Curve-Based and Empirical Fixed-Income Risk Measures

Welcome to the World of Yield Curves!

Hello there! In previous chapters, we looked at how bond prices change when interest rates move in general. But here’s the secret: interest rates for 1-year loans don't always move the same way as interest rates for 30-year loans! The Yield Curve can twist, bend, and shift in weird ways. In this chapter, we are going to learn how to measure the risk of a bond portfolio when the shape of the yield curve changes. Don't worry if this seems a bit technical at first—we'll break it down into simple, bite-sized pieces.

Quick Review: Remember that Duration measures how sensitive a bond's price is to changes in interest rates. If duration is 5, a 1% rise in rates means a 5% drop in price. Keep that in your back pocket!

1. Yield Curve Changes: Parallel vs. Non-Parallel Shifts

In a perfect world, the whole yield curve would move up or down by the same amount. This is called a Parallel Shift. However, in the real world, the curve often moves in Non-Parallel Shifts.

A. Parallel Shifts: Every maturity (short-term, medium-term, and long-term) moves up or down by the same number of basis points (bps).

B. Non-Parallel Shifts: These come in two main flavors:
1. Twists: The slope of the curve changes. A Flattening twist happens when short-term rates rise more than long-term rates. A Steepening twist happens when long-term rates rise more than short-term rates.
2. Butterflies (Curvature): The "belly" of the curve moves differently than the ends. A Positive Butterfly means the curve becomes less humped (the middle rates fall relative to the ends). A Negative Butterfly means the curve becomes more humped (the middle rates rise relative to the ends).

Analogy: Imagine a jump rope held by two people. If they both raise their hands 1 foot, that's a Parallel Shift. If one person raises their hand while the other lowers theirs, that’s a Twist. If the middle of the rope sags or gets pushed up while the ends stay still, that’s a Butterfly.

Key Takeaway: Traditional duration assumes parallel shifts. To handle twists and butterflies, we need more surgical tools like Key Rate Duration.

2. Key Rate Duration: The Laser Surgery Approach

If regular duration is a "sledgehammer" that hits the whole curve at once, Key Rate Duration (KRD) is a "laser" that targets one specific point on the curve.

What is it?
Key Rate Duration measures the sensitivity of a bond's price to a change in a single spot rate (like the 5-year rate) while keeping all other rates the same.

Why is it useful?
If you have a portfolio of bonds and you think only the 10-year interest rate is going to spike, you look at your 10-year KRD to see how much money you might lose.

The Math Connection:
If you add up all the Key Rate Durations of a bond, they will equal the bond's Effective Duration.
\( \text{Total Effective Duration} = \sum \text{Key Rate Durations} \)

Common Mistake to Avoid:
Students often forget that for a zero-coupon bond, the KRD is only significant at the bond's own maturity. For a coupon-paying bond, there will be "sensitivity" (KRD) at every maturity where a cash flow (coupon) is received!

Key Takeaway: KRD allows us to see exactly where our interest rate risk is hiding along the curve. It is the primary tool for managing shaping risk.

3. Effective Duration and Convexity Revisited

When dealing with Complex Bonds (like those with embedded options, such as callable bonds), we cannot use simple math formulas based on yields. We must use Effective Duration and Effective Convexity because they account for changes in expected cash flows.

The Formula for Effective Duration (\(D_{eff}\)):
\( D_{eff} = \frac{V_- - V_+}{2 \times V_0 \times \Delta \text{curve}} \)

The Formula for Effective Convexity (\(C_{eff}\)):
\( C_{eff} = \frac{V_- + V_+ - 2V_0}{V_0 \times (\Delta \text{curve})^2} \)

Where:
- \(V_-\) is the price if rates fall.
- \(V_+\) is the price if rates rise.
- \(V_0\) is the starting price.
- \(\Delta \text{curve}\) is the change in the benchmark yield curve.

Did you know?
We use "Effective" measures for bonds with options because the cash flows might change. If rates drop significantly, a Callable Bond might be called away by the issuer. This "shortens" the life of the bond, and Effective Duration is the only measure that captures this behavior accurately!

Key Takeaway: For any bond with uncertain cash flows (options), always use Effective measures, not Modified measures.

4. Spread Risk Measures

Most bonds (corporate bonds) pay a "spread" over the risk-free government rate. This spread is your reward for taking on Credit Risk and Liquidity Risk. If this spread widens, your bond price falls, even if government rates stay the same!

Spread Duration:
This measures how much a bond's price changes when its Credit Spread changes. For most fixed-rate corporate bonds, the spread duration is roughly equal to its Modified Duration.

Important Distinction:
For Floating Rate Notes (FRNs), the interest rate duration is very low (because the coupon resets), but the Spread Duration can still be quite high! This is because if the company's credit rating gets worse, the spread the market demands will increase, and the price of the FRN will drop.

Key Takeaway: Even if you hedge your interest rate risk, you are still exposed to Spread Risk. Spread duration tells you how much a "bad news" day for the company will hurt your bond price.

5. Empirical Duration: Real-World Observations

So far, we’ve used formulas. That’s "Analytical Duration." But sometimes, the market doesn't follow the formulas. Empirical Duration is calculated using Historical Data (regression analysis) to see how bond prices actually moved when rates changed in the past.

Why is Empirical Duration often lower than Analytical Duration?
This is a favorite CFA exam concept! In a "Flight to Quality," two things happen at once during a crisis:
1. Government interest rates fall (investors buy safe havens).
2. Corporate credit spreads widen (investors sell risky assets).
Because these two move in opposite directions, the price of a corporate bond might not rise as much as the formula suggests when rates fall. Therefore, the observed (empirical) sensitivity is lower than the theoretical (analytical) sensitivity.

Summary Table:
- Analytical Duration: Based on math and pricing models. Perfect for "what if" scenarios.
- Empirical Duration: Based on historical market behavior. Captures the correlation between spreads and base rates.

Key Takeaway: Empirical duration is "messy" but reflects how the market actually behaves during times of stress.

Final Quick Review Box

• Parallel Shift: Everything moves together.
• Non-Parallel: Twists (slope) and Butterflies (curvature).
• Key Rate Duration: Measures sensitivity to a specific maturity; sum of KRDs = Effective Duration.
• Effective Duration: Mandatory for bonds with embedded options (callable/putable).
• Spread Duration: Sensitivity to the credit spread, not the base rate.
• Empirical Duration: Based on history; usually lower than analytical duration for high-yield bonds due to "Flight to Quality."

Great job! You've just mastered the sophisticated tools professionals use to measure fixed-income risk. Keep practicing those formulas, and you'll be ready for the exam!