Welcome to Risk Measures!

In our journey through CM2 (Economic Modelling), we’ve already looked at how to calculate returns. But as any actuary will tell you, returns are only half the story. The other half is Risk.

In this chapter, we are going to look at how we define "risk" mathematically. We aren't just looking for a single number; we are looking for a set of rules (properties) that a "good" risk measure should follow. This helps us compare different investment opportunities fairly. Don't worry if the math looks a bit abstract at first—we'll break it down into simple concepts and real-world logic.

Quick Review: What is a Risk Measure?
Think of a risk measure, denoted as \(\rho(X)\), as a "fear gauge" for a random variable \(X\) (which represents our investment loss or return). It’s a function that takes the uncertainty of an investment and turns it into a single number that tells us how much capital we might need to set aside to cover potential losses.

1. The Four Properties of a Coherent Risk Measure

To ensure a risk measure is "sensible," mathematicians (specifically Artzner, Delbaen, Eber, and Heath) proposed four properties. If a risk measure satisfies all four, we call it a Coherent Risk Measure.

Memory Aid: Think of the mnemonic "M-S-P-T" (Must Study Professional Techniques) to remember these four properties.

A. Monotonicity

The Concept: If Investment A always performs worse than Investment B (i.e., its losses are always greater or equal), then Investment A must be riskier.

The Math: If \(X \le Y\) for all possible outcomes, then \(\rho(X) \ge \rho(Y)\).
Note: In CM2, we often define \(X\) as the "payoff." If the payoff is lower, the risk is higher.

Real-World Analogy: If you have two cars, and Car A breaks down more often than Car B in every possible weather condition, Car A is clearly the riskier choice for a road trip.

B. Sub-additivity

The Concept: This is the "diversification" rule. The risk of two investments combined should be less than (or equal to) the sum of their individual risks.

The Math: \(\rho(X + Y) \le \rho(X) + \rho(Y)\).

Why it matters: This is a vital property for actuaries. If a risk measure doesn't follow this, it implies that merging two portfolios makes them more dangerous, which contradicts the fundamental principle of diversification ("don't put all your eggs in one basket").

C. Positive Homogeneity

The Concept: If you double the size of your investment, you double the risk.

The Math: \(\rho(aX) = a\rho(X)\) for any \(a > 0\).

Simple Explanation: If you bet £10 on a coin toss, your risk is "X". If you bet £20 (doubling your stake), your risk is "2X". There are no "bulk discounts" on risk here!

D. Translation Invariance

The Concept: If you add a guaranteed, risk-free amount of cash to your portfolio, your total risk decreases by exactly that amount.

The Math: \(\rho(X + c) = \rho(X) - c\), where \(c\) is a constant amount of cash.

Real-World Analogy: If you are worried about a £1,000 loss, but someone hands you £200 in cash to keep in your pocket, your "net risk" drops to £800. The cash acts as a buffer.

Summary of Coherence:

Key Takeaway: A risk measure is Coherent if it satisfies Monotonicity, Sub-additivity, Positive Homogeneity, and Translation Invariance. If it fails even one, it isn't coherent!

2. Comparing Common Risk Measures

Now that we have our "ruler" (the properties), let's look at the two most common measures you’ll encounter in the CM2 exam: Value at Risk (VaR) and Tail Value at Risk (TVaR).

Value at Risk (VaR)

What is it? VaR is the maximum loss you expect to suffer over a certain time period with a given probability (confidence level). For example, a 95% VaR of £1m means there is only a 5% chance your loss will exceed £1m.

Is it Coherent?
NO. This is a frequent exam question! While VaR usually satisfies Monotonicity, Positive Homogeneity, and Translation Invariance, it often fails Sub-additivity.

Common Mistake: Students often assume VaR is always sub-additive. It is actually only sub-additive if the underlying returns follow a Normal Distribution (or other elliptical distributions). For many "fat-tailed" distributions, VaR can suggest that diversifying actually increases risk, which is why regulators often prefer other measures.

Tail Value at Risk (TVaR) / Expected Shortfall

What is it? TVaR doesn't just look at the threshold (like VaR); it looks at the average of all losses that are worse than the VaR. It asks: "If things go wrong, how bad will they be on average?"

The Math: \(TVaR_p = E[X | X > VaR_p]\).

Is it Coherent?
YES. TVaR satisfies all four properties, including sub-additivity. This makes it a theoretically "better" measure for comparing risks than VaR.

Quick Review Box:

VaR: Easy to explain, but ignores what happens in the extreme "tail" and isn't always coherent.
TVaR: More difficult to calculate, but captures tail risk and is always coherent.

3. Using Risk Measures to Compare Investment Opportunities

In your exam, you might be asked to choose between Project A and Project B based on risk. Here is a step-by-step approach to help you decide:

Step 1: Identify the Distribution
Are the returns Normally distributed? If yes, VaR and Variance are generally reliable. If the returns are skewed (have a "heavy tail"), you should be very careful using VaR.

Step 2: Check for "Tail Risk"
If an investment has a small chance of a catastrophic loss (like a 1-in-100 year flood), VaR might completely miss it if you only look at the 95th percentile. TVaR is better here because it averages those extreme "end-of-the-world" scenarios.

Step 3: Consider the "Upside"
Important Point: Standard Deviation and Variance are often used as risk measures, but they penalize good outcomes too! (They treat a huge unexpected profit as "risk"). Risk measures like Semi-variance only look at outcomes below the mean, which many investors find more useful.

Did you know?
The Basel III and Solvency II regulations (the "rulebooks" for banks and insurers) have moved increasingly toward using TVaR-like measures because they want to ensure companies have enough capital to survive the absolute worst-case scenarios, not just "mostly bad" ones.

4. Final Summary and Key Takeaways

To master this chapter for your CM2 exam, remember these core points:

  • Coherence is the Gold Standard: A measure must be Monotonic, Sub-additive, Positively Homogeneous, and Translation Invariant to be coherent.
  • The VaR Flaw: Value at Risk is popular because it's simple, but its failure of Sub-additivity is its greatest weakness. Diversifying shouldn't look "riskier" on paper!
  • TVaR is the Hero: It captures the severity of the tail and satisfies all coherence properties.
  • Context Matters: When comparing investments, always look at the shape of the distribution. If there's a "fat tail," look beyond the VaR.

Don't worry if this seems tricky at first! The math of coherence properties is just a way of putting common sense into formulas. Keep practicing the definitions, and they will become second nature.