

Target prices. The target profit is $0.12(2{,}000)=240$. Therefore the base target premium is $1{,}100+150+240=1{,}490$, and the stress target premium is $1{,}250+150+240=1{,}640$. The $150 thousand increase in expected loss raises the target premium by exactly $150 thousand when expenses, capital, and target are held fixed.

Check. At either target price, profit before the capital charge is 240 and RAROC is $240/2{,}000=12\%$. The capital charge is 160 in both cases; economic profit at the target is $240-160=80$.

When Expenses Vary With Premium

Some acquisition or distribution expenses are modeled as a percentage of premium. Let $e$ be that expense rate and $E_0$ be fixed expenses. Then total expenses are $E_0+eP$, so the target condition becomes $P-E[L]-E_0-eP=hC$. Solving gives

P_{\mathrm{req}}=\frac{E[L]+E_0+hC}{1-e}, \qquad 0\le e<1.

The denominator matters because raising the premium also raises the variable expense. This formula assumes that the expense rate is constant over the prices being considered and that $C$ does not change with premium.

Worked Example: Solving With a Variable Expense Rate

A hypothetical product is priced in thousands of dollars. Its one-year aggregate loss $L$ is $260 thousand with probability 0.25, $360 thousand with probability 0.50, and $460 thousand with probability 0.25. Fixed expenses are $40 thousand, variable expenses are 5% of premium, allocated capital is $600 thousand, the target RAROC is 14%, and the capital charge rate is 9%. Find the required premium and the resulting economic profit.

Method and expected loss. The probabilities sum to 1. Expected loss under the specified discrete distribution is

$$ E[L]=(0.25)(260)+(0.50)(360)+(0.25)(460) =65+180+115=360. $$

The target profit is $0.14(600)=84$. Total expenses are $40+0.05P$, so the target equation is

$$ P-360-40-0.05P=84. $$

Combining premium terms gives $0.95P=484$. Thus $P=484/0.95=509.473684\ldots$ thousand dollars, or $509.47 thousand rounded to the nearest cent in thousands.

Check the result. Variable expenses are $0.05(509.473684\ldots)=25.473684\ldots$ thousand dollars, and total expenses are $40+25.473684\ldots=65.473684\ldots$. Expected underwriting profit is $509.473684\ldots-360-65.473684\ldots=84$ thousand dollars, so RAROC is $84/600=14\%$. The capital charge is $0.09(600)=54$ thousand dollars. Economic profit is $84-54=30$ thousand dollars, also equal to $(0.14-0.09)(600)=30$.

Method check. Solving the equation is essential here: the variable expense depends on the premium being determined. Ignoring that dependence would produce a price that fails the target condition.

Common Exam Trap: Mixing the Capital Charge and RAROC Target

A frequent error is to substitute the capital charge rate for the target RAROC without checking whether the two rates are intended to be equal. In the stress example, the target is 12%, while the capital charge rate is 8%. The incorrect calculation $1{,}250+150+0.08(2{,}000)=1{,}560$ thousand covers expected loss, expenses, and the 8% charge. But its pre-charge underwriting profit is $1{,}560-1{,}250-150=160$, giving RAROC $160/2{,}000=8\%$, not 12%.

The correction is to use the target return when calculating the price required to achieve a specified RAROC: $1{,}250+150+0.12(2{,}000)=1{,}640$ thousand. The 8% charge is then used separately to calculate economic profit. At that price, economic profit is $240-160=80$ thousand dollars.

Interpreting a Product-Pricing Result

A calculated premium is conditional on its assumptions. In particular, check that the loss distribution is for the same period as the premium, that expenses are classified consistently as fixed or variable, and that the capital amount corresponds to the product and portfolio context being priced. The earlier material on capital allocation explains how an allocated amount can be assigned to a business; this calculation uses that amount as given rather than selecting an allocation method.

Before presenting a price recommendation, report both the target test and the capital-charge result. A product can meet its RAROC target while producing positive economic profit if $h>k$; it can meet the target while producing negative economic profit if $h<k$. This distinction prevents a price that merely recovers a capital charge from being mistaken for a price that meets a higher return hurdle.

Key takeaway: With fixed expenses and capital independent of premium, the target price is expected loss plus expenses plus target RAROC multiplied by allocated capital. With a variable expense rate, solve the full target equation. Apply the capital charge separately when calculating economic profit.

Check Your Understanding

For each hypothetical one-year product, use the stated aggregate-loss distribution and assumptions. Monetary amounts are in thousands of dollars unless specified otherwise.

A product has losses of $300, $500, and $700 thousand with probabilities 0.25, 0.50, and 0.25. Fixed expenses are $60 thousand, capital is $250 thousand, and the target RAROC is 12%. Find the expected loss and required premium. (Answer: Expected loss = $500 thousand; required premium = $590 thousand.)
A product’s aggregate loss is $600 thousand with probability 0.70 and $1,600 thousand with probability 0.30. Fixed expenses are $100 thousand, capital is $800 thousand, and the offered premium is $1,100 thousand. Find the RAROC and economic profit after a 6% capital charge. (Answer: Expected loss = $900 thousand; RAROC = 12.5%; economic profit = $52 thousand.)
Expected loss is $720 thousand, fixed expenses are $80 thousand, variable expenses are 8% of premium, capital is $1,000 thousand, and the target RAROC is 13%. Find the required premium and economic profit after a 7% capital charge. (Answer: Required premium = $1,010.87 thousand; economic profit = $60 thousand.)
Under a base model, expected loss is $2,000 thousand; under a stress model, it is $2,400 thousand. Fixed expenses are $200 thousand, capital is $3,000 thousand, and the target is 11%. Find the required premium under each model and the stress RAROC at a premium of $2,550 thousand. (Answer: Base required premium = $2,530 thousand; stress required premium = $2,930 thousand; stress RAROC at $2,550 thousand = $-1.6667\%$.)
A product has expected loss of $420 thousand and fixed expenses of $60 thousand. Its allocated capital is either $500 thousand or $650 thousand, and its target RAROC is 12%. Find the required premium under each capital allocation and the change in required premium. (Answer: $540 thousand at $500 thousand of capital; $558 thousand at $650 thousand of capital; increase = $18 thousand.)

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