Why Subadditivity Matters
An insurer evaluating two loss positions needs to distinguish the risk of holding them together from the risks of holding each one separately. A risk measure that is subadditive makes a precise statement about that comparison: the measured risk of the aggregate cannot exceed the sum of the measured standalone risks. This is a bound on the risk measure, not a claim that the aggregate loss is always smaller in each scenario.
The inequality is useful when considering aggregation, diversification, or an allocation of risk capital. It also needs careful interpretation. Subadditivity does not say that combining positions must reduce measured risk strictly, and it is a property of a particular risk measure—not a fact that follows automatically for every method of measuring risk.
The word no-superadditivity emphasizes that the risk measure does not assign the aggregate a value greater than the sum of its standalone values. It does not rule out a large aggregate loss in a particular outcome. The loss \(X+Y\) can exceed either \(X\) or \(Y\) in an outcome even when the risk measure satisfies the inequality.
Reading the Inequality Correctly
The losses must be defined using a consistent convention. In this tutorial, \(X\) and \(Y\) are nonnegative losses, and a larger loss represents a worse outcome. The aggregate position is the pointwise sum: in each outcome, add the loss from the first position to the loss from the second. The same risk measure, confidence level, time horizon, and valuation basis must be used on both sides.
For example, comparing a one-year risk measure for \(X+Y\) with ten-year standalone measures would not test the stated property. Nor does subadditivity compare the aggregate to the larger of the two standalone risks; its benchmark is their sum.
A useful example for calculations is Tail Value at Risk. For confidence level \(\alpha\), TVaR is the average loss in the worst \(1-\alpha\) fraction of outcomes, using fractional probability mass at a boundary loss when necessary. With \(\alpha=0.50\), the relevant tail is the worst half of the probability mass. The examples below use this convention for discrete losses. Their calculations illustrate the inequality in specific models; they are not a proof of a general TVaR result.
Worked Example: Independent Losses and TVaR
Method. Find the marginal TVaR values, form the joint distribution using independence, and then average the worst half of the aggregate-loss distribution. Consider two hypothetical loss positions \(X\) and \(Y\). Each is 0 with probability 0.50 and $100 with probability 0.50. They are independent. Calculate TVaR at \(\alpha=0.50\) for each position and for their aggregate.
Standalone risks. For either loss, the worst half of outcomes consists entirely of the $100 outcomes, whose probability is exactly 0.50. Thus $$ \operatorname{TVaR}_{0.50}(X) = \frac{0.50(100)}{0.50} = 100, \qquad \operatorname{TVaR}_{0.50}(Y)=100. $$
Aggregate distribution. Independence gives four joint outcomes. Both losses are zero with probability \(0.50(0.50)=0.25\); exactly one is $100 with total probability \(2(0.50)(0.50)=0.50\); and both are $100 with probability \(0.25\). Therefore \(X+Y\) is 0 with probability 0.25, $100 with probability 0.50, and $200 with probability 0.25.
Aggregate TVaR. The worst half consists of all 0.25 probability mass at $200 and 0.25 of the mass at $100. Its total loss contribution is $$ 0.25(200)+0.25(100)=50+25=75. $$ Dividing by the tail probability, 0.50, gives $$ \operatorname{TVaR}_{0.50}(X+Y)=\frac{75}{0.50}=150. $$ The check is \(150\leq100+100=200\), so the subadditivity inequality holds. The aggregate TVaR is strictly below the sum of standalone TVaRs in this example.
Common mistake avoided. Adding the standalone values gives $200, but that is the upper bound, not the calculated aggregate TVaR. Treating $200 as the aggregate answer would ignore the joint distribution and would miss the tail calculation.
Dependence Changes the Aggregate Calculation
The standalone distributions do not determine the distribution of \(X+Y\). Dependence describes how the losses occur together, so it can change the aggregate tail even when the marginal loss distributions remain fixed. Subadditivity compares the aggregate measure, calculated using the joint outcomes, with the sum of the two marginal measures.
Worked Example: Dependent Losses and TVaR
Method. First calculate each marginal risk from its own loss distribution. Then use the specified joint probabilities to calculate aggregate TVaR at \(\alpha=0.50\). In a hypothetical portfolio, each of \(X\) and \(Y\) is either $0 or $100. The joint distribution is:
| \(X\) | \(Y\) | Probability | \(X+Y\) |
|---|---|---|---|
| $100 | $100 | 0.375 | $200 |
| $100 | $0 | 0.125 | $100 |
| $0 | $100 | 0.125 | $100 |
| $0 | $0 | 0.375 | $0 |
Standalone risks. For \(X\), the probability of a $100 loss is \(0.375+0.125=0.50\). The same is true for \(Y\). Each marginal worst-half tail therefore consists entirely of $100 losses: $$ \operatorname{TVaR}_{0.50}(X)=100, \qquad \operatorname{TVaR}_{0.50}(Y)=100. $$
Aggregate risk. The aggregate is $200 with probability 0.375, $100 with probability \(0.125+0.125=0.25\), and $0 with probability 0.375. Its worst half contains all 0.375 probability mass at $200 and 0.125 of the probability mass at $100. The tail loss contribution is $$ 0.375(200)+0.125(100)=75+12.5=87.5. $$ Therefore $$ \operatorname{TVaR}_{0.50}(X+Y)=\frac{87.5}{0.50}=175. $$ The check is \(175\leq100+100=200\). Subadditivity holds, but the aggregate risk is closer to the sum of standalone risks than it was in the independent example.
Dependence check. If the losses were independent with these same marginals, the probability of both being $100 would be \(0.50(0.50)=0.25\). Here it is 0.375, so high losses occur together more often than under independence. That changes the aggregate tail, but it does not change either marginal TVaR.
Common mistake avoided. Do not infer independence from the marginal probabilities, or calculate the aggregate distribution by multiplying marginals when a joint distribution is supplied. The stated joint probabilities, not an independence assumption, determine the aggregate tail.
A Direct Justification for a Worst-Case Measure
A simple finite-scenario risk measure assigns a position its largest loss among a specified set of scenarios. Let \(X_s\) be the loss of position \(X\) in scenario \(s\), and define \(\rho_{\max}(X)=\max_s X_s\). This measure is subadditive because the largest value of a sum cannot exceed the sum of the two separate largest values.
For each scenario \(s\), the aggregate loss is \(X_s+Y_s\).
For every \(s\), \(X_s\leq\max_t X_t\) and \(Y_s\leq\max_t Y_t\). Adding gives \(X_s+Y_s\leq\max_t X_t+\max_t Y_t\).
Since the bound holds for every \(s\), \(\max_s(X_s+Y_s)\leq\max_s X_s+\max_s Y_s\). This is exactly \(\rho_{\max}(X+Y)\leq\rho_{\max}(X)+\rho_{\max}(Y)\).
The argument depends on using the same scenario set for both positions and their aggregate. It establishes the inequality for this particular worst-case construction. It does not establish that every risk measure is subadditive.
Worked Example: Subadditivity Under a Scenario Maximum
Method. Apply the maximum-loss definition to each position and to the row-by-row aggregate, all over the same hypothetical three scenarios. Losses are shown in thousands of dollars.
| Scenario | \(X\) | \(Y\) | \(X+Y\) |
|---|---|---|---|
| 1 | 20 | 70 | 90 |
| 2 | 100 | 10 | 110 |
| 3 | 40 | 50 | 90 |
The standalone worst-case values are \(\rho_{\max}(X)=100\) and \(\rho_{\max}(Y)=70\), so their sum is 170 thousand dollars. The aggregate scenario losses are \(20+70=90\), \(100+10=110\), and \(40+50=90\) thousand dollars. Thus \(\rho_{\max}(X+Y)=110\) thousand dollars, and the check is \(110\leq170\).
Common mistake avoided. Adding the scenario maxima gives the valid upper bound 170, but the two maxima occur in different scenarios. The maximum aggregate loss is found by adding losses within each scenario first; it is 110, not 170.
What Subadditivity Does—and Does Not—Establish
Subadditivity is valuable because it places a disciplined upper bound on aggregation: the combined risk cannot exceed the sum of the separately measured risks. In the TVaR examples, the joint distribution determines the aggregate value, while the standalone values remain fixed by the marginals. In the scenario-maximum example, adding separate maxima is an upper bound even though the positions reach their maxima in different scenarios.
The inequality is not necessarily strict. If two losses always reach their worst values together under a particular measure, aggregation may produce equality. And if a measure yields equality, that does not mean the losses are independent; it only reports the relationship among the three risk values under that measure.
A frequent exam error is to confuse the subadditivity statement with the claim that losses themselves are diversified. The inequality is about \(\rho\), not about each realized value of \(X+Y\). Another is to assume that a property verified for one pair of distributions proves that the risk measure is subadditive in general. A general claim requires a general argument for the chosen measure; a numerical example only verifies the inequality for its stated setup.
Check Your Understanding
For each problem, use the stated risk measure, confidence level, and joint information. Loss amounts are hypothetical.
- A finite-scenario worst-case measure is the maximum loss across three scenarios. In thousands of dollars, \(X=(30,80,20)\) and \(Y=(50,10,40)\). Find each standalone risk and the aggregate risk, then check subadditivity. (Answer: \(\rho_{\max}(X)=80\), \(\rho_{\max}(Y)=50\), \(\rho_{\max}(X+Y)=90\); \(90\leq130\).)
- Independent losses \(X\) and \(Y\) are each $0 with probability 0.70 and $100 with probability 0.30. Using worst-tail-average TVaR at \(\alpha=0.70\), find the standalone risks and aggregate risk. (Answer: each standalone TVaR is $100; aggregate TVaR is $130; \(130\leq200\).)
- Each of two dependent losses is either $0 or $100, with joint probabilities \(P(X=100,Y=100)=0.375\), \(P(X=100,Y=0)=0.125\), \(P(X=0,Y=100)=0.125\), and \(P(X=0,Y=0)=0.375\). Using worst-tail-average TVaR at \(\alpha=0.50\), calculate the aggregate value and compare it with the sum of standalone values. (Answer: aggregate TVaR is $175; each standalone TVaR is $100; \(175\leq200\).)
- For any two finite loss positions on a common scenario set, let \(\rho_{\max}\) be the maximum loss over that set. State the scenario-by-scenario bound that proves subadditivity, and explain why the common scenario set matters. (Answer: \(X_s+Y_s\leq\max_tX_t+\max_tY_t\) for every \(s\), hence \(\max_s(X_s+Y_s)\leq\max_sX_s+\max_sY_s\); the same scenarios must be used to compare all three maxima.)
- A proposed risk measure gives standalone values \(\rho(X)=\$90\) and \(\rho(Y)=\$60\), but gives \(\rho(X+Y)=\$155\) for a specified pair. Does this pair satisfy subadditivity, and by how much does the aggregate value exceed the bound? (Answer: No; \(155>90+60=150\), so the aggregate value exceeds the sum by $5.)