Start With the Decision
A risk measure is useful only insofar as it summarizes risk relevant to a decision. A limit on the chance of exceeding a loss threshold asks a different question from a capital buffer intended to cover severe tail losses. An objective that penalizes large forecast errors asymmetrically is different again from one that applies an exponential penalty to losses. The measures can therefore rank the same loss distribution differently without contradiction: they answer different questions.
Let \(X\) denote a loss over a specified horizon, with larger values representing worse outcomes. The examples below use hypothetical one-period losses in dollars. They assume no discounting or mortality adjustment; those assumptions are not relevant to the one-period calculations. The choice of measure should follow the decision objective, the acceptable model assumptions, and the constraints on how the result will be used.
What Each Measure Says About Risk
The definitions differ in what they emphasize. VaR is a loss threshold at a selected percentile. It does not describe how large losses are beyond that threshold. TVaR averages the quantiles in the upper tail and therefore reflects both the frequency and severity of losses in that part of the distribution. At a point mass, TVaR’s quantile-integral definition handles the boundary consistently; simply conditioning on losses above VaR can produce a different result.
An expectile is determined by a balance between squared deviations below and above a candidate value. For a level \(\tau\) between 0 and 1, its loss-based definition uses the minimizer of an asymmetric squared-error objective. For \(\tau\geq 1/2\), the expectile is an increasing, coherent risk measure under the usual finite-mean conditions; the squared-objective calculation below assumes a finite second moment. At \(\tau=1/2\), it equals the mean. Increasing \(\tau\) gives greater weight to losses above the candidate expectile.
The entropic risk measure uses an exponential penalty. For \(\theta>0\), define \(\rho_{\theta}(X)=\frac{1}{\theta}\log E[e^{\theta X}]\), provided the moment-generating function is finite at \(\theta\). Larger positive losses receive disproportionately greater weight. The parameter \(\theta\) controls the strength of that penalty and has inverse-loss units. This measure is convex and translation equivariant, but it is not positively homogeneous: multiplying a loss by a constant does not, in general, multiply its entropic risk by the same constant.
| Decision objective | Measure to consider | Important limitation or condition |
|---|---|---|
| Set a loss threshold at a stated percentile | VaR | Does not summarize severity beyond the threshold; not generally subadditive |
| Summarize average losses in an upper tail | TVaR | Requires a stated tail level and a defensible tail distribution |
| Penalize large forecast deviations asymmetrically | Expectile | The level \(\tau\) must reflect the intended asymmetry; squared-objective definition requires a finite second moment |
| Apply a smooth, increasingly severe penalty to larger losses | Entropic risk | Requires a finite exponential moment; the choice of \(\theta\) affects the result |
Worked Comparisons
The following examples show how to calculate the measures before deciding which one fits the stated objective. Each distribution is hypothetical, and all calculations concern a single-period loss in dollars.
Worked Example: A Threshold Objective Versus a Tail-Severity Objective
A hypothetical one-year portfolio loss is \(X=0\) with probability \(0.90\) and \(X=100\) with probability \(0.10\). There is no discounting or mortality adjustment. Calculate VaR and TVaR at \(\alpha=0.90\), the expectile at \(\tau=0.90\), and entropic risk for \(\theta=\ln(2)/100\). Identify which result answers a threshold question and which reflects tail severity.
VaR method. The cumulative distribution satisfies \(F_X(0)=0.90\). By the infimum-quantile definition, \(\operatorname{VaR}_{0.90}(X)=0\). Thus, at this convention for the boundary quantile, 90% of outcomes are at or below a zero loss.
TVaR method. The quantile is zero through probability 0.90 and is 100 above 0.90. Therefore,
Expectile method. Since the expectile lies between 0 and 100, its balancing equation is
Entropic method. The specified parameter gives \(e^{100\theta}=2\), so \(E[e^{\theta X}]=0.90+0.10(2)=1.10\). Hence,
The threshold result is VaR, equal to zero at this percentile; TVaR instead reports the full loss of 100 in the upper 10% tail. The expectile and entropic results give different summaries because their weighting rules differ. The method avoids treating these numbers as competing estimates of one uniquely defined quantity: they serve distinct objectives. No rounding is needed for the exact VaR or TVaR values; the entropic result is rounded to the nearest cent.
Worked Example: A Capital Limit That Depends on Tail Severity
A hypothetical one-year liability loss is 0 with probability 0.94, 40 with probability 0.05, and 400 with probability 0.01. There is no discounting or mortality adjustment. A decision rule allows a risk measure no greater than 60 dollars. Compare VaR and TVaR at \(\alpha=0.95\), and calculate expected loss as a reference.
Step 1: Find VaR. The cumulative probabilities are \(F_X(0)=0.94\), \(F_X(40)=0.99\), and \(F_X(400)=1\). The first loss value whose cumulative probability reaches 0.95 is 40, so \(\operatorname{VaR}_{0.95}(X)=40\). This result is below the 60-dollar limit.
Step 2: Find TVaR. The upper 5% of probability consists of all of the 1% outcome at 400 and 4% of the probability at 40. Thus,
TVaR is above the limit, so the same distribution passes the VaR rule but fails the TVaR rule. The decision changes because the rule’s objective changes: VaR checks a percentile threshold, while TVaR measures average severity in the selected tail.
Step 3: Calculate expected loss. \(E[X]=0.94(0)+0.05(40)+0.01(400)=2+4=6\) dollars. The expected loss is well below both the 60-dollar limit and the TVaR result, but it is not a substitute for a tail-focused criterion. This method avoids inferring tail protection from a low mean. All probability-weighted amounts and the final measures are exact dollar values.
Worked Example: Choosing Between an Expectile and an Entropic Penalty
A hypothetical one-year loss is 0 with probability 0.75, 20 with probability 0.20, and 100 with probability 0.05. There is no discounting or mortality adjustment. A decision-maker uses an expectile with \(\tau=0.80\) to penalize high forecast deviations, and also reviews entropic risk with \(\theta=\ln(2)/100\). Calculate both measures and explain the distinction.
Expectile method. First locate \(e\). At \(m=20\), the left side of the balancing equation is positive while the right side is zero, and at \(m=100\), the left side is zero while the right side is positive; the solution is between 20 and 100. In this interval, the excess above \(e\) occurs only at loss 100, while the shortfall below \(e\) occurs at losses 0 and 20. The equation is
Expanding gives \(4-0.04e=0.19e-0.8\), so \(4.8=0.23e\) and \(e=4.8/0.23\approx20.87\). This lies in the assumed interval, confirming the calculation.
Entropic method. Since \(e^{100\theta}=2\), \(e^{20\theta}=2^{0.2}\approx1.1487\). Therefore,
The expectile emphasizes the asymmetric balance specified by \(\tau\), whereas the entropic measure applies the chosen exponential penalty to every outcome, with larger losses receiving greater weight. Their different values do not establish that one is universally preferable. The calculation avoids choosing \(\theta\) or \(\tau\) solely to obtain a desired answer. Values are rounded to the nearest cent.
Why the Expectile Balancing Equation Holds
The balancing equation is useful because it turns the expectile definition into a calculation. For \(L(m)=E[\tau(X-m)_+^2+(1-\tau)(m-X)_+^2]\), differentiate with respect to \(m\). Away from \(X=m\), the derivative of the first squared term is \(-2\tau(X-m)_+\); the derivative of the second is \(2(1-\tau)(m-X)_+\). At \(X=m\), both contributions are zero. With \(E[X^2]<\infty\), these terms are integrable, so setting the derivative of the objective to zero gives
which is precisely the stated balance. The objective is convex in \(m\), and for a non-degenerate distribution its minimizer is unique. This derivation also explains why the squared-objective definition requires care: a finite first moment alone does not ensure a finite squared objective. When using this definition and differentiation argument, assume a finite second moment.
Properties and Practical Constraints
The axioms matter when a measure will be used to compare or combine risks. Under the standard loss convention, TVaR is coherent, while VaR is not generally subadditive. Expectiles with \(\tau\geq1/2\) are coherent under the usual conditions. The entropic measure is convex and translation equivariant, but not positively homogeneous, so it is convex rather than coherent. These distinctions are not a ranking of business usefulness: they indicate which mathematical behaviors are guaranteed.
Use the objective and constraints to narrow the choice. A contractual or internal rule defined in terms of a percentile threshold may call for VaR, even though it says nothing about tail severity. A tail-loss limit is more directly represented by TVaR. An asymmetric squared-error criterion may support an expectile, provided its level and moment assumptions are defensible. An exponential utility or penalty framework may motivate an entropic measure, but the parameter and finiteness of the exponential moment must be addressed.
Other practical constraints can be decisive. A measure may be mathematically attractive but difficult to estimate reliably from sparse data, or its parameter may be hard to explain to decision-makers. Conversely, a familiar percentile can be easy to communicate but inadequate if the decision depends on the magnitude of extreme losses. State what loss is being measured, over what horizon, and which property of the distribution drives the decision before presenting a number.
Common Exam Trap
A frequent error is to calculate TVaR at a discrete boundary as \(E[X\mid X\geq\operatorname{VaR}_{\alpha}(X)]\), without checking the mass at the boundary. In the first worked example, \(\operatorname{VaR}_{0.90}(X)=0\), so this incorrect shortcut gives \(E[X\mid X\geq0]=10\). But the upper 10% tail defined by the quantile integral consists entirely of loss 100, giving \(\operatorname{TVaR}_{0.90}(X)=100\). The shortcut includes outcomes below the intended tail through the atom at zero. Use the quantile-integral definition, or an equivalent formula that handles boundary mass correctly.
Check Your Understanding
For each original scenario, identify the relevant calculation or decision and check your result against the answer.
- A hypothetical one-period loss is 0 with probability 0.80 and 30 with probability 0.20. With no discounting or mortality adjustment, calculate VaR and TVaR at \(\alpha=0.80\), using the infimum-quantile convention. (Answer: \(\operatorname{VaR}_{0.80}=0\), \(\operatorname{TVaR}_{0.80}=30\).)
- A hypothetical one-period loss is 0 with probability 0.90, 50 with probability 0.08, and 200 with probability 0.02. With no discounting or mortality adjustment, calculate VaR and TVaR at \(\alpha=0.95\). (Answer: \(\operatorname{VaR}_{0.95}=50\), \(\operatorname{TVaR}_{0.95}=110\).)
- A hypothetical one-period loss is 0 with probability 0.60 and 40 with probability 0.40. With no discounting or mortality adjustment, find the expectile at \(\tau=0.75\), showing which interval contains the solution. (Answer: \(e_{0.75}=26.67\), rounded to the nearest cent; the solution lies between 0 and 40.)
- A hypothetical one-period loss is 0 or 80, each with probability 0.50. With no discounting or mortality adjustment, calculate entropic risk for \(\theta=\ln(2)/80\). (Answer: \(\rho_{\theta}=80\ln(1.5)/\ln(2)\approx46.80\).)
- A decision rule is specifically framed as “the average loss among the worst 5% of outcomes,” and the loss distribution has a point mass at the percentile boundary. Which measure directly matches the objective, and what definition should be used to avoid mishandling the boundary? (Answer: TVaR at \(\alpha=0.95\), using the quantile-integral definition or an equivalent formula that accounts for boundary mass.)