Sensitivity Analysis for Operational Loss Capital

Why Sensitivity Analysis Matters

An operational loss capital estimate depends on assumptions about how often events occur, how large losses can be, how risks move together, and how expert scenarios are represented. Sparse observations and uncertain tails make it important to ask not only, “What is the estimate?” but also, “Which assumptions make it change, and by how much?”

Sensitivity analysis answers that question by recalculating a chosen capital measure under specified alternative inputs. A one-way analysis changes one input at a time and holds the others fixed. A combined analysis changes two or more inputs together. Neither method proves that a changed assumption is more likely; each quantifies the model’s response to a stated alternative.

Definition: Let $L_\theta$ be annual aggregate operational loss under model assumptions $\theta$. For confidence level $q$, define $\operatorname{VaR}_q(L_\theta)=\inf\{x:F_{L_\theta}(x)\ge q\}$. In this tutorial, operational loss capital is $K_q(\theta)=\operatorname{VaR}_q(L_\theta)-E[L_\theta]$. State the confidence level, loss basis, and capital definition whenever reporting a sensitivity result.

This capital measure is unexpected loss at the selected quantile above expected loss. It is not the quantile itself. Some frameworks use different measures or adjustments; the sensitivity comparison must use the measure specified for the analysis. For discrete loss distributions, the infimum definition means the quantile is the smallest loss amount whose cumulative probability is at least $q$. Small input changes can therefore produce a large quantile jump, or no quantile change at all.

A Consistent Comparison Method

A useful sensitivity calculation begins with a documented baseline, changes a defined assumption, and recalculates both the quantile and expected loss. The difference in capital is then attributed to the changed model input as a conditional comparison, not as proof of a real-world cause.

Specify the baseline.
Record the annual loss definition, confidence level, frequency and severity assumptions, dependence structure, scenario inputs, and baseline capital.

Choose a controlled alternative.
For a one-way test, change one assumption and keep the rest fixed. For a combined test, state every changed assumption and its value.

Recalculate both components.
Find $\operatorname{VaR}_q$ and $E[L]$ for the alternative model, using the same quantile convention and loss basis as in the baseline.

Report and interpret the change.
Calculate the new capital, its absolute change, and, where meaningful, its percentage change. Explain any quantile jump or threshold effect.

The reason both components must be recalculated follows directly from the capital definition. For baseline assumptions $\theta_0$ and alternative assumptions $\theta_1$,

K_q(\theta_1)-K_q(\theta_0) = \left[\operatorname{VaR}_q(L_{\theta_1})-\operatorname{VaR}_q(L_{\theta_0})\right] - \left[E[L_{\theta_1}]-E[L_{\theta_0}]\right].

Thus a higher quantile does not by itself imply a higher capital result: expected loss may also increase. Conversely, capital can change even when the quantile stays fixed, because expected loss changes. For discrete models, the quantile term can move in steps, so a smooth percentage-change interpretation may be misleading.

Frequency and Severity-Tail Sensitivities

Frequency sensitivities change how often losses occur while holding severity assumptions fixed. Severity-tail sensitivities change the probability or size of large losses while holding the frequency assumptions fixed. These are separate tests: if both inputs change at once, the result is a combined sensitivity and should be labeled as such.

Worked Example: Increasing Operational Event Frequency

Consider a hypothetical annual model with $N\sim\operatorname{Binomial}(2,p)$. Each event produces a fixed loss of $100,000, and event severities do not vary. Calculate $K_{0.95}$ for $p=0.20$ and $p=0.30$. Use the exact discrete quantile definition.

Method. For two independent event opportunities, $P(N=0)=(1-p)^2$, $P(N=1)=2p(1-p)$, and $P(N=2)=p^2$. Annual loss is $L=100{,}000N$, and $E[L]=100{,}000(2p)$.

At $p=0.20$, the probabilities for $N=0,1,2$ are $0.80^2=0.64$, $2(0.20)(0.80)=0.32$, and $0.20^2=0.04$. The cumulative probabilities at losses $0, $100,000, and $200,000 are respectively $0.64$, $0.64+0.32=0.96$, and $1.00$. Since $0.64<0.95\le0.96$, $\operatorname{VaR}_{0.95}=$100{,}000$. Expected loss is $100{,}000(2)(0.20)=$40{,}000$, so $K_{0.95}=$100{,}000-$40{,}000=$60{,}000$.

At $p=0.30$, the probabilities are $0.70^2=0.49$, $2(0.30)(0.70)=0.42$, and $0.30^2=0.09$. Cumulative probability through $100,000 is $0.49+0.42=0.91$, which is below $0.95$. Therefore $\operatorname{VaR}_{0.95}=$200{,}000$. Expected loss is $100{,}000(2)(0.30)=$60{,}000$, giving $K_{0.95}=$200{,}000-$60{,}000=$140{,}000$. The frequency increase raises capital by $80,000.

Check. The $200,000 quantile at $p=0.30$ is required because the cumulative probability at $100,000 is only $0.91$, not $0.95$. A common mistake is to use the expected event count as though it determined the tail quantile.

Worked Example: Changing the Severity Tail

Suppose a hypothetical model has exactly one annual event, with loss in thousands of dollars. Under the baseline, loss is 10, 50, or 200 with probabilities 0.80, 0.15, and 0.05. Under a tail stress, the probabilities become 0.80, 0.09, and 0.11. Calculate the change in $K_{0.90}$.

Baseline. The cumulative probability through 10 is 0.80, and through 50 it is $0.80+0.15=0.95$. Thus $\operatorname{VaR}_{0.90}=50$ thousand dollars. Expected loss is $10(0.80)+50(0.15)+200(0.05)=8+7.5+10=25.5$ thousand dollars. Baseline capital is $50-25.5=24.5$ thousand dollars.

Tail stress. The cumulative probability through 50 is $0.80+0.09=0.89$, below 0.90, so $\operatorname{VaR}_{0.90}=200$ thousand dollars. Expected loss is $10(0.80)+50(0.09)+200(0.11)=8+4.5+22=34.5$ thousand dollars. Stressed capital is $200-34.5=165.5$ thousand dollars.

The capital increase is $165.5-24.5=141.0$ thousand dollars. The quantile jumps by 150 thousand because the cumulative probability at 50 falls just below the selected confidence level. The expected loss increase is only 9 thousand, so it does not offset most of the quantile jump.

Check. The tail-stressed probabilities sum to $0.80+0.09+0.11=1.00$. A common mistake is to report the stressed VaR, 200 thousand dollars, as capital without subtracting the stressed expected loss.

Thresholds, Dependence, and Scenario Assumptions

A reporting threshold requires particular care. If losses below a threshold are omitted from a data set or set to zero in a model, that changes the modeled observation distribution. It does not, by itself, eliminate the underlying economic losses. State whether the threshold represents data capture, a contractual retention, or a genuine change in loss payment; these interpretations are not interchangeable.

Dependence sensitivities hold individual risk distributions fixed while changing how their outcomes occur together. In operational risk, changing dependence can alter the probability of coincident losses and therefore the aggregate tail, even when the expected aggregate loss is unchanged. Scenario sensitivities similarly compare alternative expert assumptions, such as event probability or scenario severity; a combined scenario stress changes multiple assumptions together.

Worked Example: Applying a Reporting Threshold

A hypothetical gross annual loss, in thousands of dollars, is 0, 30, or 150 with probabilities 0.75, 0.15, and 0.10. A reporting rule excludes losses below $40,000 by recording them as zero. Calculate gross and thresholded $K_{0.90}$. Treat this rule as an observation transformation, not as a reduction in economic loss.

Gross loss. The cumulative probabilities at 0, 30, and 150 are 0.75, $0.75+0.15=0.90$, and 1.00. Thus gross $\operatorname{VaR}_{0.90}=30$ thousand dollars. Gross expected loss is $0(0.75)+30(0.15)+150(0.10)=4.5+15=19.5$ thousand dollars. Gross capital is $30-19.5=10.5$ thousand dollars, or $10,500.

Thresholded record. The 30-thousand loss is below the 40-thousand reporting threshold, so the recorded values are 0 with probability $0.75+0.15=0.90$, and 150 with probability 0.10. The thresholded $\operatorname{VaR}_{0.90}$ is 0 thousand dollars, and the thresholded expected loss is $150(0.10)=15$ thousand dollars. Thresholded capital is $0-15=-15$ thousand dollars, or $-$15,000$.

The negative result is possible because this defined measure subtracts expected loss from VaR; it does not mean that losses create a benefit. Gross capital is $10,500, not $15,000: the gross expected loss is $19,500, and $30{,}000-19{,}500=10{,}500$. The thresholded result describes the transformed record and should not be interpreted as the capital for the underlying gross economic loss.

Worked Example: Changing Dependence Between Two Risks

Two hypothetical operational risks each produce a fixed loss if an event occurs. Risk A has probability 0.10 and loss $100,000; risk B has probability 0.10 and loss $80,000. Their marginal probabilities remain fixed, but the probability $r$ that both events occur is varied. Calculate $K_{0.95}$ at $r=0$ and $r=0.10$.

Method. The probability of A only and B only is $0.10-r$ for each risk, and the probability of neither is $1-0.10-0.10+r=0.80+r$. Expected aggregate loss is unchanged at $100{,}000(0.10)+80{,}000(0.10)=$18{,}000$.

At $r=0$, neither event occurs with probability 0.80, B only occurs with probability 0.10, and A only occurs with probability 0.10. Aggregate losses are $0, $80,000, and $100,000, with cumulative probabilities 0.80, 0.90, and 1.00. Thus $\operatorname{VaR}_{0.95}=$100{,}000$ and $K_{0.95}=$100{,}000-$18{,}000=$82{,}000$.

At $r=0.10$, both events occur with probability 0.10 and neither occurs with probability 0.90; neither event occurs alone. Aggregate losses are $0 with probability 0.90 and $180,000 with probability 0.10. Therefore $\operatorname{VaR}_{0.95}=$180{,}000$ and $K_{0.95}=$180{,}000-$18{,}000=$162{,}000$. The dependence change increases capital by $80,000 while expected aggregate loss stays fixed.

Check. The joint probability must satisfy $0\le r\le0.10$ for these equal marginal probabilities. A common mistake is to change the marginals while intending to test dependence alone.

Combined Sensitivities and Interpretation

One-way tests help identify which inputs drive a result, but combined stresses can reveal interactions. The combined change need not equal the sum of individual changes, particularly when a quantile crosses a probability boundary. For transparency, report the baseline, each one-way alternative, and the combined alternative rather than presenting only the combined result.

Worked Example: Combining Scenario Probability and Severity Stresses

A hypothetical annual scenario produces loss $s$ with probability $p$, and zero otherwise. Use $q=0.995$. The baseline has $p=0.01$ and $s=$2{,}000{,}000$. Consider a probability-only stress $p=0.02$, a severity-only stress $s=$3{,}000{,}000$, and a combined stress with both changes. Calculate capital in each case.

Method. Since each value of $p$ exceeds $1-q=0.005$, the cumulative probability at zero, $1-p$, is below 0.995. Therefore the quantile is $s$ in all four cases, and $K_{0.995}=s-ps=(1-p)s$.

Baseline capital is $(1-0.01)($2{,}000{,}000)=$1{,}980{,}000$. Under the probability-only stress, capital is $(1-0.02)($2{,}000{,}000)=$1{,}960{,}000$, a change of $-$20{,}000$. Under the severity-only stress, capital is $(1-0.01)($3{,}000{,}000)=$2{,}970{,}000$, a change of $+$990{,}000$.

Under the combined stress, capital is $(1-0.02)($3{,}000{,}000)=$2{,}940{,}000$, an increase of $960,000 from baseline. The two one-way changes sum to $970,000, which is $10,000 more than the combined change. The interaction arises because expected loss is $ps$, so changing probability and severity together changes their product as well as each input separately.

Check. The quantile remains the scenario severity because $1-p$ is 0.99 or 0.98, both below 0.995. A common mistake is to assume the quantile must change whenever scenario probability changes; here it does not, although expected loss and capital do.

Sensitivity results are most useful when they preserve the distinction between model response and model validation. A large change can identify an assumption that deserves scrutiny, documentation, or additional evidence. It does not establish that the stressed assumption is correct. For operational loss capital, document at least the loss basis, quantile convention, frequency and severity changes, dependence treatment, scenario changes, and whether thresholds alter the observed data or the economic loss itself.

Key takeaway: Recalculate both VaR and expected loss under every stated alternative. Keep one-way and combined sensitivities distinct, use a consistent annual loss basis, and explain discrete quantile jumps rather than treating capital as a smooth function of every assumption.

Check Your Understanding

For each problem, use $K_q=\operatorname{VaR}_q-E[L]$ and the smallest-loss quantile whose cumulative probability is at least $q$.

A hypothetical model has $N\sim\operatorname{Binomial}(2,p)$, with a fixed loss of $50,000 per event. At $p=0.25$, calculate $K_{0.90}$. (Answer: VaR = $50,000; expected loss = $25,000; capital = $25,000.)
A single annual loss, in thousands of dollars, is 5, 40, or 120 with probabilities 0.70, 0.20, and 0.10. Calculate $K_{0.85}$. (Answer: VaR = 40 thousand dollars; expected loss = $3.5+8+12=23.5$ thousand dollars; capital = 16.5 thousand dollars.)
A hypothetical gross annual loss, in thousands, is 0, 25, or 100 with probabilities 0.80, 0.12, and 0.08. A rule records losses below 30 as zero. Calculate gross and thresholded $K_{0.90}$. (Answer: Gross VaR = 25, expected loss = 11, capital = 14 thousand dollars; thresholded VaR = 0, expected loss = 8, capital = $-8$ thousand dollars.)
Two risks each have event probability 0.10 and fixed loss $60,000. Their joint event probability changes from 0 to 0.10. Find $K_{0.95}$ at each dependence assumption. (Answer: At joint probability 0, VaR = $60,000 and capital = $48,000; at joint probability 0.10, VaR = $120,000 and capital = $108,000.)
A scenario causes a loss of $s$ with probability $p$, otherwise zero. At $q=0.99$, compare baseline $p=0.02,s=$1{,}000{,}000$ with combined stress $p=0.04,s=$1{,}500{,}000$. (Answer: Baseline capital = $980,000; stressed capital = $1,440,000; increase = $460,000.)

What You'll Learn