RAROC and Economic Profit Rankings

Why Two Measures Can Rank the Same Businesses Differently

A business with a high return per dollar of capital may create less total value than a business with a lower return on a much larger capital base. RAROC measures a rate; economic profit measures a dollar amount after charging for capital. The distinction matters whenever management compares business lines or investment opportunities using both measures.

The comparison is meaningful only if the measures use a consistent basis. In this tutorial, earnings are annual, risk-adjusted earnings are measured before the cost-of-capital charge, and economic capital is the amount assigned to the business for the same period. All amounts in the worked examples are hypothetical.

Definition: Let $R$ be a business's annual risk-adjusted earnings before the cost of economic capital, $K$ its allocated economic capital, and $k$ the annual cost-of-capital hurdle rate. For $K>0$, its RAROC is $R/K$, and its economic profit is $R-kK$. Risk-adjusted earnings should be defined consistently across the businesses being compared; for example, expected losses and other agreed adjustments should be treated consistently.

The hurdle rate $k$ represents the required annual return on economic capital. It is not the RAROC itself: RAROC is the business's measured rate, while $k$ is the benchmark used to determine whether that rate is sufficient. Economic profit is positive when RAROC exceeds the hurdle, zero when it equals the hurdle, and negative when it falls short, provided $K>0$.

\text{RAROC}=\frac{R}{K}, \qquad \text{Economic profit}=R-kK

Why the Rankings Can Diverge

Write RAROC as $r=R/K$. Since $R=Kr$, economic profit can also be expressed as capital multiplied by the return in excess of the hurdle:

\text{Economic profit} =R-kK =K\left(\frac{R}{K}-k\right) =K(r-k)

This identity explains the ranking difference. RAROC compares the excess return rate $r-k$, whereas economic profit multiplies that rate by the amount of capital $K$. A smaller business can have the larger excess return per dollar, while a larger business can generate more total excess dollars.

Ranking result: If two businesses have the same positive economic capital, the same hurdle rate, and consistently defined earnings, their RAROC and economic-profit rankings agree. With different capital amounts, their rankings can differ. A higher RAROC does not by itself imply higher economic profit.

For businesses A and B using a common hurdle rate, their economic-profit difference is:

$$ EP_A-EP_B =K_A(r_A-k)-K_B(r_B-k) $$

Thus, even when $r_A>r_B$, business A has lower economic profit if $K_A(r_A-k)<K_B(r_B-k)$. The hurdle rate matters because it determines how much of each business's return remains after the capital charge. The amount of capital matters because that excess rate is applied to the full capital base.

Worked Comparisons

For each comparison, first calculate RAROC, then calculate the capital charge and economic profit. Do not infer the dollar-value ranking from the rate ranking.

Worked Example: The Higher RAROC Does Not Produce Higher Economic Profit

Two hypothetical business units report annual risk-adjusted earnings and allocated economic capital in millions of dollars. Unit A has $R_A=\$24$ million and $K_A=\$240$ million. Unit B has $R_B=\$7.2$ million and $K_B=\$48$ million. The common annual cost-of-capital hurdle is $k=8\%$. Calculate and rank both measures.

Method. Calculate each return as earnings divided by capital, then subtract the capital charge from earnings to obtain economic profit. This avoids treating a percentage return as if it were a dollar contribution.

Unit A RAROC.

\text{RAROC}_A =\frac{\$24\text{ million}}{\$240\text{ million}} =0.10 =10\%

Unit B RAROC.

\text{RAROC}_B =\frac{\$7.2\text{ million}}{\$48\text{ million}} =0.15 =15\%

By RAROC, B ranks above A. Now compute the dollar capital charges. A's charge is $0.08\times \$240$ million, or $\$19.2$ million. B's charge is $0.08\times \$48$ million, or $\$3.84$ million.

EP_A=\$24-\$19.2=\$4.8\text{ million}, \qquad EP_B=\$7.2-\$3.84=\$3.36\text{ million}

By economic profit, A ranks above B. The two rankings differ: B earns a greater excess return rate, $15\%-8\%=7\%$, while A applies a smaller excess rate, $10\%-8\%=2\%$, to a much larger capital base. The check is $0.02\times \$240=\$4.8$ million and $0.07\times \$48=\$3.36$ million, matching the direct calculations.

Common mistake avoided: Ranking B first on both measures because it has the higher RAROC. The RAROC ranking is correct, but it does not determine the economic-profit ranking.

Worked Example: Equal Capital Makes the Rankings Agree

Two hypothetical units each have $\$75$ million of allocated economic capital. Unit C has annual risk-adjusted earnings of $\$9$ million, while Unit D has $\$7.5$ million. The annual hurdle rate is $8\%$. Find both measures and explain why they rank the units consistently.

Method. Calculate the two rates and then apply the same capital charge to each unit. This checks the equal-capital condition rather than assuming that the measures always agree.

For C, RAROC is $\$9/\$75=0.12=12\%$. For D, RAROC is $\$7.5/\$75=0.10=10\%$. Each unit's capital charge is $0.08\times \$75=\$6$ million.

EP_C=\$9-\$6=\$3\text{ million}, \qquad EP_D=\$7.5-\$6=\$1.5\text{ million}

C ranks above D by both RAROC and economic profit. Equivalently, the excess returns are 4% for C and 2% for D; multiplying each by the same $\$75$ million capital gives $\$3$ million and $\$1.5$ million. The direct and excess-return calculations agree.

Common mistake avoided: Claiming that the ranking agreement proves RAROC and economic profit are interchangeable. Here the capital amounts are equal; with unequal amounts, the dollar ranking need not follow the rate ranking.

Worked Example: Finding the Capital Level That Reverses the Ranking

Hypothetical unit E has capital of $\$60$ million and RAROC of $16\%$. Unit F has RAROC of $13\%$, but its capital amount is not yet fixed. The common annual hurdle rate is $9\%$. Determine which unit has higher economic profit if F has $\$150$ million of capital, then find the F capital amount at which their economic profits are equal.

Method. Convert each RAROC into an excess return over the hurdle. Economic profit is that excess rate times capital, so the equality condition can be solved directly for F's capital. This avoids comparing rates without accounting for scale.

E's excess return is $16\%-9\%=7\%$. At $\$60$ million of capital, its economic profit is $0.07\times \$60=\$4.2$ million. F's excess return is $13\%-9\%=4\%$. At $\$150$ million of capital, its economic profit is $0.04\times \$150=\$6$ million. Therefore, E has the higher RAROC, but F has the higher economic profit at the stated capital level.

To find the equality capital $K_F^*$, set the two economic profits equal:

0.07(\$60\text{ million})=0.04K_F^*

The left side is $\$4.2$ million, so $K_F^*=\$4.2/0.04=\$105$ million. At $\$105$ million, both units have $\$4.2$ million of economic profit. Below $\$105$ million, E's economic profit is higher; above $\$105$ million, F's is higher. The stated $\$150$ million exceeds the threshold, consistent with F's $\$6$ million profit.

Common mistake avoided: Assuming E must also lead in economic profit because $16\%>13\%$. The threshold calculation shows exactly how F's larger capital base can reverse the dollar ranking.

Conditions and Interpretation

The ranking result depends on the inputs being comparable. In practice, the analyst should check that:

The earnings basis is consistent. Both figures should represent the same type of risk-adjusted earnings for the same period, with expected losses and other adjustments treated consistently.
Capital is measured consistently. The capital amounts should use a comparable economic-capital definition and allocation basis. A change in allocation can alter both the ratio and the dollar-profit comparison.
The hurdle is appropriate and clearly stated. A common hurdle supports the direct ranking comparison above. If different units have different required rates, calculate each economic profit using its own stated hurdle; do not use the common-hurdle ranking condition.
Capital is positive. The usual RAROC interpretation assumes $K>0$. If capital is zero or negative, the ratio is undefined or can be misleading, and the standard ranking interpretation does not apply.

A frequent exam trap is to confuse passing a hurdle with ranking highest. If both units have RAROC above a common hurdle, both have positive economic profit, but one can still have lower economic profit than the other. Likewise, the unit with the highest economic profit need not have the highest RAROC. The measures answer different questions: RAROC is a rate comparison, and economic profit is a value comparison after the capital charge.

Key takeaway: With a common hurdle, economic profit equals capital times the excess of RAROC over that hurdle. Equal capital makes the two rankings agree; unequal capital can make them diverge. A sound comparison reports both the rate and dollar result when both questions matter. This treatment is aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus.

Check Your Understanding

For each hypothetical comparison, calculate the requested measures before checking the final answer.

A business has annual risk-adjusted earnings of $\$8$ million, allocated capital of $\$50$ million, and an annual hurdle of $9\%$. Calculate its RAROC, capital charge, and economic profit. (Answer: RAROC $=16\%$; capital charge $=\$4.5$ million; economic profit $=\$3.5$ million.)
Unit G has risk-adjusted earnings of $\$18$ million and capital of $\$120$ million. Unit H has earnings of $\$8$ million and capital of $\$40$ million. At a common $10\%$ hurdle, rank both units by RAROC and economic profit. (Answer: G RAROC $=15\%$, economic profit $=\$6$ million; H RAROC $=20\%$, economic profit $=\$4$ million; H ranks higher by RAROC and G by economic profit.)
Two units each have $\$90$ million of capital. Unit J earns $\$10.8$ million and Unit L earns $\$9$ million in annual risk-adjusted earnings. With a common $8\%$ hurdle, calculate and compare their RAROCs and economic profits. (Answer: J RAROC $=12\%$, economic profit $=\$3.6$ million; L RAROC $=10\%$, economic profit $=\$1.8$ million; J ranks higher on both.)
Unit M has $\$80$ million of capital and a $14\%$ RAROC. Unit N has a $12\%$ RAROC and $\$120$ million of capital. At a common $9\%$ hurdle, which unit has higher RAROC and which has higher economic profit? (Answer: M has higher RAROC; M economic profit $=0.05\times \$80=\$4$ million, N economic profit $=0.03\times \$120=\$3.6$ million, so M also has higher economic profit.)
Unit P has $\$100$ million of capital and a $15\%$ RAROC. Unit Q has a $12\%$ RAROC and an unknown capital amount. If the common hurdle is $9\%$, determine the Q capital at which economic profits are equal and state which unit leads above that amount. (Answer: P economic profit $=0.06\times \$100=\$6$ million; equality requires $0.03K_Q=\$6$ million, so $K_Q=\$200$ million; above $\$200$ million, Q has higher economic profit.)

What You'll Learn