Value at Risk, Tail Value at Risk, and risk measure properties

Welcome to the World of Risk Measures!

In your journey through Exam FAM, you've looked at how to model losses using Severity (how much) and Frequency (how often). But once we have those models, how do we actually "measure" the risk? How does an insurance company decide exactly how much money to keep in the bank to avoid going bankrupt?

In this chapter, we will learn about Value at Risk (VaR), Tail Value at Risk (TVaR), and the formal Properties that make a risk measure "good" (which we call "coherence"). Don't worry if these sound like scary financial terms—at their heart, they are just fancy ways of looking at percentiles and averages!

1. Value at Risk (VaR)

Value at Risk (VaR) is simply a specific percentile of a loss distribution. If you are asked to find the \(VaR_{0.95}\), you are looking for the dollar amount such that there is a 95% chance the loss will be less than that amount, and only a 5% chance it will be higher.

The Formula

For a loss random variable \(X\), the VaR at confidence level \(p\) is:
\(VaR_p(X) = \pi_p = F_X^{-1}(p)\)
Where \(\pi_p\) is the \(100p^{th}\) percentile of the distribution.

An Everyday Analogy

Imagine you are commute to work. If we say the 95th percentile (VaR) of your commute time is 45 minutes, it means that 95% of the time, you get to work in 45 minutes or less. Only on the 5% "worst" days (accidents, snowstorms) does it take longer than 45 minutes.

Common Mistakes to Avoid

Confusing Probability and Dollars: VaR is a dollar amount (the loss), not a percentage. If \(p = 0.99\), the VaR is the dollar amount at that 99th percentile, not the 0.99 itself.

Quick Review: VaR

• What is it? A percentile (\(\pi_p\)).
• What does it tell us? The maximum loss we expect to see, excluding the very worst \(1-p\) cases.
• Calculations: For a continuous distribution, just solve \(F(x) = p\) for \(x\).

2. Tail Value at Risk (TVaR)

VaR is great, but it has a weakness: it doesn't care *how bad* the losses are once they cross the threshold. Tail Value at Risk (TVaR) fixes this. It asks: "If we are in that worst 5% of cases, what is the average loss we should expect?"

The Formula

TVaR is the expected value of the loss, given that the loss has exceeded the VaR:
\(TVaR_p(X) = E[X | X > VaR_p(X)]\)
For continuous distributions, this can also be calculated as:
\(TVaR_p(X) = \frac{1}{1-p} \int_{p}^{1} VaR_u(X) du\)

Useful Calculation Shortcuts

On the exam, you often don't want to do a hard integral. Use these relationships instead:
1. Using the Mean Excess Loss function \(e(d)\):
\(TVaR_p(X) = VaR_p(X) + e(VaR_p(X))\)
2. Using the Limited Expected Value \(E[X \wedge x]\):
\(TVaR_p(X) = VaR_p(X) + \frac{E[X] - E[X \wedge VaR_p(X)]}{1-p}\)

Did you know? TVaR is also sometimes called "Expected Shortfall" or "Conditional Value at Risk (CVaR)" in other finance textbooks!

Quick Review: TVaR

• What is it? The average loss in the "tail" of the distribution.
• Key Insight: \(TVaR_p(X)\) is always greater than or equal to \(VaR_p(X)\).
• Memory Trick: VaR is the "boundary" of the bad zone; TVaR is the "average" of the bad zone.

3. Properties of Risk Measures (Coherence)

In the late 1990s, mathematicians decided that for a risk measure (\(\rho\)) to be "logical" or "coherent," it must follow four specific rules. These are common exam questions!

1. Translation Invariance

\(\rho(X + c) = \rho(X) + c\)
Simple meaning: If you add a guaranteed loss of \(c\) dollars to your portfolio, your total risk measure goes up by exactly \(c\).

2. Positive Homogeneity

\(\rho(cX) = c\rho(X)\) for \(c > 0\)
Simple meaning: If you double the size of your portfolio (double the losses), you double your risk measure. There are no "bulk discounts" or "penalties" for scale here.

3. Subadditivity

\(\rho(X + Y) \leq \rho(X) + \rho(Y)\)
Simple meaning: Diversification is good! The risk of two things combined should be less than or equal to the sum of their individual risks.
CRITICAL POINT: VaR is NOT always subadditive, which is why many experts prefer TVaR. TVaR IS subadditive.

4. Monotonicity

If \(X \leq Y\) for all possible outcomes, then \(\rho(X) \leq \rho(Y)\).
Simple meaning: If Loss B is always worse than Loss A, then the risk measure for B must be higher than the risk measure for A.

Memory Aid: To remember the four properties, think of "MSPT": Monotonicity, Subadditivity, Positive Homogeneity, and Translation Invariance. ("My Stats Professor is Tough")

Key Takeaway

A risk measure is called Coherent if it satisfies all four of these properties. TVaR is coherent, but VaR is not (because it can fail subadditivity).

4. Working with Common Distributions

The exam will often ask you to calculate VaR or TVaR for specific distributions. Here is a "cheat sheet" for the two most common ones:

The Normal Distribution

If \(X \sim N(\mu, \sigma^2)\):
• \(VaR_p(X) = \mu + \sigma \cdot \Phi^{-1}(p)\)
• \(TVaR_p(X) = \mu + \sigma \cdot \frac{\phi(\Phi^{-1}(p))}{1-p}\)
(Where \(\phi\) is the standard normal PDF and \(\Phi^{-1}\) is the z-score from your tables).

The Exponential Distribution

If \(X \sim Exp(\theta)\) (where \(\theta\) is the mean):
• \(VaR_p(X) = -\theta \ln(1-p)\)
• \(TVaR_p(X) = VaR_p(X) + \theta\)
Note: This is because the Exponential distribution is "memoryless," so the average amount above the threshold is always just the mean \(\theta\)!

Summary Table: VaR vs. TVaR

VaR:
- Logic: "What is the minimum loss of the worst-case 5%?"
- Calculation: Simple Percentile.
- Coherent? No (fails subadditivity).
- Information: Ignores the severity of the "extreme" tail.

TVaR:
- Logic: "What is the average loss of the worst-case 5%?"
- Calculation: Conditional Expectation.
- Coherent? Yes.
- Information: Accounts for how heavy the tail is.

Keep practicing! Risk measures are one of the more "intuitive" parts of Exam FAM once you get past the notation. You've got this!