Value at Risk and Tail Value at Risk

A pension fund asks: "What is our worst-case loss at the 99th percentile?" That is VaR. "If we land in that worst 1%, how bad on average?" That is TVaR. The FAM exam tests whether you can compute both for continuous and discrete distributions, and whether you know that the discrete case requires the stop-loss formula, not a naive conditional expectation.

HIGH-FREQUENCY: VaR definition and computation at a given confidence level appear on virtually every FAM exam.

$${\text{VaR}}_p(X)=F_X^{-1}(p)=\inf \{x:F_X(x)\ge p\}$$

For continuous distributions with strictly increasing CDF, solve $F_X(x)=p$.

Exponential (mean $\theta$):

$${\text{VaR}}_p(X)=-\theta \ln (1-p)$$

Normal ($\mu ,\sigma$):

$${\text{VaR}}_p(X)=\mu +\sigma z_p$$

Pareto ($\alpha ,\theta$):

$${\text{VaR}}_p(X)=\theta \left[(1-p{)}^{-1/\alpha }-1\right]$$

Discrete VaR: Find the smallest $x$ where $F_X(x)\ge p$. If $F_X(3)=0.93$ and $F_X(4)=0.97$, then ${\text{VaR}}_{0.95}=4$.

Common mistakes

• Using $\ln (p)$ instead of $\ln (1-p)$ for exponential VaR. Solving $1-e^{-x/\theta }=p$ gives $x=-\theta \ln (1-p)$. Using $\ln (0.95)$ at the 95th percentile gives 256 instead of the correct 14,979. Trap: a VaR near the mean instead of deep in the tail.
• Using the wrong CDF direction for discrete VaR. VaR is the smallest $x$ where $F_X(x)\ge p$, not the largest where $F_X(x)<p$. With CDF at 0.85 and 0.95, ${\text{VaR}}_{0.90}=50$, not 20. Trap: reporting 20 because the CDF at 20 is "close to" 0.90.
• Forgetting the $1/(1-p)$ factor in TVaR. The stop-loss premium alone is $E[(X-d{)}_{+}]$. Dividing by $1-p$ converts it to a conditional expectation. Without division, you get 2.50 instead of 75. Trap: reporting the stop-loss premium as TVaR.

Bottom line

• VaR: ${\text{VaR}}_p(X)=F_X^{-1}(p)$, the $p$-th quantile (the tail boundary)
• TVaR (continuous): ${\text{TVaR}}_p(X)=E[X\mid X>{\text{VaR}}_p(X)]$, the average loss beyond the boundary
• TVaR (discrete): ${\text{TVaR}}_p(X)={\text{VaR}}_p(X)+\pi ({\text{VaR}}_p(X))/(1-p)$ via stop-loss
• Exponential: ${\text{VaR}}_p=-\theta \ln (1-p)$ and ${\text{TVaR}}_p={\text{VaR}}_p+\theta$ (memoryless, gap independent of $p$)

Exam shortcut

Before computing, identify whether the distribution is continuous or discrete. For continuous, VaR is a straightforward quantile inversion and TVaR is a conditional expectation. For discrete, VaR requires checking the CDF at each jump, and TVaR requires the stop-loss formula. Write "continuous" or "discrete" at the top of your work. Remember: "VaR = boundary, TVaR = beyond." For the exponential, "add one mean" ($\text{TVaR}=\text{VaR}+\theta$.

The full lesson (about 2,083 words, 14 min read) adds 2 worked examples, all 5 common mistakes, a self-check, free in the app.

Learning objectives

• 2m

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