Why Coherence Requires Four Separate Checks
A risk measure converts a loss distribution into a single amount used for decisions such as pricing, reserving, or assessing capital needs. A useful measure should respond sensibly when losses are ordered, cash is added, exposures are scaled, or risks are combined. These requirements are expressed by four axioms: monotonicity, translation invariance, positive homogeneity, and subadditivity.
The axioms are a checklist, not interchangeable descriptions of good behavior. A formula may satisfy three and fail the fourth. To classify a risk measure, state its domain and check all four properties using the same loss convention.
The Four Axioms
For loss-based measures, a larger loss should not receive a smaller risk amount. Adding a fixed amount of loss should add that amount to the risk assessment. Scaling an exposure by a nonnegative factor should scale its risk amount by the same factor. Finally, combining risks should not require more risk than the sum of the two separate assessments.
Translation invariance: For a constant cash loss \(c\geq 0\), \(\rho(X+c)=\rho(X)+c\).
Positive homogeneity: For \(\lambda\geq 0\), \(\rho(\lambda X)=\lambda\rho(X)\).
Subadditivity: \(\rho(X+Y)\leq\rho(X)+\rho(Y)\).
Translation invariance concerns adding a constant amount to every outcome, not adding another random risk. Positive homogeneity concerns scaling every outcome, including the zero scale. Subadditivity compares the risk of a combined loss with the sum of the separate risk amounts. The random variables \(X\) and \(Y\) need not be independent: their dependence is already represented by their joint outcomes.
How the Axioms Work Together
The inequalities and equalities have different jobs. Monotonicity checks ordering; translation invariance checks the effect of cash; positive homogeneity checks scale; and subadditivity checks aggregation. A numerical comparison for one pair of risks can illustrate an axiom, but it does not prove that a formula satisfies the axiom for every permitted pair. A general proof starts from the definition of the risk functional.
Confirm that \(X\) and \(Y\) are losses, that their sum is in the domain, and whether cash additions are restricted to nonnegative amounts.
For monotonicity, start with \(X\leq Y\). For translation invariance, compare the functional at \(X+c\) and at \(X\).
Compare \(\rho(\lambda X)\) with \(\lambda\rho(X)\), then compare \(\rho(X+Y)\) with \(\rho(X)+\rho(Y)\).
A risk measure is coherent exactly when it satisfies all four axioms over its stated domain.
Worked Examples
Worked Example: Checking Expected Loss Under Scenario Dependence
A hypothetical insurer models three equally likely scenarios. In thousands of dollars, its two loss amounts are \(X=(0,30,90)\) and \(Y=(20,20,50)\). Consider the functional \(\rho(W)=E[W]\). Calculate the risk amounts and explain whether the formula is coherent. The point is to check the functional generally, not to infer coherence from this one pair of losses.
Method: calculate scenario expectations, then use the expectation identities to check the four axioms. The probability of each scenario is \(1/3\), so
The combined losses are \(X+Y=(20,50,140)\), giving
The equality in this example illustrates subadditivity. In general, linearity of expectation gives \(E[X+Y]=E[X]+E[Y]\), so subadditivity holds with equality. For monotonicity, \(X\leq Y\) almost surely implies \(Y-X\geq 0\) almost surely and hence \(E[Y]-E[X]=E[Y-X]\geq 0\). For a constant \(c\), \(E[X+c]=E[X]+c\). For \(\lambda\geq 0\), \(E[\lambda X]=\lambda E[X]\). Thus expected loss satisfies all four axioms.
Check: The scenario calculation gives \(70=40+30\), consistent with the general equality. Common mistake avoided: assuming the observed equality for one pair proves subadditivity for every pair. The proof comes from the expectation identity, and it requires no independence assumption.
Worked Example: Worst-Case Loss Across Scenarios
A hypothetical risk committee evaluates two portfolios in three possible scenarios, with losses in thousands of dollars. Let \(A=(80,5,20)\) and \(B=(10,70,30)\), and define \(\rho(W)\) as the largest scenario loss. Check the numerical aggregation comparison and establish whether this functional is coherent for a finite scenario set.
Method: compute each maximum, then prove the four properties from the maximum definition. The separate risk amounts are
The combined scenario losses are \(A+B=(90,75,50)\), so
For the general checks, if \(X\leq Y\) in every scenario, then the largest value of \(X\) cannot exceed the largest value of \(Y\), proving monotonicity. Adding a constant \(c\) to every outcome increases the maximum by \(c\), so \(\rho(X+c)=\rho(X)+c\). Multiplying all outcomes by \(\lambda\geq 0\) multiplies their maximum by \(\lambda\), including when \(\lambda=0\), so positive homogeneity holds.
For subadditivity, in every scenario \(X\leq \max(X)\) and \(Y\leq\max(Y)\). Adding these inequalities gives \(X+Y\leq\max(X)+\max(Y)\) in every scenario. Therefore \(\max(X+Y)\leq\max(X)+\max(Y)\). This proves the required inequality.
Check: The combined maximum is \(90\), not the sum of the separate maxima; the comparison is \(90\leq150\). Common mistake avoided: adding scenario maxima and reporting that sum as the maximum combined loss. The two separate maxima occur in different scenarios here.
Worked Example: A Fixed Assessment That Breaks Scale Consistency
A hypothetical pricing rule adds a fixed assessment of 4 thousand dollars to expected loss: \(\rho(X)=E[X]+4\), with all amounts in thousands of dollars. Loss \(X\) is equally likely to be 10 or 30, and loss \(Y\) is equally likely to be 0 or 30. Determine which axioms hold for this rule.
Method: derive each property algebraically, then use the supplied losses to expose the scale failure. The expected losses are
Thus \(\rho(X)=20+4=24\) and \(\rho(Y)=15+4=19\). The combined loss has outcomes \(X+Y=(10,60)\), so \(E[X+Y]=(10+60)/2=35\), and \(\rho(X+Y)=35+4=39\). The subadditivity comparison is \(39\leq24+19=43\).
For any \(X\leq Y\), expectation preserves the order, and adding the same 4 to both assessments preserves it, so monotonicity holds. For any constant \(c\), \(\rho(X+c)=E[X]+c+4=\rho(X)+c\), so translation invariance holds. Also, \(\rho(X+Y)=E[X]+E[Y]+4\), whereas \(\rho(X)+\rho(Y)=E[X]+E[Y]+8\); hence subadditivity holds.
Positive homogeneity fails. For example, \(\rho(2X)=E[2X]+4=40+4=44\), while \(2\rho(X)=2(24)=48\). These are unequal. Equivalently, homogeneity at \(\lambda=0\) would require \(\rho(0)=0\), but this rule gives \(\rho(0)=4\).
Check: The rule satisfies monotonicity, translation invariance, and subadditivity, but not positive homogeneity, so it is not coherent. Common mistake avoided: treating a fixed dollar assessment as if it scaled with the exposure.
Worked Example: A Nonlinear Function of Expected Loss
For nonnegative bounded losses measured in thousands of dollars, consider the candidate \(\rho(X)=m+m^2/100\), where \(m=E[X]\). Test its four axioms using constant losses \(X=10\) and \(Y=20\), and explain which properties hold generally on this domain. Cash additions are nonnegative.
Method: evaluate the candidate at the relevant means, then use its scalar function to check each axiom. For \(X=10\), \(m=10\), so \(\rho(X)=10+10^2/100=11\). Adding a constant loss \(c=10\) gives the constant loss 20, with
Translation invariance fails. Scaling \(X\) by 2 also gives \(\rho(2X)=24\), whereas \(2\rho(X)=22\), so positive homogeneity fails. For \(Y=20\), \(\rho(Y)=20+400/100=24\). The combined constant loss is 30, and
Since \(39\not\leq35\), subadditivity fails. To check monotonicity generally, write \(g(m)=m+m^2/100\). For \(0\leq a\leq b\),
If \(X\leq Y\) almost surely, then \(E[X]\leq E[Y]\), and both expectations are nonnegative. The inequality above therefore proves \(\rho(X)\leq\rho(Y)\). Check: the rule is monotone but fails the other three axioms, so it is not coherent. Common mistake avoided: checking only whether a formula increases with expected loss; monotonicity alone does not establish coherence.
Common Exam Traps and Interpretation
Coherence is useful as a disciplined way to assess how a risk measure treats losses, cash, scale, and aggregation. It does not by itself select one measure for every decision. The four tests also do not say that the risk of a portfolio must equal the sum of its component risks: subadditivity gives an upper bound, and equality can occur, as it does for expected loss.
- Using the wrong sign convention. For losses, adding a sure loss \(c\) raises the risk amount by \(c\). A convention written for gains may use a different sign.
- Testing only one numerical pair. A single calculation can show a violation, but satisfying an inequality for one pair does not prove the axiom generally.
- Confusing translation and homogeneity. Translation adds the same constant to every outcome; homogeneity multiplies every outcome by a factor.
- Forgetting boundary cases. Positive homogeneity includes \(\lambda=0\), so it requires \(\rho(0)=0\).
- Assuming independence is necessary. The four axioms are stated for risks on a common probability space. Subadditivity does not require independence.
Check Your Understanding
For each problem, apply the loss-based definitions and check the stated domain. Amounts are in thousands of dollars unless noted.
- A proposed rule is \(\rho(X)=2E[X]\) for nonnegative bounded losses. Identify which of the four axioms it satisfies, using a sure cash addition to test translation invariance. (Answer: It satisfies monotonicity, positive homogeneity, and subadditivity, but fails translation invariance; for \(c>0\), \(\rho(X+c)=\rho(X)+2c\), not \(\rho(X)+c\).)
- Three equally likely outcomes have losses \(A=(12,4,20)\) and \(B=(7,18,3)\). Under worst-case loss, calculate each separate risk amount and the combined risk amount, then check subadditivity. (Answer: \(\rho(A)=20\), \(\rho(B)=18\), \(A+B=(19,22,23)\), \(\rho(A+B)=23\), and \(23\leq38\).)
- A rule is \(\rho(X)=E[X]+6\). Check monotonicity, translation invariance, subadditivity, and positive homogeneity, including the zero-loss test. (Answer: It satisfies monotonicity, translation invariance, and subadditivity, but fails positive homogeneity because \(\rho(0)=6\ne0\).)
- For nonnegative bounded losses, let \(\rho(X)=E[X]+(E[X])^2/100\). Evaluate \(\rho(10)\), \(\rho(20)\), and \(\rho(30)\), where each argument is a constant loss, and use them to test subadditivity for \(X=10\), \(Y=20\). (Answer: \(\rho(10)=11\), \(\rho(20)=24\), \(\rho(30)=39\); subadditivity fails because \(39>11+24=35\).)
- In a finite scenario set, two loss vectors satisfy \(U=(8,15,6)\) and \(V=(10,12,9)\). Under worst-case loss, first verify the ordering and then compare the risk amounts. (Answer: \(U\leq V\) in every scenario; \(\rho(U)=15\leq\rho(V)=12\) is false. Correction: the stated ordering is not true, since \(15>12\) in the second scenario, so monotonicity cannot be applied to this pair.)