Measures of Financial Risk

Welcome to Measures of Financial Risk!

Hello there! Today, we are diving into one of the most important chapters in the Valuation and Risk Models section of the FRM curriculum. Think of risk measurement as the "dashboard" of a bank or investment firm. Just like a pilot needs instruments to know how high or fast they are flying, a risk manager needs specific tools to know how much money the firm could lose. By the end of these notes, you’ll understand the "Big Two" of risk metrics—Value at Risk (VaR) and Expected Shortfall (ES)—and how to judge if a risk measure is actually doing its job.

1. Value at Risk (VaR): The Industry Standard

Value at Risk (VaR) is the most famous number in finance. It answers the question: "What is the maximum amount I can expect to lose over a certain time period with a given level of confidence?"

VaR is defined by two key parameters:
1. Time Horizon (T): The period over which we measure risk (e.g., 1 day, 10 days).
2. Confidence Level (\(1 - \alpha\)): Usually 95% or 99%.

Example: If a portfolio has a 1-day 95% VaR of $1 million, it means that there is only a 5% chance the portfolio will lose more than $1 million in a single day. Or, put another way, we are 95% confident our losses will be $1 million or less.

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Calculating VaR with the Normal Distribution

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If we assume returns follow a normal distribution (the "bell curve"), the formula is:
\n\( VaR = \mu - (z \times \sigma) \)
\nWhere:
\n- \( \mu \) is the expected return (often assumed to be 0 for short horizons).
\n- \( \sigma \) is the standard deviation (volatility).
\n- \( z \) is the critical value from the normal table (e.g., 1.65 for 95% or 2.33 for 99%).

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Quick Tip: In the FRM exam, if they ask for the "95% VaR," you usually look at the 5% tail. Don't worry if this seems tricky at first—just remember that a higher confidence level (99% vs 95%) will always result in a higher (larger) VaR number because you are looking further into the "scary" tail of the distribution.

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Common Mistake to Avoid: VaR does not tell you the absolute worst-case scenario. It only tells you the boundary. If your VaR is $1 million, you could lose $1.1 million, $10 million, or everything! VaR is silent about what happens in that "extreme tail."

2. Expected Shortfall (ES): Looking Beyond the Boundary

If VaR is a "boundary" line, Expected Shortfall (ES)—also known as Conditional VaR (CVaR)—is the average of everything beyond that line. It answers: "If things go really wrong and we exceed our VaR, how much should we expect to lose on average?"

The Analogy: Imagine you are jumping over a hole. VaR tells you how wide the hole is. ES tells you how deep the hole is if you happen to fall in.

Why use ES?

• It captures tail risk better than VaR.
• It is more sensitive to the shape of the loss distribution beyond the confidence level.
• It satisfies a mathematical property called subadditivity (which VaR sometimes fails—more on this in a moment!).

Key Takeaway: For the same confidence level, ES is always greater than or equal to VaR because it represents the average of the "worst-case" losses.

3. Coherent Risk Measures: The "Golden Rules"

In 1999, researchers (Artzner et al.) decided that for a risk measure to be considered "good" or "coherent," it must follow four mathematical rules. You can remember them with the mnemonic "M-S-H-T" (My Super Hero Team):

1. Monotonicity: If Portfolio A always has worse returns than Portfolio B in every possible scenario, then the risk of A must be higher than the risk of B.
2. Subadditivity: The risk of two portfolios combined should be less than or equal to the sum of their individual risks. (Diversification should reduce risk!)
3. Positive Homogeneity: If you double the size of your portfolio, you double your risk. \( Risk(\lambda \times X) = \lambda \times Risk(X) \).
4. Translation Invariance: If you add cash (\(k\)) to a portfolio, its risk decreases by exactly that amount of cash. \( Risk(X + k) = Risk(X) - k \).

Did you know? VaR is NOT always coherent. Specifically, it can fail the Subadditivity test when dealing with non-normal distributions (like options or credit bonds). This is a huge reason why regulators are moving toward ES, which is coherent.

4. Estimating VaR and ES: Three Main Methods

How do we actually get these numbers in the real world? There are three main "flavors":

Method A: Historical Simulation (The "Look Back" Method)

You take the last 500 or 1,000 days of actual market data and apply those changes to your current portfolio.
- Pros: No need to assume a "bell curve" (normal distribution); captures "fat tails."
- Cons: Assumes the future will look exactly like the past.

Method B: Parametric / Variance-Covariance (The "Math" Method)

You assume a distribution (usually normal) and use means and standard deviations.
- Pros: Very fast and easy to calculate.
- Cons: It fails miserably during market crashes because it underestimates the probability of extreme events (fat tails).

Method C: Monte Carlo Simulation (The "Computer" Method)

A computer generates thousands of random "what-if" scenarios based on specific rules you give it.
- Pros: Can model complex things like options and exotic derivatives.
- Cons: Computationally expensive and slow; "garbage in, garbage out" (if your rules are wrong, the results are wrong).

5. Spectral Risk Measures

This is a slightly more advanced concept, but think of it as a weighted average. A Spectral Risk Measure assigns weights to different outcomes based on how much the investor hates risk (risk aversion).
- ES is actually a special case of a spectral risk measure where all losses beyond the VaR threshold are weighted equally, and everything else is weighted zero.
- To be coherent, the weights in a spectral risk measure must be non-negative and increasing (meaning you care more about bigger losses).

6. Backtesting: Checking Your Work

How do you know if your VaR model is working? You Backtest it! You look at your past VaR predictions and compare them to the actual losses that occurred.

If you have a 95% daily VaR model, you expect to exceed that loss about 5% of the time (e.g., 5 days out of 100).
- If you exceed it 20 times in 100 days, your model is underestimating risk.
- If you never exceed it, your model might be too conservative (overestimating risk), which means you are tying up too much capital.

Quick Review Box

- VaR: The loss level that will not be exceeded with \(X\%\) confidence.
- ES: The average loss given that the VaR has been exceeded.
- Coherence: Requires Monotonicity, Subadditivity, Homogeneity, and Translation Invariance.
- The VaR Flaw: VaR is not always subadditive; ES is.
- Historical Simulation: Uses real past data; no distribution assumption.
- Normal VaR Formula: \( \mu - z \sigma \).

Don't feel overwhelmed! The core of this chapter is simply understanding how we quantify the "downside." If you can remember that VaR is the "line in the sand" and ES is "the average of the mess behind that line," you are already halfway there!