When VaR and TVaR Rank Portfolios Differ

Why Portfolio Rankings Can Reverse

Value-at-Risk and Tail Value-at-Risk summarize different parts of a loss distribution. VaR identifies a loss quantile at a chosen confidence level. TVaR averages losses in the upper tail beyond that level, accounting carefully for any probability concentrated at the quantile. A portfolio can therefore have a lower VaR but a higher TVaR than another portfolio: its selected quantile is lower, but its losses farther into the tail are sufficiently severe.

A useful comparison does more than report two rankings. It asks what creates them. Probability mass near the quantile determines where the cutoff falls and how much of an atom must be included in the tail. Losses beyond the cutoff determine the tail excess and can make TVaR much larger than VaR. The calculations below use hypothetical portfolios and discrete loss distributions.

Definition: Let $X$ be a loss with cumulative distribution function $F_X(x)=P(X\le x)$, and let $0<p<1$ be a confidence level. Under the lower-quantile convention, $$ \operatorname{VaR}_p(X)=\inf\{x:F_X(x)\ge p\}. $$ For an integrable loss, define $$ \operatorname{TVaR}_p(X)=\frac{1}{1-p}\int_p^1 \operatorname{VaR}_q(X)\,dq. $$ The integral definition applies to both continuous and discrete distributions. In a discrete distribution, it includes only the amount of probability needed to make the upper tail have probability $1-p$; it does not generally equal a conditional mean given $X>\operatorname{VaR}_p(X)$.

Calculating TVaR at a Discrete Quantile

Write $v=\operatorname{VaR}_p(X)$. The probability strictly above $v$ is $1-F_X(v)$, which may be less than $1-p$. If so, the upper tail must be completed using probability mass at $v$. The required amount of that mass is $F_X(v)-p$. This is a probability amount, not necessarily all the mass at $v$.

\operatorname{TVaR}_p(X) = \frac{E[X\,\mathbf{1}_{\{X>v\}}]+v\bigl(F_X(v)-p\bigr)}{1-p}

Here $\mathbf{1}_{\{X>v\}}$ equals 1 when $X>v$ and 0 otherwise. The formula can also be written in terms of excess loss above the quantile:

\operatorname{TVaR}_p(X) = v+\frac{E[(X-v)_+]}{1-p}, \qquad (X-v)_+=\max(X-v,0).

This second form separates the quantile level from the average excess contributed by losses above it. The excess is not determined by the size of the quantile alone: both the probability of exceeding the quantile and the sizes of those exceedances matter.

Find the quantile.
Accumulate probabilities from the smallest loss upward. The first loss whose cumulative probability is at least $p$ is $v$.

Measure the probability above it.
Calculate $P(X>v)$. If this is less than $1-p$, the shortfall is supplied by the atom at $v$.

Average the exact upper-tail amount.
Include all losses strictly above $v$, plus probability $F_X(v)-p$ at $v$, then divide the corresponding loss total by $1-p$.

Worked Comparisons

Worked Example: A Higher VaR but a Lower TVaR

At confidence level $p=0.90$, compare two hypothetical annual portfolio losses, measured in dollars. Portfolio A has loss 0 with probability 0.89, loss 100 with probability 0.02, and loss 1,000 with probability 0.09. Portfolio B has loss 0 with probability 0.89, loss 200 with probability 0.02, and loss 500 with probability 0.09. Find both risk measures and compare the rankings.

Method. Accumulate the probabilities to find each lower quantile, then average the largest 10% of losses. This avoids treating the entire probability at the quantile as tail probability when only part of it is needed.

Portfolio A VaR. Its cumulative probability is 0.89 at loss 0 and 0.91 at loss 100. Thus the first cumulative probability at least 0.90 occurs at 100:

\operatorname{VaR}_{0.90}(A)=100.

The upper tail needs probability 0.10. Loss 1,000 contributes probability 0.09, so the remaining 0.01 is taken at loss 100. Its total loss amount is $0.09(1{,}000)+0.01(100)=90+1=91$. Therefore:

\operatorname{TVaR}_{0.90}(A)=\frac{91}{0.10}=910.

Portfolio B VaR. The cumulative probability is 0.89 at loss 0 and 0.91 at loss 200, so $\operatorname{VaR}_{0.90}(B)=200$. The upper 10% consists of probability 0.09 at loss 500 and probability 0.01 at loss 200. Its loss total is $0.09(500)+0.01(200)=45+2=47$, giving:

\operatorname{TVaR}_{0.90}(B)=\frac{47}{0.10}=470.

Portfolio B has the higher VaR, $200>100$, but Portfolio A has the higher TVaR, $910>470$. The reason is the severity beyond each quantile. For A, the excess above 100 has expectation $0.09(1{,}000-100)=81$, so its TVaR is $100+81/0.10=910$. For B, the excess above 200 has expectation $0.09(500-200)=27$, so its TVaR is $200+27/0.10=470$. These results agree with the direct tail averages.

Worked Example: Mass at the Quantile Versus a Severe Tail

At $p=0.90$, Portfolio C has loss 0 with probability 0.89 and loss 100 with probability 0.11. Portfolio D has loss 0 with probability 0.80, loss 90 with probability 0.10, and loss 1,000 with probability 0.10. Compare their VaR and TVaR.

Portfolio C. The cumulative probability reaches 0.89 at 0 and 1.00 at 100, so $\operatorname{VaR}_{0.90}(C)=100$. All losses are at most 100, and the upper 10% can be selected from the probability mass at 100. Thus $\operatorname{TVaR}_{0.90}(C)=100$. In the discrete formula, $F_C(100)-p=1.00-0.90=0.10$, so the selected tail loss total is $100(0.10)=10$, and $10/0.10=100$.

Portfolio D. Its cumulative probability is exactly 0.90 at loss 90. The lower-quantile convention therefore gives $\operatorname{VaR}_{0.90}(D)=90$, not 1,000. The probability strictly above 90 is 0.10, all at loss 1,000; there is no additional probability at 90 to include. Hence:

\operatorname{TVaR}_{0.90}(D)=\frac{0.10(1{,}000)}{0.10}=1{,}000.

C has the higher VaR, $100>90$, while D has the higher TVaR, $1{,}000>100$. C's 10% upper tail is drawn from a large atom exactly at its quantile. D has a 10% probability of a much more severe loss strictly above its quantile. The two measures are ranking portfolios using different features of the distribution.

Worked Example: A Reversal at a Different Confidence Level

At confidence level $p=0.95$, Portfolio E has loss 0 with probability 0.93, loss 100 with probability 0.04, and loss 1,000 with probability 0.03. Portfolio F has loss 0 with probability 0.93, loss 150 with probability 0.03, and loss 400 with probability 0.04. Find the rankings.

Portfolio E. Its cumulative probability is 0.93 at 0 and 0.97 at 100, so its VaR is 100. The upper tail has probability 0.05: it includes all 0.03 probability at 1,000 and 0.02 at 100. The loss total is $0.03(1{,}000)+0.02(100)=30+2=32$, giving:

\operatorname{TVaR}_{0.95}(E)=\frac{32}{0.05}=640.

Portfolio F. Its cumulative probability is 0.93 at 0 and 0.96 at 150, so its VaR is 150. The upper 5% includes all 0.04 probability at 400 and 0.01 at 150. The loss total is $0.04(400)+0.01(150)=16+1.5=17.5$, and:

\operatorname{TVaR}_{0.95}(F)=\frac{17.5}{0.05}=350.

F has the higher VaR, $150>100$, but E has the higher TVaR, $640>350$. Using tail excess makes the comparison checkable: E's expected excess above 100 is $0.03(900)=27$, giving $100+27/0.05=640$. F's expected excess above 150 is $0.04(250)=10$, giving $150+10/0.05=350$. The deeper losses in E outweigh its lower quantile.

How to Diagnose a Reversal

A ranking reversal is a signal to inspect the distributions at and beyond the chosen quantiles. First compare the VaR values. Then calculate $P(X>v)$, the amount of probability at $v$ needed to complete the upper tail, and the excess $E[(X-v)_+]$. A large excess can arise from a relatively frequent set of moderate exceedances, a small probability of very large losses, or a combination of the two. The excess formula summarizes their joint effect; inspecting the individual probabilities and severities explains where it comes from.

In particular, a higher VaR does not guarantee a higher TVaR. Conversely, a lower VaR does not imply a safer portfolio throughout the tail. The confidence level also matters: changing $p$ changes both the quantile and the portion of the distribution averaged by TVaR. The comparison is meaningful only when the portfolios use the same loss definition, units, time horizon, and confidence level.

Derivation: In the upper $1-p$ probability of the quantile distribution, all outcomes above $v$ are included. They contribute $E[X\mathbf{1}_{\{X>v\}}]$ to the loss total. Since the cumulative probability at or below $v$ is $F_X(v)$, the amount of probability at $v$ needed to fill the upper tail is $F_X(v)-p$. This contributes $v(F_X(v)-p)$. Dividing their sum by $1-p$ gives the discrete TVaR formula. Subtract $v(1-p)$ from its numerator: the result is $E[(X-v)\mathbf{1}_{\{X>v\}}]=E[(X-v)_+]$. Dividing through yields $\operatorname{TVaR}_p(X)=v+E[(X-v)_+]/(1-p)$.

Common Mistakes

Using the next loss after the quantile. If $F_X(v)=p$, then $v$ already meets the lower-quantile definition. For example, if losses are 0 with probability 0.90 and 200 with probability 0.10, then $\operatorname{VaR}_{0.90}=0$, not 200. The upper 10% is entirely the loss 200, so $\operatorname{TVaR}_{0.90}=200$.
Averaging all losses at or above VaR without checking probability. The atom at $v$ may be larger than the amount needed to make a $1-p$ tail. Include only $F_X(v)-p$ probability at $v$.
Using $E[X\mid X>v]$ for a discrete distribution. This omits the portion at $v$ needed to complete the tail. It is not generally TVaR when there is an atom at the quantile.
Explaining a reversal using VaR alone. VaR gives the cutoff, not the severity beyond it. Compute the tail excess or the exact tail average before drawing a conclusion.

Careful quantile conventions and exact tail probabilities are central to reliable comparisons and are aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus. In practice, reporting both the ranking and the tail calculation makes clear whether a difference comes from the cutoff, the probability around it, or losses deeper in the tail.

Check Your Understanding

For each hypothetical loss distribution, use the lower-quantile convention and include only the required probability at an atom.

At $p=0.80$, losses are 0 with probability 0.75, 40 with probability 0.10, and 300 with probability 0.15. Find VaR and TVaR. (Answer: VaR = $40; TVaR = $235.)
At $p=0.90$, Portfolio A has losses 0, 80, and 500 with probabilities 0.88, 0.04, and 0.08. Portfolio B has losses 0, 120, and 250 with probabilities 0.88, 0.02, and 0.10. Find both rankings. (Answer: A: VaR = $80, TVaR = $416; B: VaR = $120, TVaR = $250; B has higher VaR and A has higher TVaR.)
At $p=0.95$, losses are 0 with probability 0.90, 200 with probability 0.06, and 900 with probability 0.04. Find VaR and TVaR. (Answer: VaR = $200; TVaR = $760.)
At $p=0.90$, losses are 0 with probability 0.90, 75 with probability 0.05, and 1,000 with probability 0.05. Find VaR and TVaR, and explain why the quantile is not 75. (Answer: VaR = $0; TVaR = $537.50; the cumulative probability at 0 is already 0.90.)
At $p=0.90$, Portfolio C has losses 0, 100, and 600 with probabilities 0.85, 0.10, and 0.05. Portfolio D has losses 0, 150, and 400 with probabilities 0.88, 0.03, and 0.09. Find both pairs of risk measures and state whether the rankings reverse. (Answer: C: VaR = $100, TVaR = $350; D: VaR = $150, TVaR = $375; the rankings do not reverse.)

What You'll Learn