Sensitivity of Business Rankings to Capital Allocation

Why Capital Allocation Can Change a Ranking

A business line’s RAROC depends on both its profit and the capital assigned to it. When total portfolio risk and business profits are held fixed, changing the allocation method can still change each line’s capital charge and RAROC. It can therefore change which line appears to perform better, or whether a line meets a RAROC hurdle.

That sensitivity is important for management decisions. A ranking change caused by reallocating a fixed amount of portfolio capital is not, by itself, evidence that a line’s profit or risk has changed. A sound comparison keeps the profit definition, measurement period, total capital basis, and other assumptions consistent, then varies the allocation assumption separately.

Definition: Let $P_i$ be the profit measure for business line $i$, and let $K_i$ be the capital allocated to that line. Its RAROC is $P_i/K_i$, provided $K_i>0$. An allocation vector $(K_1,\ldots,K_n)$ is reconciled when $\sum_i K_i=K$, where $K$ is the portfolio’s total economic capital on the chosen basis. An alternative allocation changes the $K_i$ values while preserving the same portfolio capital $K$; it does not, on its own, change the portfolio’s underlying risk or profit.

The definition uses a common profit numerator and a consistent capital basis. If a comparison changes the profit measure, the risk horizon, or the total capital estimate at the same time as it changes the allocation, the resulting RAROC difference cannot be attributed to allocation alone.

Comparing Rankings and Deriving the Switch Point

For two lines with positive profits and positive allocations, the RAROC ranking can be tested directly. Line A ranks above line B when $P_A/K_A>P_B/K_B$. Cross-multiplication gives a useful condition that makes the role of the allocation explicit.

\frac{P_A}{K_A}>\frac{P_B}{K_B} \quad\Longleftrightarrow\quad P_AK_B>P_BK_A \quad\Longleftrightarrow\quad \frac{K_A}{K_B}<\frac{P_A}{P_B}.

The last equivalence assumes $P_A>0$, $P_B>0$, $K_A>0$, and $K_B>0$. It shows that the ranking depends on the ratio of allocated capital as well as the ratio of profits. When profits are zero or negative, do not use the final ratio condition without checking signs; compare the RAROCs directly, and consider whether a ratio-based measure is informative for the decision.

Justification: Starting from $P_A/K_A>P_B/K_B$, multiply both sides by the positive quantity $K_AK_B$. This preserves the inequality and gives $P_AK_B>P_BK_A$. Dividing by the positive quantity $P_BK_B$ gives $P_A/P_B>K_A/K_B$, which is the stated condition. Equality occurs at the ranking switch point.

If total capital for these two lines is fixed at $K$, write $K_A=x$ and $K_B=K-x$, where $0<x<K$. At equal RAROC, $P_A/x=P_B/(K-x)$. Solving gives the allocation to A at the switch point:

x=\frac{KP_A}{P_A+P_B}.

This threshold is a sensitivity result, not an allocation recommendation. It identifies how much capital must be assigned to A, under the stated fixed-total assumption, to make the two RAROCs equal.

Worked Example: A Ranking Reversal Under Two Reconciled Allocations

A hypothetical insurer has three business lines. Annual profit is $12 million for Line A, $10 million for Line B, and $6 million for Line C. Portfolio economic capital is $100 million under either allocation method. Method 1 assigns $45 million, $35 million, and $20 million to A, B, and C. Method 2 assigns $30 million, $50 million, and $20 million. Compare capital charges and RAROC rankings, then assess a 25% RAROC hurdle. Assume the profit figures and total portfolio capital do not change between methods.

Method. First check that each allocation sums to the same portfolio capital. Then divide each line’s profit by its own allocated capital. A line meets the hurdle if its RAROC is at least 25%.

Reconciliation. Method 1 assigns $45+35+20=100$ million dollars. Method 2 assigns $30+50+20=100$ million dollars. Both allocations therefore reconcile to the same $100 million total.

Method 1 RAROCs. Line A has $12/45=0.266667$, or 26.67%. Line B has $10/35=0.285714$, or 28.57%. Line C has $6/20=0.30$, or 30.00%. The ranking is C, then B, then A. All three lines meet the 25% hurdle.

Method 2 RAROCs. Line A has $12/30=0.40$, or 40.00%. Line B has $10/50=0.20$, or 20.00%. Line C remains at $6/20=0.30$, or 30.00%. The ranking is A, then C, then B. Lines A and C meet the hurdle; B does not.

Capital charges at the hurdle. At 25%, the capital charge is 25% of allocated capital. Under Method 1, the charges are $0.25(45)=\$11.25$ million for A, $0.25(35)=\$8.75$ million for B, and $0.25(20)=\$5.00$ million for C. Under Method 2, they are $0.25(30)=\$7.50$ million, $0.25(50)=\$12.50$ million, and $0.25(20)=\$5.00$ million, respectively. For example, Line B’s profit exceeds its Method 1 charge by $10-8.75=\$1.25$ million, but falls short of its Method 2 charge by $10-12.50=-\$2.50$ million.

Check. Profits still sum to $12+10+6=\$28$ million and allocated capital still sums to $100 million under both methods. The ranking and hurdle results change because capital assigned to A and B changes, not because either line’s profit or the portfolio’s total capital changed.

Common mistake avoided. Dividing each line’s profit by the full $100 million portfolio capital would give every line the wrong denominator. RAROC here uses that line’s allocated capital.

Allocation Sensitivity at a Break-Even Point

A ranking can be summarized by a threshold rather than by comparing only two selected allocation methods. The next example varies the allocation continuously while keeping total capital and both profits fixed. This makes clear exactly where the ranking switches.

Worked Example: Finding the Allocation That Makes RAROCs Equal

Two hypothetical business lines generate profits of $15 million for A and $12 million for B. Their combined economic capital is $100 million. Let $x$ million dollars be allocated to A and $100-x$ million dollars to B. Find the allocation at which their RAROCs are equal, and determine the ranking at allocations of $50 million and $60 million to A.

Method. Set the two RAROCs equal to find the switch point, then compare each test allocation with that point or calculate the RAROCs directly.

Switch point. Equality requires $15/x=12/(100-x)$. Multiplying by the positive denominators gives $15(100-x)=12x$. Expanding gives $1{,}500-15x=12x$, so $27x=1{,}500$ and $x=55.5556$ million dollars, rounded to four decimal places. Line B receives $100-55.5556=44.4444$ million dollars at the switch point. At the exact, unrounded value, both RAROCs equal $15/55.5556$, approximately 27.00%.

Allocation of $50 million to A. A receives $50 million and B receives $100-50=50$ million dollars. A’s RAROC is $15/50=30.00\%$; B’s is $12/50=24.00\%$. A ranks above B.

Allocation of $60 million to A. A receives $60 million and B receives $100-60=40$ million dollars. A’s RAROC is $15/60=25.00\%$; B’s is $12/40=30.00\%$. B ranks above A.

Check. The $50 million allocation is below the $55.5556 million switch point and A ranks higher; the $60 million allocation is above it and A ranks lower. At every allocation considered, the two capital amounts sum to $100 million and the profits remain $15 million and $12 million.

Common mistake avoided. Comparing the profit amounts alone would rank A higher because $15 million exceeds $12 million. RAROC ranks profit relative to allocated capital, so the capital allocation can reverse that comparison.

Separate Allocation Effects from Risk and Profit Changes

A practical sensitivity analysis should make controlled comparisons. First hold profit and total capital fixed and change only the allocation. Separately, hold the allocation rule fixed and update the risk estimate or profit inputs. A two-way comparison can then show whether either change, or their combination, drives the observed ranking.

Set the baseline.
Record each line’s profit, total portfolio capital, allocated capital, and RAROC. Confirm that allocated capital reconciles to total capital.

Change allocation alone.
Keep profits and total portfolio capital fixed. Recalculate each line’s allocated capital and RAROC under the alternative allocation.

Change risk or profit separately.
Apply the revised total capital or profit assumptions while holding the allocation rule fixed. This comparison measures the effect of changed inputs rather than a changed allocation method.

Interpret the combined result.
If the ranking changes only when both factors change together, report that interaction rather than assigning the entire change to allocation or to business performance.

For a common RAROC hurdle $h$, line $i$ meets the hurdle when $P_i/K_i\ge h$, equivalently $P_i-hK_i\ge0$, assuming $K_i>0$. The amount $P_i-hK_i$ is the line’s profit after a hurdle-based capital charge. When total capital and profits stay fixed and allocations reconcile, the sum of these amounts is unchanged: $\sum_i(P_i-hK_i)=\sum_iP_i-hK$. The allocation can redistribute the amounts across lines without changing their portfolio total.

Worked Example: Distinguishing an Allocation Reversal from a Risk-and-Profit Change

Two hypothetical lines, A and B, initially have profits of $14 million and $10 million, respectively, and total economic capital of $100 million. Allocation method M assigns 60% of capital to A and 40% to B; method N assigns 50% to each. Later, a separate stress assessment changes profits to $15 million for A and $9 million for B, and total economic capital to $120 million. Under stress, retain the same allocation shares for each method. Calculate RAROCs in all four cases and identify which comparisons isolate allocation effects.

Baseline, method M. Allocated capital is $0.60(100)=\$60$ million for A and $0.40(100)=\$40$ million for B. Their RAROCs are $14/60=23.33\%$ and $10/40=25.00\%$. B ranks above A.

Baseline, method N. Each line receives $0.50(100)=\$50$ million. Their RAROCs are $14/50=28.00\%$ for A and $10/50=20.00\%$ for B. A ranks above B. Comparing the two baseline methods isolates the allocation effect: profits and total capital are unchanged, but the ranking reverses.

Stress, method M. Allocated capital is $0.60(120)=\$72$ million for A and $0.40(120)=\$48$ million for B. RAROCs are $15/72=20.83\%$ and $9/48=18.75\%$. A ranks above B. Comparing baseline M with stressed M holds the allocation shares fixed; the profits and total capital changed, so this is not an allocation-only comparison.

Stress, method N. Each line receives $0.50(120)=\$60$ million. RAROCs are $15/60=25.00\%$ for A and $9/60=15.00\%$ for B. A ranks above B. Comparing stressed M with stressed N isolates the allocation effect under the stressed inputs: A remains above B, though the RAROC levels differ.

Check. In each case, allocations sum to the relevant total: $60+40=100$, $50+50=100$, $72+48=120$, and $60+60=120$, all in millions of dollars. The baseline ranking reverses under a change in allocation method; the stress comparison under method M also reverses the ranking, but for a different reason. The four results therefore do not support attributing every ranking change to one cause.

Common mistake avoided. Comparing baseline method M directly with stressed method N changes both allocation and business inputs. That comparison cannot identify either effect by itself; use the controlled comparisons above.

Common Traps and Interpretation

Using total capital as every line’s denominator. Use the capital allocated to the line under the stated method, not the portfolio total.
Comparing unreconciled allocations. If allocated capital does not sum to the portfolio total, identify the unexplained difference before interpreting rankings.
Calling an allocation-driven reversal a change in risk. A different distribution of fixed portfolio capital among lines does not establish that aggregate risk changed.
Changing several inputs at once. If profit, total capital, and allocation all change, RAROC differences combine these effects. Use controlled comparisons to distinguish them.
Ignoring sign and denominator conditions. RAROC requires positive allocated capital. The simple profit-ratio threshold derived above also assumes positive profits; with nonpositive profits, compare the actual ratios and interpret them cautiously.

Sensitivity analysis does not establish which allocation method is preferable. It reveals how dependent a business-line ranking or hurdle decision is on the capital assignment. A robust decision process reports the allocation basis, tests reasonable alternatives, and distinguishes a redistribution of fixed capital from changes in underlying risk or profit. This treatment is aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus.

Check Your Understanding

For each problem, keep the profit definition and capital basis consistent unless the problem explicitly changes them.

Two hypothetical lines earn $9 million and $8 million. Their total capital is $80 million, allocated $40 million to each. Calculate both RAROCs and state the ranking. (Answer: A earns 22.50%, B earns 20.00%; A ranks higher.)
Lines A and B earn $18 million and $12 million, with $120 million of total capital allocated between them. Find the allocation to A at which RAROCs are equal. (Answer: $72 million to A and $48 million to B; each RAROC is 25.00%.)
Line A earns $16 million and Line B earns $10 million. Under method X, they receive $50 million and $30 million of a fixed $80 million total; under method Y, they receive $40 million each. Calculate both rankings and identify the allocation effect. (Answer: Under X, A is 32.00% and B is 33.33%, so B ranks higher; under Y, A is 40.00% and B is 25.00%, so A ranks higher. The ranking reversal is caused by allocation, with profits and total capital fixed.)
Two lines earn $11 million and $9 million and receive $44 million and $36 million of total capital. At a 22% hurdle, calculate each profit after the hurdle-based capital charge and identify which lines meet the hurdle. (Answer: A: $11-0.22(44)=\$1.32$ million and 25.00%, so it meets the hurdle; B: $9-0.22(36)=\$1.08$ million and 25.00%, so it also meets the hurdle.)
Initially, A earns $12 million and B earns $8 million; total capital is $100 million and the allocation is $60 million to A and $40 million to B. A revised assessment changes profits to $15 million and $7 million and total capital to $110 million, retaining the same allocation shares. Calculate both sets of RAROCs and explain whether the comparison isolates allocation. (Answer: Initially A is 20.00% and B is 20.00%; revised capital is $66 million and $44 million, giving A 22.73% and B 15.91%. The comparison does not isolate allocation: the shares are fixed, but profits and total capital change.)

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