Ranking Alternatives With Risk-Adjusted Returns

Why Risk-Adjusted Returns Matter

A choice that increases expected earnings does not necessarily improve the use of capital. It may also require substantially more risk capital. Conversely, a reinsurance purchase can reduce expected earnings while lowering capital by a greater proportion. A return-on-risk-adjusted-capital measure puts the two quantities on a common basis, so alternatives can be compared by how much expected earnings they generate per unit of risk capital.

The ratio is useful only when its numerator, denominator, and measurement period are defined consistently. In particular, compare alternatives using the same earnings basis, risk-capital methodology, and time horizon. A higher ratio is informative about capital efficiency, but it is not by itself a complete decision rule: the absolute earnings, the amount of capital required, and other constraints also matter.

Definition: For an alternative $j$, define its return on risk-adjusted capital (RORAC) as

\operatorname{RORAC}_j = \frac{E_j}{K_j},

where $E_j$ is expected net earnings over the stated period after the costs and benefits included in the comparison, and $K_j$ is the corresponding risk capital for that alternative. If $E_j$ is in dollars and $K_j$ is in dollars, RORAC is a rate. State the earnings definition and capital measure; terminology and numerator conventions can vary between organizations.

For the examples below, earnings are expected net earnings over one year, after the specifically stated operating and risk-transfer costs. Risk capital is the one-year amount supplied for the whole portfolio under the same stipulated risk model across alternatives. No mortality assumption is relevant. Amounts are measured at year-end with a stipulated effective annual interest rate $i=0\%$, so no discounting or investment-income adjustment is made unless explicitly included in expected earnings. The capital figures are supplied model outputs; no loss distribution or parameters are required to calculate the ratios. In practice, a decision analysis should document how those capital outputs were obtained.

Comparing Whole-Portfolio Alternatives

Calculate the ratio for each complete alternative, not just for the transaction in isolation. For reinsurance, earnings should reflect the treaty premium and expected recoveries, while capital should reflect the post-treaty portfolio. For a securitization, include its expected recoveries and costs in earnings, and use the capital for the portfolio with the securitization. A capital raise changes the capital base and may also change expected earnings, for example through investment income if that is included consistently.

Worked Example: Retention Versus a Reinsurance Treaty

A hypothetical insurer estimates one-year expected net earnings of $20.00 million if it retains a specified catastrophe exposure. Its risk capital on that basis is $100.00 million. A proposed treaty has a $5.00 million premium and $3.00 million of expected recoveries over the year; after the treaty, the portfolio’s risk capital is estimated at $70.00 million. Compare the two RORACs. Assume all other earnings are unchanged, the figures are expected year-end amounts, $i=0\%$, and there is no mortality exposure. The capital values are stipulated outputs of the same one-year risk model; no distribution is needed for this ratio calculation.

Method. First calculate the treaty’s net effect on expected earnings. The insurer pays $5.00 million and expects to receive $3.00 million, so earnings decrease by $2.00 million:

\Delta E = 3.00 - 5.00 = -2.00 \text{ million}.

Treaty earnings are therefore $20.00-2.00=$$18.00 million. The retained alternative has RORAC $20.00/100.00=0.20$, or 20.00%. The treaty alternative has RORAC $18.00/70.00=0.257142\ldots$, or 25.71% to two decimal places.

The treaty reduces expected earnings, but it reduces risk capital proportionally more, so the portfolio’s RORAC rises from 20.00% to 25.71%. This comparison does not say the treaty creates more expected earnings: it does not. It says the stated earnings are higher relative to the remaining risk capital. The common mistake is to subtract the premium but omit expected recoveries, or to calculate the treaty ratio using the original $100.00 million capital. The correct calculation includes both cash-flow effects and the post-treaty capital of $70.00 million.

Incremental Returns and the Portfolio Test

For a proposed change, let the current portfolio have expected earnings $E_0$ and risk capital $K_0$. Let the change produce incremental earnings $\Delta E$ and change portfolio capital by $\Delta K$. The new portfolio has earnings $E_0+\Delta E$ and capital $K_0+\Delta K$. When $\Delta K>0$, the incremental ratio $\Delta E/\Delta K$ can be compared with the current portfolio’s RORAC to determine whether the change raises the total ratio.

The comparison follows directly from the definition. Assume $K_0>0$ and $K_0+\Delta K>0$. The new portfolio has a higher RORAC precisely when

\frac{E_0+\Delta E}{K_0+\Delta K} > \frac{E_0}{K_0} \quad\Longleftrightarrow\quad K_0(E_0+\Delta E) > E_0(K_0+\Delta K) \quad\Longleftrightarrow\quad K_0\Delta E > E_0\Delta K.

If $\Delta K>0$, dividing by $K_0\Delta K$ gives

\frac{\Delta E}{\Delta K} > \frac{E_0}{K_0}.

Thus, for a positive addition to capital, the change raises portfolio RORAC only if the change’s earnings per unit of marginal portfolio capital exceed the current RORAC. The marginal capital is $K_{\text{new}}-K_0$, not automatically the standalone capital assigned to the new business. If $\Delta K\leq 0$, avoid relying on the incremental ratio as a general ranking device; use the cross-multiplied comparison or calculate both whole-portfolio ratios directly.

Fix the basis.
Specify the horizon, expected-earnings definition, and risk-capital measure for every alternative.

Calculate portfolio effects.
Include all stated earnings changes and use capital for the whole portfolio after the proposed change.

Calculate and interpret.
Compute each portfolio’s RORAC. For a positive capital increment, compare incremental earnings per marginal capital with the current RORAC as a diagnostic.

Worked Example: Adding a Diversified Portfolio

A hypothetical insurer has one-year expected net earnings of $24.00 million and risk capital of $160.00 million. A proposed portfolio is expected to add $4.80 million of net earnings. Its standalone risk capital is $30.00 million, but the combined portfolio model gives total risk capital of $184.00 million after diversification. Compare the current and proposed portfolio returns. Assume $i=0\%$, no mortality exposure, and that earnings and capital are on the same one-year basis. The supplied capital values come from the same portfolio risk model; the loss distribution is not needed to calculate these ratios.

Method. Current RORAC is $24.00/160.00=0.15$, or 15.00%. Proposed portfolio earnings are $24.00+4.80=$$28.80 million. Its total risk capital is given as $184.00 million, so the marginal portfolio capital is $184.00-160.00=$$24.00 million.

The proposed portfolio RORAC is $28.80/184.00=0.1565217\ldots$, or 15.65% to two decimal places. As a check using the incremental test, the change’s earnings per unit of marginal portfolio capital are $4.80/24.00=20.00\%$, which exceeds the current 15.00%. The whole-portfolio calculation confirms that the RORAC increases to 15.65%.

The common mistake is to divide $4.80 million by the new portfolio’s $30.00 million standalone capital, producing 16.00%, and call that the incremental portfolio return. That denominator ignores the fact that portfolio capital rises by only $24.00 million. The 16.00% figure may describe a standalone comparison under a different basis, but it does not give the change’s return per marginal portfolio capital in this problem.

Comparing Securitization and Capital Raising

A securitization and a capital raise affect the numerator and denominator differently. A securitization can reduce retained risk capital while adding expected recoveries and transaction costs to earnings. A capital raise increases the funds available to absorb losses and therefore may increase the capital denominator; it can also change expected earnings if the additional funds earn investment income. To make a fair comparison, include only the earnings effects defined in the problem, and do not compare one alternative’s earnings before costs with another’s after costs.

Worked Example: Catastrophe Bond Versus Additional Capital

A hypothetical insurer’s current one-year expected net earnings are $18.00 million, with risk capital of $120.00 million. A catastrophe bond is expected to provide $4.00 million of recoveries and incur $3.00 million of coupon and issuance costs; post-transaction risk capital is $90.00 million. Alternatively, the insurer could raise $20.00 million of additional capital, with $1.20 million of expected net investment earnings attributable to it; total risk capital would then be $140.00 million. Compare both alternatives with the current portfolio. Assume $i=0\%$ for discounting year-end amounts, no mortality exposure, and that the supplied capital values are outputs from a common one-year risk model. No loss distribution is assumed in this return calculation.

Method. Current RORAC is $18.00/120.00=15.00\%$. For the catastrophe bond, the net earnings effect is $4.00-3.00=$$1.00 million. Bond-alternative earnings are $18.00+1.00=$$19.00 million, so its RORAC is $19.00/90.00=0.211111\ldots$, or 21.11%.

For the capital raise, earnings are $18.00+1.20=$$19.20 million. The stated total risk capital is $140.00 million, giving $19.20/140.00=0.137142\ldots$, or 13.71%. The catastrophe bond has the highest RORAC among the stated options; the capital raise has a lower RORAC than the current portfolio.

The common mistake is to count the $20.00 million raised as earnings, or to omit the capital it adds to the denominator. The capital raised is not itself profit. Under the stated assumptions, only the $1.20 million of expected net investment earnings is added to earnings, while total capital becomes $140.00 million. These calculations rank the alternatives by RORAC, not by every consideration relevant to a capital decision.

Interpretation and Common Exam Traps

RORAC is a comparison measure, not a guarantee that a transaction is desirable. A high ratio on a small amount of capital may produce little total earnings, while a lower ratio can accompany a larger absolute contribution. A complete recommendation may need to consider the amount of earnings, risk tolerance, contractual features, and constraints specified in the decision problem. Keep that broader context separate from the ratio calculation.

Mixing horizons: Do not divide annual earnings by a capital amount defined on a different time basis without a stated conversion.
Mixing numerator definitions: Compare earnings after equivalent expenses and risk-transfer costs; identify whether investment income or financing costs are included.
Using standalone capital as marginal capital: When assessing a portfolio addition, calculate $\Delta K$ from the combined portfolio amounts if those are supplied.
Ranking only by earnings: A larger earnings amount does not imply a larger RORAC if it also requires much more capital.
Overinterpreting the ratio: A higher RORAC indicates more expected earnings per unit of the specified capital measure, not necessarily higher total profit or lower risk in every respect.

Key takeaway: Define expected earnings and risk capital consistently, calculate each alternative’s whole-portfolio RORAC, and use marginal portfolio capital—not an assumed standalone amount—when testing whether a positive capital addition improves the existing return.

Check Your Understanding

For each problem, use the stated one-year earnings and risk-capital basis. Unless specified otherwise, assume $i=0\%$, no mortality exposure, and that capital amounts are stipulated outputs from a common risk model rather than quantities to be derived from a loss distribution.

A hypothetical insurer has expected net earnings of $10.00 million and risk capital of $80.00 million. A treaty changes earnings to $9.60 million and capital to $60.00 million. Calculate both RORACs and state which is higher. (Answer: Current 12.50%; treaty 16.00%, so the treaty alternative is higher.)
A portfolio earns $15.00 million on $100.00 million of capital. A securitization is expected to recover $2.00 million and cost $2.80 million, and post-transaction capital is $80.00 million. Calculate the new earnings and RORAC, then compare with the current RORAC. (Answer: Earnings $14.20 million; new RORAC 17.75% versus current 15.00%, so the securitization alternative is higher.)
A portfolio has earnings of $30.00 million and capital of $200.00 million. A proposed addition contributes $6.00 million of earnings; standalone capital is $50.00 million, while combined portfolio capital is $238.00 million. Calculate marginal capital, the incremental return per marginal capital, and the new whole-portfolio RORAC. (Answer: Marginal capital $38.00 million; incremental return 15.7895%; new RORAC $36/238=15.13\%$, above the current 15.00%.)
A current portfolio earns $18.00 million on $120.00 million of capital. Alternative A has earnings of $22.00 million and capital of $110.00 million; Alternative B has earnings of $21.00 million and capital of $100.00 million; Alternative C has earnings of $23.00 million and capital of $130.00 million. Calculate each ratio and rank the alternatives from highest to lowest. (Answer: A 20.00%, B 21.00%, C 17.69%; ranking is B, A, C. The current portfolio is 15.00%.)
A portfolio earns $12.00 million on $75.00 million of capital. A proposed change adds $2.80 million of earnings and $20.00 million of marginal portfolio capital. Determine whether it improves RORAC by comparing the incremental return with the current return, then verify using the new whole-portfolio ratio. (Answer: Current 16.00%; incremental return 14.00%; new ratio $14.8/95=15.58\%$. It does not improve RORAC.)

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