Measuring Returns, Volatility, and Correlation

Q: What are the two primary advantages of using log returns in financial modeling?

Log returns are advantageous because of their time additivity and because they often fit normal distribution models better than simple returns.

Q: What is the difference between covariance and correlation?

Covariance indicates the direction of a relationship, while correlation is a standardized measure between -1 and +1 indicating both direction and strength.

FRM Part I · FRM · 5 min read

Welcome to the World of Risk Measurement!

Hi there! Welcome to one of the most important chapters in your FRM Part I journey. Before we can manage risk, we have to measure it. Think of this chapter as the "thermometer" of finance. Just as a doctor checks your temperature and heart rate to see how you're doing, risk managers check returns, volatility, and correlation to see how a portfolio is "feeling."

Don't worry if these terms sound a bit math-heavy at first. We’re going to break them down into simple, bite-sized pieces using real-world examples. By the end of this, you’ll be talking about GARCH and EWMA like a pro!

1. Returns: Discrete vs. Continuous

Before we can talk about risk, we need to know how much money we made or lost. In finance, we look at returns.

Simple (Discrete) Returns

This is the return you probably used in school. If you bought a stock for \$100 and it’s now \$110, your return is 10%.
Formula: $ R_t = \frac{P_t - P_{t-1}}{P_{t-1}} $

Logarithmic (Continuously Compounded) Returns

In the FRM world, we love Log Returns. Why? Because they make math easier over long periods! If you have a series of log returns, you can just add them up to find the total return.
Formula: $ r_t = \ln(\frac{P_t}{P_{t-1}}) $

Quick Review Box: Why use Log Returns?
• Time Additivity: You can add daily log returns to get a weekly return.
• Normal Distribution: They often fit our statistical models better than simple returns.

Common Mistake to Avoid: Don't mix these up! If the exam asks for a "continuously compounded return," always reach for the ln (natural log) button on your calculator.

2. Volatility: The Measure of Fear

Volatility (standard deviation) tells us how much a stock's price "wiggles" around its average. High volatility means high risk (and potentially high reward). Low volatility means a smoother, steadier ride.

The Standard Formula:
Historically, we calculate variance ($ \sigma^2 $) by looking at the average of squared deviations from the mean:
$ \sigma^2 = \frac{1}{m-1} \sum_{i=1}^{m} (u_{n-i} - \bar{u})^2 $

Analogy: Imagine two drivers. Driver A stays at exactly 60 mph. Driver B constantly speeds up to 80 and slows down to 40. Both have an average speed of 60, but Driver B is much more "volatile" and risky!

Key Takeaway: In risk management, we often assume the mean return ($ \bar{u} $) is zero for very short periods (like one day) to simplify the math. This is a common FRM convention!

3. EWMA: Weighting the Recent Past

Standard volatility models treat every day in the past as equally important. But does a market crash from three years ago matter as much as what happened yesterday? Probably not! This is where the Exponentially Weighted Moving Average (EWMA) comes in.

The Concept: We give the most weight to the most recent data and "fade" the importance of older data as we go back in time.

The Formula:
$ \sigma_n^2 = \lambda \sigma_{n-1}^2 + (1-\lambda) u_{n-1}^2 $

Memory Aid (The "Lambda" Rule):
• $\lambda$ (Lambda): Usually 0.94 for daily data (RiskMetrics standard).
• If $\lambda$ is high (close to 1), volatility changes slowly (it's "sticky").
• If $\lambda$ is low, volatility reacts very quickly to new information.

Did you know? The EWMA is actually a special case of the GARCH model we are about to learn!

4. GARCH (1,1): Adding Mean Reversion

If EWMA is the "cool younger sibling," GARCH (1,1) is the "sophisticated older sibling." GARCH stands for Generalized Autoregressive Conditional Heteroskedasticity (don't worry, you don't need to say that three times fast!).

The Big Difference: GARCH assumes that volatility eventually returns to a long-run average level. This is called Mean Reversion.

The GARCH (1,1) Recipe:
The variance is made of three ingredients:
1. A weighted long-run average variance ($ \omega $)
2. Yesterday’s "news" or squared return ($ \alpha u_{n-1}^2 $)
3. Yesterday’s predicted volatility ($ \beta \sigma_{n-1}^2 $)

Formula:
$ \sigma_n^2 = \omega + \alpha u_{n-1}^2 + \beta \sigma_{n-1}^2 $

Important Condition: For the model to be stable and return to the mean, $ \alpha + \beta $ must be less than 1. If $ \alpha + \beta = 1 $, you actually have the EWMA model!

Key Takeaway: GARCH is better for long-term forecasting because it acknowledges that "crazy" markets eventually calm down and return to normal.

5. Correlation and Covariance: How Assets Dance Together

Risk isn't just about how one stock moves; it's about how they move together. This is the heart of diversification.

Covariance: Tells us the direction of the relationship. Positive means they move together; negative means they move in opposite directions.
Correlation ($ \rho $): A "standardized" version of covariance that always stays between -1 and +1.

The Scale of Correlation:
• +1.0: Perfect Harmony. They move exactly together.
• 0: No relationship. They are like strangers on a subway.
• -1.0: Perfect Opposites. When one goes up, the other goes down.

Step-by-Step Calculation:
To find Correlation ($ \rho_{xy} $):
1. Calculate the Covariance ($ Cov_{xy} $).
2. Divide by the product of the two Standard Deviations ($ \sigma_x \times \sigma_y $).
$ \rho_{xy} = \frac{Cov_{xy}}{\sigma_x \sigma_y} $

Encouraging Note: If the math for Covariance looks scary, just remember it’s just the "average of the products of the deviations." You are just checking if both assets "missed their average" in the same direction at the same time.

Summary: Putting it All Together

• Returns are the starting point. Log returns are preferred for their additive properties.
• Volatility is our measure of risk. We use EWMA and GARCH to estimate it because market "fear" changes over time.
• EWMA is simple and reactive; GARCH is more complex and includes a "pull" back to the long-run average.
• Correlation tells us if our portfolio is diversified. Aim for low correlation to reduce overall risk!

Final Tip for the Exam: Be ready to calculate a "new" volatility using the EWMA or GARCH formulas. The exam loves giving you "yesterday's" data and asking for "today's" volatility estimate. Just plug the numbers into the formula step-by-step, and you'll do great!

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What is the formula for calculating a simple (discrete) return?

The formula for a simple return is $ R_t = \frac{P_t - P_{t-1}}{P_{t-1}} $.

What is the formula for a logarithmic (continuously compounded) return?

The formula for a log return is $ r_t = \ln\left(\frac{P_t}{P_{t-1}}\right) $.

What are the two primary advantages of using log returns in financial modeling?

Log returns are advantageous because of their time additivity and because they often fit normal distribution models better than simple returns.

What is the standard historical formula for calculating variance ($ \sigma^2 $)?

$ \sigma^2 = \frac{1}{m-1} \sum_{i=1}^{m} (u_{n-i} - \bar{u})^2 $

What common simplification is used for the mean return when calculating daily volatility in risk management?

For very short periods (like one day), risk managers often assume the mean return ($ \bar{u} $) is zero.

What is the main conceptual goal of the Exponentially Weighted Moving Average (EWMA) model?

The EWMA model gives more weight to recent data and less weight to older data when estimating volatility.

What is the EWMA formula for updating variance, and what does the parameter $ \lambda $ represent?

$ \sigma_n^2 = \lambda \sigma_{n-1}^2 + (1-\lambda) u_{n-1}^2 $. $ \lambda $ is the decay factor.

What is the GARCH (1,1) formula and its three core components?

$ \sigma_n^2 = \omega + \alpha u_{n-1}^2 + \beta \sigma_{n-1}^2 $. Components are: long-run average variance ($ \omega $), yesterday's news ($ \alpha u_{n-1}^2 $), and yesterday's predicted volatility ($ \beta \sigma_{n-1}^2 $).

What condition must be met for a GARCH (1,1) model to be stable and exhibit mean reversion?

The sum of the weights $ \alpha + \beta $ must be less than 1.

What is the difference between covariance and correlation?

Covariance indicates the direction of a relationship, while correlation is a standardized measure between -1 and +1 indicating both direction and strength.

How is the correlation coefficient ($ \rho_{xy} $) calculated using covariance and standard deviation?

$ \rho_{xy} = \frac{Cov_{xy}}{\sigma_x \sigma_y} $

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