Coherence and Diversification in Aggregate Risk

Risk Measures and Diversification

A risk measure turns a loss distribution into a number used to describe risk or required capital. When a portfolio combines two losses, a natural question is whether its measured risk is no greater than the sum of the risks measured separately. That inequality is called subadditivity; it formalizes one aspect of diversification. Dependence between losses matters: two risks that rarely incur losses together can have a different aggregate risk from two risks that tend to incur losses at the same time.

This tutorial compares value at risk (VaR) and tail value at risk (TVaR) using a loss convention: larger losses should correspond to larger risk. The examples use discrete hypothetical losses, for which probability mass at a quantile makes the calculation especially important. The results are aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Definition: A risk measure $\rho$ is coherent if it has all four properties below, for losses on a common probability space:

Monotonicity: if $X\leq Y$ with probability 1, then $\rho(X)\leq\rho(Y)$.
Translation invariance: for a constant $c$, $\rho(X+c)=\rho(X)+c$.
Positive homogeneity: for a constant $a\geq 0$, $\rho(aX)=a\rho(X)$.
Subadditivity: $\rho(X+Y)\leq\rho(X)+\rho(Y)$.

Here $X$ and $Y$ are loss amounts. Subadditivity says that combining risks cannot produce a measured risk greater than the sum of the separate measurements. It does not say that the combined risk must be smaller.

VaR and TVaR for Discrete Losses

For a loss $X$, let $F_X(x)=P(X\leq x)$ be its distribution function. At confidence level $p$, where $0<p<1$, define VaR as the lower $p$-quantile. We use the quantile-integral definition of TVaR so that calculations remain well-defined when there is probability mass at the quantile.

Definition: For $0<p<1$, define $$ \operatorname{VaR}_p(X)=\inf\{x:F_X(x)\geq p\}, \qquad \operatorname{TVaR}_p(X)=\frac{1}{1-p}\int_p^1\operatorname{VaR}_u(X)\,du. $$ For an integrable discrete loss, let $q=\operatorname{VaR}_p(X)$. Then $$ \operatorname{TVaR}_p(X) = \frac{E[X\,\mathbf{1}_{\{X>q\}}]+q\big(F_X(q)-p\big)}{1-p}. $$ The second term includes just enough of the probability mass at $q$ to fill the upper tail of probability $1-p$.

The distinction between VaR and TVaR is useful in the presence of a large atom. VaR reports the threshold at the selected quantile. TVaR averages the worst $1-p$ fraction of the distribution, using only the needed fraction of the mass at the threshold. In particular, when there is an atom at $q$, TVaR is not necessarily $E[X\mid X>q]$.

Worked Example: VaR Fails Subadditivity Under Exclusive Losses

A hypothetical insurer has two loss amounts, $X$ and $Y$. With probability $0.05$, only $X$ is 100; with probability $0.05$, only $Y$ is 100; and with probability $0.90$, both are 0. Calculate each risk’s VaR and TVaR at $p=0.95$, then compare the combined loss with the separate risks.

Method. Find each distribution and the aggregate distribution, then apply the quantile definition and the discrete TVaR formula. This example also tests the endpoint convention: if the distribution function is already at the confidence level at zero, the lower quantile is zero.

For $X$, the probability of zero is $0.95$, so $F_X(0)=0.95$. Therefore $\operatorname{VaR}_{0.95}(X)=0$. The worst five percent of outcomes all have loss 100, so $\operatorname{TVaR}_{0.95}(X)=100$. By symmetry, $\operatorname{VaR}_{0.95}(Y)=0$ and $\operatorname{TVaR}_{0.95}(Y)=100$.

The aggregate $X+Y$ is 100 with probability $0.10$ and 0 with probability $0.90$. Its distribution function at zero is $0.90$, below $0.95$, so $\operatorname{VaR}_{0.95}(X+Y)=100$. The worst five percent of aggregate outcomes also all have loss 100, giving $\operatorname{TVaR}_{0.95}(X+Y)=100$.

Thus the VaR comparison is $\operatorname{VaR}_{0.95}(X+Y)=100>0+0=\operatorname{VaR}_{0.95}(X)+\operatorname{VaR}_{0.95}(Y)$. VaR fails subadditivity in this example. For TVaR, $100\leq100+100=200$, so the subadditivity inequality holds. The combined risk’s TVaR is strictly less than the sum of the separate TVaRs.

Check. The aggregate has a 10% chance of loss 100, so a worst 5% slice consists entirely of loss-100 outcomes. This confirms its TVaR of 100. A common mistake is to conclude that each individual VaR is 100 because 100 is its only positive loss; the correct quantile check is $F_X(0)=0.95$, which makes the lower 95th percentile zero.

Why TVaR Is Subadditive

For integrable losses, TVaR has a useful representation as a maximum over bounded weights. The weights can emphasize adverse outcomes, but cannot put more than a specified amount of weight on any outcome. This representation explains why TVaR satisfies subadditivity even when VaR does not.

\operatorname{TVaR}_p(X) = \sup\left\{E[ZX]:0\leq Z\leq\frac{1}{1-p},\ E[Z]=1\right\}.

Here $Z$ is a random weight on the same probability space as $X$. To see why the representation gives TVaR, assign the maximum weight $1/(1-p)$ to outcomes in the worst upper tail. Those outcomes have total probability $1-p$, so the resulting weights have expectation 1. If the tail boundary cuts through an atom, assign only the required fraction of the maximum weight to that atom. The weighted expectation is then exactly the average of the worst $1-p$ fraction, which is the quantile-integral TVaR. No other permitted weights can produce a larger expectation: they cannot exceed the maximum weight on the worst outcomes, and their total expected weight is fixed at 1.

Now use the representation for two integrable losses $X$ and $Y$. Each admissible $Z$ is allowed for both losses, so

\begin{aligned} \operatorname{TVaR}_p(X+Y) &=\sup_Z E[Z(X+Y)]\\ &=\sup_Z\big(E[ZX]+E[ZY]\big)\\ &\leq \sup_Z E[ZX]+\sup_Z E[ZY]\\ &=\operatorname{TVaR}_p(X)+\operatorname{TVaR}_p(Y). \end{aligned}

The inequality follows because the supremum of a sum over a common set cannot exceed the sum of the two separate suprema. The two losses need not have the same adverse outcomes. This proves TVaR subadditivity for integrable losses under the stated quantile-integral convention.

Choose the tail level.
Set the confidence level $p$; the upper tail to be averaged has probability $1-p$.

Find the quantile.
Use $q=\inf\{x:F_X(x)\geq p\}$, checking the distribution function at each possible loss value.

Fill the upper tail.
Include all probability above $q$, plus only enough probability mass at $q$ to make the total tail probability $1-p$.

Compare aggregate and separate risks.
Calculate the measure for $X+Y$ and compare it with the sum of the individual measures. Do not infer the result from the marginals alone when dependence is relevant.

Coherence Properties in Practice

VaR satisfies monotonicity, translation invariance, and positive homogeneity for the quantile convention used here. For example, adding a sure loss $c$ shifts every quantile by $c$; multiplying losses by a nonnegative constant $a$ scales every quantile by $a$. If $X\leq Y$ with probability 1, the distribution of $Y$ cannot have a lower quantile than $X$. However, the exclusive-loss example shows that VaR is not subadditive for all loss distributions.

TVaR satisfies all four coherence properties for integrable losses. Its tail-weight representation gives subadditivity; its quantile definition gives monotonicity, translation invariance, and positive homogeneity. These properties do not mean TVaR always gives a smaller number than VaR. Rather, TVaR describes average severity within a selected upper tail and has a guaranteed subadditivity property that VaR lacks.

Worked Example: TVaR with Mass at the Quantile

A hypothetical portfolio loss $X$ is 0 with probability $0.80$, 10 with probability $0.15$, and 40 with probability $0.05$. A second component has a certain loss of 7, so the combined loss is $X+7$. At $p=0.90$, calculate VaR and TVaR for both the original and combined losses.

Method. Find the quantile from the cumulative probabilities, then use the discrete tail formula. This method avoids treating only outcomes strictly above the quantile as the TVaR tail.

For $X$, $F_X(0)=0.80$ and $F_X(10)=0.95$. Since $0.80<0.90\leq0.95$, $\operatorname{VaR}_{0.90}(X)=10$. The tail probability is $1-0.90=0.10$. It contains probability $0.05$ at loss 40 and requires another $0.05$ at loss 10. Thus $\operatorname{TVaR}_{0.90}(X)=(0.05(40)+0.05(10))/0.10=(2+0.5)/0.10=25$.

For $X+7$, the loss values are 7 with probability $0.80$, 17 with probability $0.15$, and 47 with probability $0.05$. The cumulative probability at 7 is $0.80$, and at 17 it is $0.95$, so $\operatorname{VaR}_{0.90}(X+7)=17$. The worst 10% contains $0.05$ at 47 and $0.05$ at 17: $\operatorname{TVaR}_{0.90}(X+7)=(0.05(47)+0.05(17))/0.10=(2.35+0.85)/0.10=32$.

As a check, translation invariance gives $\operatorname{VaR}_{0.90}(X+7)=10+7=17$ and $\operatorname{TVaR}_{0.90}(X+7)=25+7=32$, matching the direct calculations. A common incorrect calculation is $E[X\mid X>10]=40$. That conditions on losses strictly above the quantile and ignores the required 0.05 probability at 10; it is not the quantile-integral TVaR here.

Common Traps When Comparing Risk Measures

Assuming VaR is always subadditive. Its three other coherence properties do not imply the fourth. Check the combined distribution when dependence may matter.
Using separate distributions to determine aggregate VaR. Marginal distributions alone do not specify how losses occur together. The joint distribution, or a dependence assumption, is needed to calculate the distribution of $X+Y$.
Ignoring mass at the quantile. For discrete losses, the upper tail may require only part of the probability at $q$. Include that fraction when calculating TVaR.
Confusing the two tail conventions. $E[X\mid X>q]$ excludes the quantile atom. The quantile-integral TVaR includes the needed fraction at $q$, so the two quantities can differ.
Interpreting subadditivity as guaranteed savings. Subadditivity gives an upper bound; equality is possible, and the actual benefit depends on the joint loss behavior.

Key takeaway: VaR can fail subadditivity when dependence separates the individual tail events, so it is not coherent for arbitrary losses. For integrable losses, quantile-integral TVaR is coherent: its upper-tail average respects monotonicity, translation invariance, positive homogeneity, and subadditivity.

Check Your Understanding

For each discrete loss, use the quantile-integral convention and check any probability mass at the quantile.

A hypothetical loss is 0 with probability 0.92 and 30 with probability 0.08. Find its VaR and TVaR at $p=0.90$. (Answer: $\operatorname{VaR}_{0.90}=0$, $\operatorname{TVaR}_{0.90}=24$.)
A loss is 0 with probability 0.70, 12 with probability 0.20, and 50 with probability 0.10. Find VaR and TVaR at $p=0.80$. (Answer: $\operatorname{VaR}_{0.80}=12$, $\operatorname{TVaR}_{0.80}=31$.)
A loss $X$ is 0 with probability 0.80 and 20 with probability 0.20. Let $W=3+2X$. Find VaR and TVaR at $p=0.90$ for $X$ and $W$. (Answer: for $X$, VaR $=20$ and TVaR $=20$; for $W$, VaR $=43$ and TVaR $=43$.)
At $p=0.95$, two hypothetical risks each pay 50 with probability 0.05, and their positive-loss events are mutually exclusive. Otherwise each pays 0. Find each individual VaR, the aggregate VaR, and determine whether VaR is subadditive. (Answer: each individual VaR is 0, aggregate VaR is 50, so VaR is not subadditive.)
At $p=0.95$, two independent hypothetical risks each pay 20 with probability 0.05 and 0 otherwise. Find the individual and aggregate TVaRs. The aggregate is 40 with probability 0.0025, 20 with probability 0.095, and 0 with probability 0.9025. (Answer: each individual TVaR is 20; aggregate TVaR is 21; $21\leq20+20$, so subadditivity holds.)

What You'll Learn