Why Coherence Matters
A risk measure assigns a numerical amount of risk to a random loss. That number may be used as a capital requirement or as a way to compare exposures. But two risk measures can behave differently when losses increase, when a fixed amount is added to every outcome, or when separate losses are combined. The coherence axioms describe four properties that make these behaviors transparent and economically interpretable.
Throughout, losses are represented by random variables on a common probability space. A larger value means a larger loss. A risk measure is written \(\rho(X)\), where \(X\) is a loss and \(\rho(X)\) is a number measured in the same monetary units. Unless stated otherwise, assume the losses are integrable, and use the same joint scenario or probability model when comparing \(X\), \(Y\), and \(X+Y\).
Monotonicity: If \(X\leq Y\) with probability 1, then \(\rho(X)\leq\rho(Y)\).
Translation invariance: For every constant \(c\), \(\rho(X+c)=\rho(X)+c\).
Positive homogeneity: For every \(\lambda\geq 0\), \(\rho(\lambda X)=\lambda\rho(X)\).
Subadditivity: \(\rho(X+Y)\leq\rho(X)+\rho(Y)\).
The direction of these inequalities follows the loss convention. If \(Y\) is never smaller than \(X\), monotonicity says its risk cannot be smaller. Translation invariance says that adding a certain dollar amount to every outcome adds exactly that amount to measured risk. Positive homogeneity concerns scaling a loss by a nonnegative factor. Subadditivity compares the risk of a combined loss with the sum of the separate risk assessments.
Subadditivity is often associated with diversification: combining exposures should not require more risk capital than the sum of their stand-alone requirements. It is an inequality, not a claim that the combined risk must be strictly lower. If losses are dependent, their joint distribution matters when evaluating \(\rho(X+Y)\).
How to Test the Axioms
For a proposed risk measure, test each property using its definition. A formula that looks reasonable, or that satisfies one or two properties, is not thereby coherent. The following outline keeps the checks separate.
Confirm that larger values represent worse losses and that the risk measure is defined for the losses being considered.
Compare \(X\) and \(Y\) outcome by outcome for monotonicity, then compare \(X+c\) with \(X\) for a constant addition.
Compare \(\rho(\lambda X)\) with \(\lambda\rho(X)\), and compare \(\rho(X+Y)\) with \(\rho(X)+\rho(Y)\).
A risk measure is coherent only if every axiom holds throughout its stated domain, not merely in one numerical example.
Expected Loss as a Coherent Risk Measure
For integrable losses, the expected loss risk measure is \(\rho(X)=E[X]\). Its coherence follows directly from the properties of expectation. The justification is useful because it shows which mathematical facts support each axiom, rather than relying on a few numerical checks.
Worked Example: Testing Expected Loss on Paired Scenarios
A hypothetical insurer evaluates two losses across four equally likely scenarios. The outcomes are \(X=(0,2,4,6)\) and \(Y=(1,1,3,3)\), in thousands of dollars. Let \(\rho(Z)=E[Z]\). Test the four axioms for these losses, including the combined loss.
Method. Compute the scenario averages, then compare them with the values required by each axiom. This example checks the calculations; the general justification above establishes the properties for all integrable losses.
The expected loss of \(X\) is
The expected loss of \(Y\) is
Monotonicity. Define \(Z=(1,3,5,7)\), so \(X\leq Z\) in every scenario. Then \(E[Z]=(1+3+5+7)/4=4\), and \(\rho(X)=3\leq4=\rho(Z)\).
Translation invariance. Adding \(2\) to every outcome of \(X\) gives \(X+2=(2,4,6,8)\). Thus \(\rho(X+2)=(2+4+6+8)/4=5\), while \(\rho(X)+2=3+2=5\).
Positive homogeneity. Scaling \(X\) by \(3\) gives \(3X=(0,6,12,18)\). Therefore \(\rho(3X)=36/4=9\), and \(3\rho(X)=3(3)=9\).
Subadditivity. The scenario sums are \(X+Y=(1,3,7,9)\), so \(\rho(X+Y)=(1+3+7+9)/4=20/4=5\). Also, \(\rho(X)+\rho(Y)=3+2=5\). The inequality holds with equality.
The calculation checks out: for expected loss, combining these particular exposures produces the sum of their expected losses. The axiom is satisfied generally because expectation is additive, not because this one set of scenarios proves it.
Worst-Case Loss
For a loss that can take only finitely many values, the worst-case risk measure is the largest possible loss among the scenarios being considered. More generally, the corresponding measure is the essential supremum, which ignores outcomes that have probability zero. For bounded losses, this measure also satisfies the four coherence axioms: increasing a loss cannot reduce its maximum, a constant shift shifts the maximum by that constant, nonnegative scaling scales it, and the maximum of a sum is no greater than the sum of the separate maxima.
Worked Example: Checking Worst-Case Risk for Two Losses
A hypothetical portfolio is evaluated in four equally likely states. Its two loss components have outcomes \(X=(0,8,3,5)\) and \(Y=(2,1,6,4)\), in thousands of dollars. Let \(\rho(W)\) be the largest outcome of \(W\). Check the axioms using these outcomes and a specified monotonicity comparison.
Method. Find each maximum and the scenario-by-scenario sum. The comparison tests the inequalities directly; it does not replace a general proof of the worst-case measure’s properties.
The separate risk values are \(\rho(X)=8\) and \(\rho(Y)=6\). For the combined loss,
For a translation check, add \(3\) to each outcome of \(X\). The shifted outcomes are \((3,11,6,8)\), whose maximum is \(11\). This equals \(\rho(X)+3=8+3=11\). For a scaling check, multiply by \(0.5\): the outcomes become \((0,4,1.5,2.5)\), with maximum \(4=0.5(8)\).
For monotonicity, take \(Z=(1,9,4,8)\). Each outcome of \(Z\) is at least the corresponding outcome of \(X\), and \(\rho(Z)=9\geq8=\rho(X)\). The specific checks are consistent with coherence. In general, for finite scenario sets, \(\max(X+Y)\leq\max(X)+\max(Y)\), which supplies the subadditivity inequality.
VaR and the Difference Between Testing and Proving
For a loss \(X\), \(\operatorname{VaR}_p(X)\) is the loss quantile at confidence level \(p\), using the lower-quantile convention \(\inf\{x:F_X(x)\geq p\}\). The earlier material on VaR establishes how to calculate quantiles, including when a distribution has atoms. A quantile measure responds to constant shifts and nonnegative scaling in the same way as the loss: shifting all outcomes shifts the quantile, and multiplying by a positive constant multiplies it. Quantiles also preserve order when \(X\leq Y\) almost surely. These properties alone do not establish coherence: subadditivity must be checked separately, and VaR is not subadditive for every possible loss model.
Worked Example: Shifts and Scaling of a Discrete VaR
A hypothetical one-period loss \(X\), in thousands of dollars, is \(0\) with probability \(0.5\), \(4\) with probability \(0.3\), and \(10\) with probability \(0.2\). Use the lower-quantile convention to calculate \(\operatorname{VaR}_{0.75}(X)\), then calculate the VaR after adding \(3\) and after multiplying the loss by \(2\). State what these calculations do and do not establish about coherence.
Method. Accumulate the probabilities in increasing order of loss. The cumulative probabilities are \(F_X(0)=0.5\), \(F_X(4)=0.8\), and \(F_X(10)=1\). The first loss value whose cumulative probability is at least \(0.75\) is \(4\), so \(\operatorname{VaR}_{0.75}(X)=4\).
Adding \(3\) produces outcomes \(3,7,13\) with the same probabilities. Their cumulative probabilities remain \(0.5,0.8,1\), so \(\operatorname{VaR}_{0.75}(X+3)=7=4+3\). Multiplying by \(2\) produces outcomes \(0,8,20\), again with cumulative probabilities \(0.5,0.8,1\). Thus \(\operatorname{VaR}_{0.75}(2X)=8=2(4)\).
These calculations confirm the shift and scaling results for this example. They do not test subadditivity, which compares a quantile of a combined loss with the sum of separate quantiles. Do not infer coherence from the two equalities alone; the subadditivity requirement is a separate condition.
A Risk Measure That Fails an Axiom
Testing a deliberately modified functional helps identify why each axiom is a distinct requirement. For example, define \(\rho(X)=E[X]+1\), with the added \(1\) measured in the same units as the loss. The constant offset does not affect comparisons between ordered losses, but it interferes with scaling. A single failed axiom is enough to show that a measure is not coherent.
Worked Example: Detecting a Failure of Positive Homogeneity
Let \(X\) have outcomes \(0,2,4,6\) in four equally likely scenarios, and define \(\rho(W)=E[W]+1\), with losses measured in thousands of dollars. Determine whether this functional is coherent by checking its response to doubling \(X\).
Method. Calculate both sides of positive homogeneity for \(\lambda=2\). First, \(E[X]=(0+2+4+6)/4=3\), so \(\rho(X)=3+1=4\). Therefore \(2\rho(X)=8\).
The doubled outcomes are \(2X=(0,4,8,12)\), with expectation \((0+4+8+12)/4=24/4=6\). Consequently, \(\rho(2X)=6+1=7\). Since \(7\ne8\), positive homogeneity fails, so this functional is not coherent.
Common mistake. It is incorrect to calculate \(\rho(2X)\) as \(2E[X]+2=8\). The definition adds \(1\) once to the value of the functional, regardless of how \(X\) is scaled. The correct calculation is \(\rho(2X)=E[2X]+1=6+1=7\).
Common Exam Traps
- Reversing monotonicity. Under the loss convention, a larger loss should not receive a smaller risk value: \(X\leq Y\) implies \(\rho(X)\leq\rho(Y)\).
- Shifting only selected outcomes. Translation invariance concerns adding the same certain amount \(c\) to every outcome, not changing only one scenario.
- Using negative scaling in the homogeneity axiom. Positive homogeneity requires \(\lambda\geq0\). A negative multiplier reverses the meaning of losses and is not covered.
- Confusing subadditivity with equality. The requirement is \(\rho(X+Y)\leq\rho(X)+\rho(Y)\); equality is permitted but is not required.
- Checking only separate losses. Subadditivity requires evaluating the risk of \(X+Y\) under the joint model. Separate marginal distributions alone do not specify the outcomes of the combined loss.
- Calling a measure coherent after three checks. Coherence requires all four axioms on the stated domain.
Check Your Understanding
For each problem, keep losses in the stated units and distinguish numerical checks from general conclusions.
- A loss \(X\) takes values \(1,3,5\) with equal probabilities. For \(\rho(X)=E[X]\), calculate \(\rho(X)\) and \(\rho(X+4)\), and verify translation invariance for this loss. (Answer: \(\rho(X)=3\), \(\rho(X+4)=7=\rho(X)+4\).)
- For the same \(X\), define \(\rho(X)=E[X]+2\). Compare \(\rho(3X)\) with \(3\rho(X)\). Does the functional satisfy positive homogeneity? (Answer: \(\rho(3X)=11\), while \(3\rho(X)=15\); no.)
- A loss \(W\) has outcomes \(2,6,10\) with probabilities \(0.4,0.4,0.2\). Using the lower-quantile convention, find \(\operatorname{VaR}_{0.7}(W)\). If \(W\) is shifted upward by \(5\), find the shifted VaR. (Answer: \(\operatorname{VaR}_{0.7}(W)=6\); the shifted VaR is \(11\).)
- For a finite scenario set, suppose \(\max X=12\) and \(\max Y=7\). What upper bound does the worst-case risk measure give for \(\max(X+Y)\), and which axiom does this illustrate? (Answer: \(\max(X+Y)\leq19\); subadditivity.)
- A proposed risk measure satisfies monotonicity, translation invariance, and positive homogeneity on its domain, but violates subadditivity for one pair of losses. Is it coherent? Explain using the definition. (Answer: No; coherence requires all four axioms, and the subadditivity violation is sufficient to disqualify it.)