Deriving a Risk-Adjusted Price Requirement

Why a RAROC Hurdle Creates a Price Requirement

A price can cover expected claims and expenses yet still fail to earn an adequate return on the capital assigned to support the business. A risk-adjusted price requirement combines these components: the price must fund expected costs and produce enough earnings to meet the required return on allocated capital. This tutorial derives that requirement for a one-period underwriting decision.

The calculation takes the expected loss and allocated capital as inputs. It does not determine how capital should be allocated, estimate a loss distribution, or choose the hurdle rate. Those are separate decisions. The focus here is the algebra that turns those inputs into a minimum premium.

Definition: For a one-period product or portfolio, let $P$ be premium, $L=E[X]$ be expected aggregate loss, $F$ be fixed expenses, $e$ be the variable expense fraction of premium, and $K$ be allocated capital. Let $h$ be the required RAROC hurdle. In the model used here, underwriting earnings are $P-L-F-eP$, and RAROC is underwriting earnings divided by $K$, provided $K>0$.

All quantities must use a consistent time period and currency basis. The model assumes that the stated expected loss and expenses correspond to the period for which premium and capital are measured. Investment income, taxes, discounting, and other income or costs are excluded unless explicitly added to the earnings definition. No mortality assumption or loss distribution is needed when aggregate expected loss $L$ is supplied directly. The examples below use a one-year period and assume no investment income.

Deriving the Minimum Premium

A product meets the hurdle when its RAROC is at least $h$. In the model above, this means underwriting earnings must be at least $hK$. The required earnings are a return on capital; they are not a repayment of the capital principal.

\operatorname{RAROC}=\frac{P-L-F-eP}{K}

Write the hurdle condition.
For $K>0$, meeting the hurdle means $\frac{P-L-F-eP}{K}\ge h$.

Convert the return target to earnings.
Multiplying by positive $K$ preserves the inequality, so $P-L-F-eP\ge hK$.

Collect the premium terms.
Rearranging gives $P(1-e)\ge L+F+hK$.

Solve for the minimum premium.
If $0\le e<1$, then $1-e>0$, so division preserves the inequality and $P\ge\frac{L+F+hK}{1-e}$.

P_{\min}=\frac{L+F+hK}{1-e}, \qquad 0\le e<1

The derivation also explains why the condition $e<1$ matters: only then does each additional dollar of premium leave a positive amount, $1-e$, after variable expenses. If variable expenses equal or exceed 100% of premium, this model cannot produce a finite premium satisfying a positive earnings target when costs are otherwise positive. If there are no premium-based expenses, set $e=0$, giving $P_{\min}=L+F+hK$.

Key interpretation: With fixed expenses only, the premium covers expected loss, fixed expenses, and earnings of $hK$. With variable expenses, the required total must be grossed up by $1/(1-e)$, because only the fraction $1-e$ of each premium dollar remains after those expenses.

Worked Applications

Worked Example: Fixed Expenses and a Capital Return Target

A hypothetical one-year commercial cover has expected aggregate losses of $720,000, fixed expenses of $90,000, and allocated capital of $500,000. The required RAROC hurdle is 12%. There are no variable expenses. Find the minimum premium and verify the result. Investment income is excluded; the expected aggregate loss is supplied, so no loss distribution is assumed.

Method. With $e=0$, use $P_{\min}=L+F+hK$. First calculate the required earnings on capital:

$$ hK=0.12(500{,}000)=60{,}000 $$

Then add expected losses, fixed expenses, and required earnings:

P_{\min}=720{,}000+90{,}000+60{,}000=870{,}000

At this premium, underwriting earnings are $870{,}000-720{,}000-90{,}000=60{,}000$. Therefore, RAROC is $60{,}000/500{,}000=0.12$, or 12.00%, exactly the hurdle. The calculation avoids the common mistake of adding the full $500,000 of capital to the premium: the target is a 12% return on that capital, or $60,000 of earnings.

Worked Example: Premium-Based Expenses Require a Gross-Up

A hypothetical one-year liability program has expected aggregate losses of $1,260,000, fixed expenses of $120,000, and allocated capital of $900,000. Variable expenses are 8% of premium, and the RAROC hurdle is 11%. Determine the minimum premium and verify it. Investment income is excluded, and the expected loss is supplied without a distributional assumption.

Method. First calculate the capital return requirement:

$$ hK=0.11(900{,}000)=99{,}000 $$

The premium must leave enough after the 8% variable expense to pay $1,260,000 of expected losses, $120,000 of fixed expenses, and $99,000 of earnings. Thus:

P_{\min}(1-0.08)=1{,}260{,}000+120{,}000+99{,}000=1{,}479{,}000

Since $1-0.08=0.92$,

P_{\min}=\frac{1{,}479{,}000}{0.92}=1{,}607{,}608.6957

Rounded to the nearest cent, the minimum premium is $1,607,608.70. The variable expense is $0.08(1{,}607{,}608.70)=$128{,}608.70$, to the nearest cent. Using the unrounded premium, earnings are $1{,}607{,}608.6957-1{,}260{,}000-120{,}000-128{,}608.6957=$99{,}000$. The resulting RAROC is $99{,}000/900{,}000=11.00\%$.

Common mistake and correction. Adding the costs and capital return but omitting the gross-up gives a supposed premium of $1,479,000. At that price, variable expenses are $0.08(1{,}479{,}000)=$118{,}320$, leaving earnings of $1{,}479{,}000-1{,}260{,}000-120{,}000-118{,}320=-$19{,}320$, not the required $99,000. The correction is to divide by $1-e=0.92$.

Worked Example: Check a Proposed Quote Against the Hurdle

A hypothetical one-year specialty product is quoted at $2,000,000. Expected aggregate losses are $1,500,000, fixed expenses are $75,000, and variable expenses are 10% of premium. The product has $450,000 of allocated capital, and the hurdle is 13%. Does the quote meet the hurdle? If not, find the minimum premium. Investment income is excluded; expected losses are given directly.

Method. First calculate earnings at the quoted premium. Variable expenses are $0.10(2{,}000{,}000)=$200{,}000$. Earnings are therefore:

$$ 2{,}000{,}000-1{,}500{,}000-75{,}000-200{,}000=225{,}000 $$

The quote's RAROC is $225{,}000/450{,}000=0.50$, or 50.00%, so it exceeds the 13% hurdle. To calculate the minimum, required earnings are $0.13(450{,}000)=$58{,}500$. The price requirement is:

P_{\min}=\frac{1{,}500{,}000+75{,}000+58{,}500}{1-0.10} =\frac{1{,}633{,}500}{0.90} =1{,}815{,}000

At that premium, variable expenses are $0.10(1{,}815{,}000)=$181{,}500$, and earnings are $1{,}815{,}000-1{,}500{,}000-75{,}000-181{,}500=$58{,}500$. Dividing by $450,000 gives 13.00%, confirming the minimum. The proposed quote is $185,000 above the minimum and meets the hurdle.

Using the Requirement Correctly

A risk-adjusted price requirement is a threshold conditional on the inputs. In particular, $K$ is the capital amount assigned to the business for the pricing decision, and $h$ is the required return for the relevant period. The formula does not calculate either one. If the allocated capital changes, the required earnings change by $h$ times that change in capital, and the required premium changes accordingly.

Keep the expense basis explicit. An expense stated as a fixed dollar amount belongs in $F$; an expense stated as a proportion of premium belongs in $eP$. If a problem supplies only a total expense amount at a stated premium, do not automatically treat it as fixed or variable. Use the problem's stated expense basis, or state the assumption needed to proceed.

The formula is a pricing condition under the stated earnings model, not a guarantee that the price is suitable under every risk objective. It uses expected losses and allocated capital as given. It does not replace assessment of the underlying risk, selection of an appropriate capital measure, or review of whether the hurdle and the one-year earnings definition are consistent.

Common exam trap: The required capital contribution to earnings is $hK$, not $K$, and it is not $h$ dollars per policy unless the capital input is itself stated per policy. Also, when expenses are a fraction of premium, the denominator is $1-e$, not $e$. Verify any proposed price by recalculating earnings and dividing by allocated capital.

Check Your Understanding

For each problem, use the stated one-year underwriting model. Investment income and taxes are excluded; expected losses are supplied directly, so no loss distribution is required.

A hypothetical product has expected losses of $310,000, fixed expenses of $40,000, allocated capital of $200,000, and a 9% hurdle. There are no variable expenses. Find its minimum premium. (Answer: $368,000.00)
A hypothetical program has expected losses of $640,000, fixed expenses of $50,000, allocated capital of $400,000, a 10% hurdle, and variable expenses equal to 5% of premium. Find the minimum premium to the nearest cent. (Answer: $768,421.05)
A hypothetical policy has expected loss of $480 per policy, fixed expenses of $30 per policy, allocated capital of $250 per policy, and a 14% hurdle. Variable expenses are 12% of premium. Find the minimum premium per policy to the nearest cent. (Answer: $619.32)
A hypothetical book has a proposed premium of $1,200,000, expected losses of $900,000, fixed expenses of $60,000, variable expenses of 5% of premium, and allocated capital of $1,500,000. Calculate its RAROC and determine the minimum premium for a 13% hurdle, to the nearest cent. (Answer: RAROC = 12.00%; minimum premium = $1,215,789.47)
A hypothetical product has a premium of $2,800,000, expected losses of $2,100,000, fixed expenses of $140,000, and variable expenses of 7% of premium. What is the largest allocated capital consistent with a 13% RAROC hurdle at this premium? (Answer: $2,800,000.00)

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