Selecting Portfolios Under Capital Constraints

Portfolio Selection Begins With the Objective

A business line can have a strong return per unit of capital and still be a poor choice for a portfolio with a fixed capital limit. The decision depends on the total profit and capital of each feasible portfolio, not only on the standalone ranking of its components. In particular, when opportunities are indivisible, the highest standalone RAROC need not lead to the combination with the greatest total expected profit.

Before comparing candidates, specify the decision horizon, the profit measure, the capital measure, and the capital limit. Profit might be expected underwriting profit or another consistently defined risk-adjusted result. Capital must be measured on the same basis for every candidate and portfolio. The examples below use annual amounts in millions of dollars and assume that expected profit is already net of the expected losses and expenses relevant to the decision.

Definition: For a portfolio $S$, let $P(S)$ be its expected profit over the stated horizon and $C(S)$ its allocated capital, measured consistently with that profit. If $C(S)>0$, its RAROC is $P(S)/C(S)$. Under a fixed capital limit $K$, a portfolio is feasible when $C(S)\leq K$. If the objective is to maximize expected profit, select a feasible $S$ with the greatest $P(S)$.

A portfolio may consist of business lines that are indivisible at the decision scale, such as a proposed product launch or a whole reinsurance program. Expected profits of components add when the profit measure is the sum of their results. Capital need not add: diversification or other features of the capital model may cause $C(S)$ to differ from the sum of standalone capital amounts. The portfolio capital must therefore be the figure appropriate to the portfolio, not an automatic sum of its standalone figures.

Why a Standalone RAROC Ranking Can Mislead

RAROC is a useful measure of profit per unit of capital, but a ranking by that ratio does not by itself solve a selection problem with indivisible opportunities. The capital limit may prevent the selected high-ratio item from being combined with other attractive items, while a set of lower-ratio opportunities may use the available capacity more effectively.

The distinction can be seen by comparing indivisible opportunities with a divisible, linear allocation. If two opportunities have constant returns per unit of capital $r_i$ and $r_j$, and $r_i>r_j$, shifting a small amount $\epsilon$ of capital from $j$ to $i$ changes profit by $\epsilon(r_i-r_j)>0$. That exchange supports ranking by return when capital can be divided freely and there are no other constraints. With indivisible opportunities, the shift may not be possible: the decision is to take or leave a whole opportunity. The ratio ranking is then a guide, not a guarantee of the best feasible combination.

Worked Example: A High RAROC Opportunity Blocks a Better Combination

A hypothetical insurer can pursue any subset of three indivisible annual business opportunities. Their expected profits and standalone allocated capital are: A, profit $18 million and capital $60 million; B, profit $14 million and capital $50 million; C, profit $14 million and capital $50 million. Assume capital is additive for this decision, and the limit is $100 million. Select the feasible portfolio with the highest expected profit.

Method. Calculate each standalone RAROC, then compare the total profit of feasible combinations against the capital limit. Because the opportunities are indivisible, checking only the first item in a ratio ranking is insufficient.

The standalone ratios are:

\operatorname{RAROC}_A=\frac{18}{60}=30.00\%,\qquad \operatorname{RAROC}_B=\frac{14}{50}=28.00\%,\qquad \operatorname{RAROC}_C=\frac{14}{50}=28.00\%.

A ranks first. The feasible portfolios are the empty portfolio; A, B, or C alone; and B+C. Their relevant totals are:

\begin{aligned} C(A)&=60,\quad &P(A)&=18,\\ C(B)&=50,\quad &P(B)&=14,\\ C(C)&=50,\quad &P(C)&=14,\\ C(B+C)&=50+50=100,\quad &P(B+C)&=14+14=28. \end{aligned}

A+B and A+C each require $60+50=110$ million of capital, so neither is feasible. The largest feasible expected profit is $28 million from B+C. The RAROC-first choice of A alone earns $18 million, so it is not optimal.

Check. B+C uses exactly the $100 million limit, and its $28 million profit exceeds the profit of each feasible single opportunity. The common mistake is to treat the first-ranked standalone RAROC as the answer: that gives A and $18 million, while overlooking the feasible B+C portfolio.

Evaluate Capital at the Portfolio Level

The previous example assumed additive capital so that each combination could be evaluated by summing the stated figures. When portfolio capital reflects diversification, standalone capital amounts cannot be used to determine feasibility by addition. The capital model or allocation method must supply the capital for each candidate portfolio on a consistent basis.

Worked Example: Diversification Changes Which Combinations Fit

A hypothetical insurer is considering three indivisible business proposals, X, Y, and Z. Their annual expected profits are $18 million, $16 million, and $14 million. Their standalone allocated capital amounts are $70 million, $60 million, and $50 million, respectively. A portfolio capital model gives $C(X+Y)=\$115$ million, $C(X+Z)=\$105$ million, $C(Y+Z)=\$90$ million, and $C(X+Y+Z)=\$130$ million. The capital limit is $100 million. Find the feasible portfolio with the greatest expected profit.

Method. Use standalone capital only for a single proposal; for combinations, use the stated portfolio capital. Profits add across proposals. First, the standalone RAROCs are:

\operatorname{RAROC}_X=\frac{18}{70}=25.71\%,\qquad \operatorname{RAROC}_Y=\frac{16}{60}=26.67\%,\qquad \operatorname{RAROC}_Z=\frac{14}{50}=28.00\%.

Thus Z ranks first, followed by Y and X. Z+Y has portfolio capital of $90 million, not the $110 million obtained by adding their standalone capital amounts. It is feasible, with profit:

P(Y+Z)=16+14=30\text{ million}.

For completeness, X+Y requires $115 million and X+Z requires $105 million, so both exceed the limit. All three proposals require $130 million. Each single proposal is feasible; the largest single-business profit is $18 million from X. Therefore, the best feasible portfolio is Y+Z, with $30 million of expected profit.

Check. The selected portfolio has capital $90\leq100$, and its profit of $30 million exceeds every other feasible combination’s profit. Adding standalone capital would incorrectly label Y+Z infeasible and could lead to selecting only Z, with $14 million profit. The common mistake is to apply standalone capital figures to a combined portfolio even though portfolio capital has been recalculated for diversification.

Be Clear About What “Value” Means

Maximizing expected profit and maximizing economic profit are not necessarily the same objective. If the organization uses a hurdle rate $h$ for capital, a common annual economic-profit measure is:

\operatorname{EP}(S)=P(S)-hC(S).

This measure deducts the required return on allocated capital from expected profit. When economic profit is the objective, a feasible portfolio with lower expected profit can be preferred if it uses materially less capital. Conversely, if the stated objective is simply to maximize expected profit subject to a hard capital limit, the hurdle should not be silently substituted for that objective. State which criterion governs the decision.

Worked Example: A Hurdle Rate Can Change the Preferred Portfolio

A hypothetical insurer considers indivisible opportunities A, B, and C. Their annual expected profits and allocated capital are A: $20 million and $50 million; B: $12 million and $40 million; C: $3 million and $20 million. Capital is additive, the hard limit is $70 million, and the annual capital hurdle rate is 20%. Compare the portfolio maximizing expected profit with the one maximizing economic profit.

Method. Enumerate feasible combinations, calculate expected profit, and then subtract the 20% capital charge to calculate economic profit. The standalone RAROCs are $20/50=40\%$, $12/40=30\%$, and $3/20=15\%$, respectively.

A+B requires $50+40=90$ million and is infeasible. A+C requires $50+20=70$ million and is feasible, with profit $20+3=23$ million. B+C requires $40+20=60$ million and is feasible, with profit $12+3=15$ million. The single opportunities are also feasible. Hence, among the nonempty feasible choices, A+C has the greatest expected profit, $23 million.

Now calculate economic profit for the relevant alternatives:

\begin{aligned} \operatorname{EP}(A)&=20-(0.20)(50)=10,\\ \operatorname{EP}(A+C)&=23-(0.20)(70)=9,\\ \operatorname{EP}(B)&=12-(0.20)(40)=4,\\ \operatorname{EP}(B+C)&=15-(0.20)(60)=3,\\ \operatorname{EP}(C)&=3-(0.20)(20)=-1. \end{aligned}

A alone maximizes economic profit at $10 million, while A+C maximizes expected profit at $23 million. The difference arises because C has a 15% RAROC, below the 20% hurdle: it adds $3 million of expected profit but also incurs a $4 million capital charge. The common mistake is to assume every positive-profit opportunity should be added under every objective. Here it increases expected profit but decreases economic profit.

Check. A+C exactly meets the capital limit. The calculations use annual profit and annual capital charges consistently, and each feasible alternative has been compared under both stated objectives.

A Practical Comparison Process

For a real portfolio decision, the most reliable approach is to make the constraint and the objective explicit before interpreting ratios. A compact process is:

Define the decision.
State the horizon, the business units or proposals that can be selected, and whether choices are indivisible or can be scaled.

Align the measures.
Use expected profit and capital figures on consistent bases. Identify whether portfolio capital is additive or must be recalculated for each combination.

Apply feasibility.
For each relevant portfolio $S$, check $C(S)\leq K$. A high RAROC does not make a portfolio feasible if its capital exceeds the limit.

Compare the stated objective.
Compare $P(S)$ when maximizing expected profit, or the consistently defined economic-profit measure when that is the objective.

Use RAROC as context.
Review the ratios to understand returns per unit of capital, but do not treat a standalone ranking as a substitute for comparing feasible portfolios.

The number of possible combinations can become large. The central discipline remains the same: evaluate candidates using the portfolio-level capital measure and the objective actually specified. If only a few proposals are under consideration, listing feasible combinations is often enough to expose a ratio-ranking trap. For more complex choices, an optimization method may be needed, but it must use the same definitions of profit, capital, and feasibility.

Common exam trap: Ranking by standalone RAROC and selecting in that order can fail when opportunities are indivisible or portfolio capital is nonadditive. For an expected-profit objective under a hard capital limit, compare the total expected profit of feasible portfolios. Do not confuse a capital charge or hurdle rate with the hard limit, and do not add standalone capital amounts when the capital model supplies portfolio-level values.

Check Your Understanding

For each problem, state the feasible choice that maximizes the requested objective. Unless stated otherwise, amounts are annual millions of dollars and proposals are indivisible.

A capital limit is $80 million. Proposal J has capital $50 million and profit $15 million; K has capital $40 million and profit $11 million; L has capital $30 million and profit $8.4 million. Assume additive capital. Which feasible subset maximizes expected profit? (Answer: J+L, with capital $80 million and profit $23.4 million.)
A limit of $120 million applies to three indivisible opportunities. A has capital $70 million and profit $21 million; B has capital $60 million and profit $16 million; C has capital $60 million and profit $16.2 million. Assume additive capital. Which portfolio maximizes expected profit, and why is taking the highest standalone RAROC first insufficient? (Answer: B+C, with capital $120 million and profit $32.2 million; A ranks first at 30%, but A cannot be combined with B or C.)
Standalone capital and profit are X: $65 million and $19 million; Y: $55 million and $16 million; Z: $45 million and $12 million. Portfolio capital is $C(X+Y)=\$105$ million, $C(X+Z)=\$95$ million, and $C(Y+Z)=\$85$ million. The limit is $100 million, and profits add. Which feasible portfolio maximizes expected profit? (Answer: X+Z, with capital $95 million and profit $31 million.)
Two additive opportunities have capital and expected profit M: $30 million and $9 million; N: $50 million and $11 million. The limit is $80 million and the objective is maximum expected profit. What portfolio is selected, and what is its economic profit at a 15% annual hurdle rate? (Answer: M+N; expected profit is $20 million, capital is $80 million, and economic profit is $20-(0.15)(80)=\$8$ million.)
Two additive opportunities have capital and expected profit R: $40 million and $14 million; S: $30 million and $6 million. The capital limit is $70 million. Compare the maximum-expected-profit choice with the maximum-economic-profit choice at a 20% annual hurdle rate. (Answer: R+S maximizes expected profit at $20 million and uses $70 million; R alone maximizes economic profit, $14-(0.20)(40)=\$6\) million, versus $20-(0.20)(70)=\$6\) million for R+S—a tie, so both are economic-profit maximizers.)

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