Welcome to Bond Convexity!

In your Fixed Income journey so far, you’ve learned about Duration—the tool we use to estimate how much a bond's price changes when interest rates move. But there’s a catch: Duration assumes the relationship between price and yield is a straight line. In reality, it’s a curve!

In this chapter, we are going to learn about Convexity. Think of Duration as your "best guess" and Convexity as the "correction" that makes that guess much more accurate. Whether you’re a math whiz or someone who prefers visual concepts, don’t worry—we’ll break this down step-by-step!

1. Why Duration Isn't Enough: The Concept of Convexity

If you look at a graph of a bond's price versus its yield, you’ll notice it isn't a straight line; it’s a curve that bows toward the origin. This "bowed" shape is what we call Convexity.

The Problem: Modified Duration is a linear approximation. It works well for very small changes in yield. However, for large changes in yield, Duration starts to give us wrong answers. It understates the price increase when yields fall and overstates the price decrease when yields rise.

The Solution: Convexity is the "second-order" effect. It measures the rate of change of duration. By adding a convexity adjustment to our duration calculation, we get a much better estimate of the actual price change.

Analogy: Imagine you are driving a car. Duration is like your speed (how far you go in one hour). Convexity is like your acceleration (how your speed is changing). To know exactly where you'll be in an hour, you need to account for both!

Key Takeaway:

Convexity accounts for the curvature of the price-yield relationship. For a standard (option-free) bond, convexity is always a positive thing for the investor!

2. The Full Price Change Formula

To calculate the total percentage change in a bond's price, we combine the duration effect and the convexity effect.

\( \Delta\% P \approx [-\text{AnnModDur} \times \Delta y] + [\frac{1}{2} \times \text{AnnConv} \times (\Delta y)^2] \)

Breaking down the formula:
1. The Duration Part: \( [-\text{AnnModDur} \times \Delta y] \). This is what you already know. Notice the negative sign: when yields (\(y\)) go up, price goes down.
2. The Convexity Part: \( [\frac{1}{2} \times \text{AnnConv} \times (\Delta y)^2] \). Because \(\Delta y\) is squared, this term is always positive (for option-free bonds), regardless of whether yields go up or down. This "adds back" some value.

Common Mistake to Avoid: Don't forget the \( \frac{1}{2} \) in the convexity adjustment! Many students lose points by simply multiplying convexity by the change in yield squared without dividing by two.

3. Calculating Approximate Convexity

Just like we have "Approximate Modified Duration," we have a formula to estimate convexity using bond prices at three different points.

The Formula:
\( \text{ApproxConv} = \frac{P_- + P_+ - 2P_0}{P_0 \times (\Delta y)^2} \)

Where:
\(P_-\) = Price if yield decreases.
\(P_+\) = Price if yield increases.
\(P_0\) = Initial price.
\(\Delta y\) = The change in yield (in decimal form).

Quick Review: Step-by-Step Calculation

1. Start with the current price (\(P_0\)).
2. Calculate the price if yields drop by a certain amount (\(P_-\)).
3. Calculate the price if yields rise by that same amount (\(P_+\)).
4. Plug these into the formula to find the convexity statistic.

4. Money Duration and Money Convexity

Sometimes, portfolio managers don't care about "percentage" changes; they want to know the "dollar" change. This is where Money Duration and Money Convexity come in.

Money Duration: \( \text{AnnModDur} \times \text{Full Price} \)
Money Convexity: \( \text{AnnConv} \times \text{Full Price} \)

The estimated price change in currency units (like Dollars or Euros) is:
\( \Delta P \approx [-\text{MoneyDur} \times \Delta y] + [\frac{1}{2} \times \text{MoneyConv} \times (\Delta y)^2] \)

Did you know? In the US and UK markets, the Money Duration for a 1 basis point change in yield is often called PVBP (Price Value of a Basis Point).

5. Portfolio Properties: Duration and Convexity

How do we measure the risk of an entire "bucket" of bonds? There are two main ways:

Method 1: Weighted Average (The Simple Way)

We simply take the duration (or convexity) of each bond and multiply it by its weight in the portfolio.

Example: If you have 60% in Bond A (Duration 5) and 40% in Bond B (Duration 10), the portfolio duration is \( (0.60 \times 5) + (0.40 \times 10) = 7.0 \).

The Catch: This method assumes a parallel shift in the yield curve (all interest rates move by the same amount). In the real world, this rarely happens!

Method 2: Based on Portfolio Yield (The Complex Way)

This calculates the internal rate of return (IRR) of the portfolio's cash flows. It’s more theoretically accurate but harder to calculate and less commonly used for daily risk management.

Key Takeaway:

For the CFA exam, remember that the weighted average method is commonly used but relies on the assumption that the yield curve shifts in a parallel fashion.

6. What Affects Convexity?

Not all bonds have the same amount of "curve." Here are the general rules:

  • Maturity: Longer maturity = Higher convexity (and higher duration).
  • Coupon Rate: Lower coupon = Higher convexity (the "back-loaded" cash flows make the bond more sensitive).
  • Yield: Lower yield = Higher convexity.
  • Dispersion: For a portfolio, if the cash flows are spread out further in time (high dispersion), the convexity will be higher.

Mnemonic: Think of "Low and Long." The lower the coupon/yield and the longer the maturity, the higher the duration and convexity.

7. Positive vs. Negative Convexity

This is a favorite topic for exam questions!

Positive Convexity: Typical for option-free bonds. When yields fall, the price rises more than duration predicts. When yields rise, the price falls less than duration predicts. This is great for investors!

Negative Convexity: Occurs in Callable Bonds when yields are low. As yields drop, the price doesn't rise as much because the issuer is likely to "call" (buy back) the bond. The price-yield curve actually flattens out or "bends" the other way.

Putable Bonds: These have even more positive convexity than option-free bonds because the "put" option protects the investor when yields rise (prices fall).

Summary of Options:
  • Option-Free: Positive Convexity.
  • Callable: Can have Negative Convexity at low yields.
  • Putable: More Positive Convexity than option-free bonds.

Final Encouragement

Fixed Income math can feel overwhelming because of all the \(P_+\) and \(P_-\) terms, but remember the core logic: Duration is your first guess, and Convexity is your correction. Once you visualize the curve, the formulas start to make much more sense. Keep practicing those calculations!