## **Learning Outcomes**

This article explains yield curve and duration strategies for the CFA Level 3 exam, including:

- differentiating modified duration from convexity and explaining how each measure captures bond price sensitivity to yield changes;
- analyzing how parallel and non-parallel yield curve shifts affect individual bonds and multi-bond portfolios with different maturities;
- applying duration–convexity approximations to estimate price changes for small and large interest rate movements and validate results;
- using key rate duration to isolate sensitivity at specific maturities and diagnose where yield curve risk is concentrated;
- constructing and comparing barbell, bullet, static, and dynamic yield curve strategies consistent with stated return and risk objectives;
- evaluating how changes in duration, convexity, and key rate exposures alter portfolio performance under different interest rate scenarios;
- designing hedging and active positioning trades that align portfolio duration, convexity, and key rate profiles with macroeconomic views;
- interpreting exam-style questions that link bond pricing, yield curve movements, and risk metrics to portfolio management decisions.

## **CFA Level 3 Syllabus**

For the CFA Level 3 exam, you are expected to understand yield curve and duration-based strategies, as well as how convexity and key rate duration influence portfolio management, with a focus on the following syllabus points:

- Measuring and interpreting duration and convexity for bonds and portfolios
- Analyzing the impact of parallel and non-parallel yield curve shifts
- Applying key rate duration to assess exposure to yield curve movements
- Comparing static and dynamic yield curve strategies for active management
- Evaluating how duration and convexity support risk management and strategy selection

## **Test Your Knowledge**

_Attempt these questions before reading this article. If you find some difficult or cannot remember the answers, remember to look more closely at that area during your revision._

1. What is the significance of convexity in estimating bond price changes for large shifts in yields?
2. How do you use key rate duration to measure a portfolio's exposure to specific parts of the yield curve?
3. Why should portfolio convexity be considered alongside duration in risk management?
4. In what scenario would relying solely on modified duration produce a misleading estimate of price sensitivity?

## **Introduction**

Yield curve and duration strategies are fundamental to fixed-income portfolio management. At CFA Level 3, you must not only calculate measures such as duration and convexity, but also understand their practical roles in formulating and managing active yield curve strategies. This section covers how duration indicates price sensitivity to yield changes, how convexity corrects duration-based estimates, and how key rate duration enables more precise measurement and control of portfolio risk as the yield curve shifts in complex ways.

> **Key Term: duration**
> A measure of the sensitivity of a bond's price to changes in interest rates. Modified duration quantifies the percentage price change for a one percentage point change in yield.
>
> **Key Term: convexity**
> A measure of the curvature, or second-order sensitivity, of a bond's price-yield relationship. Convexity corrects estimates for large yield changes, improving upon linear duration estimates.
>
> **Key Term: key rate duration**
> A measure of the change in the value of a bond or portfolio for a one basis point change in yield at a specific maturity on the yield curve, holding other yields constant.

## **Duration: The Basis for Yield Curve Strategies**

Modified duration is a core tool for assessing the interest rate risk of a single bond or an entire portfolio. For small, parallel shifts in the yield curve, the price change can be estimated linearly:

$$\% \Delta P \approx -\text{ModDur} \times \Delta \text{Yield}$$

This relationship works well for minor yield changes or when the yield curve shifts in parallel. However, most market movements are not perfectly linear or parallel, requiring greater precision.

### **Worked Example 1.1**

A portfolio manager holds a \$1 million position in a 5-year zero-coupon bond. The bond’s yield rises by 50 bps from 3% to 3.5%. The bond’s modified duration is 4.85. Estimate the percentage price change using modified duration.

> **Answer:**
> Estimated price change ≈ -4.85 × 0.005 = -2.43%. Estimated loss on \$1 million position is \$24,300.

## **Convexity: Beyond the Linear Estimate**

While duration is accurate for small, parallel shifts, larger or non-parallel shifts introduce error. Convexity accounts for the curvature of the price-yield relationship, offering a correction factor:

$$\% \Delta P \approx -\text{ModDur} \times \Delta \text{Yield} + \tfrac{1}{2} \times \text{Convexity} \times (\Delta \text{Yield})^2$$

Convexity is always positive for plain vanilla bonds, meaning that price losses when yields rise are smaller than predicted by duration alone, and gains when yields fall are larger.

### **Worked Example 1.2**

Given the same bond as above with a convexity of 31.2, estimate the more precise price change after a 100 bp yield increase.

> **Answer:**
> Linear estimate: -4.85 × 0.01 = -4.85%.  
> Convexity adjustment: 0.5 × 31.2 × (0.01)^2 = 0.0156, or 1.56%.  
> Total: -4.85% + 1.56% = -3.29%. The price decreases by 3.29%, less than suggested by duration alone.

### **Exam Warning**

> A common error is to use duration alone for large yield movements or for bonds with significant convexity (such as long-maturity or low-coupon bonds). Always supplement duration estimates with convexity for large or volatile changes to improve accuracy.

## **Yield Curve Strategies: Static, Dynamic, and Key Rate Approaches**

Yield curve changes are not always parallel. To address this, managers employ strategies that mix duration and convexity with curve positioning and key rate duration analysis:

- **Static strategies:** Use matched duration and convexity to minimize price risk for expected changes.
- **Dynamic strategies:** Adjust portfolio allocations between key yield curve segments to profit from expected curve twists (steepening, flattening, butterfly).
- **Barbell and bullet strategies:** Allocate more to extremes (barbell) or center (bullet) of curve, affecting duration, convexity, and return profile.
- **Key rate duration:** Allows for finer control by measuring sensitivity at specified maturities.

### **Worked Example 1.3**

A manager wants to assess risk to a portfolio with weightings in 2-year, 5-year, and 10-year Treasuries. The portfolio’s key rate durations are: 0.8 (2y), 2.2 (5y), and 3.4 (10y), with an effective duration of 6.4 years. If only the 5-year yield rises by 50 bps while other yields remain constant, what is the estimated change?

> **Answer:**
> Estimated loss = 2.2 × 0.005 = 1.1% price drop, assuming minimal changes from other maturities.

## **Key Rate Duration: Targeted Sensitivity Control**

Key rate duration lets managers quantify and control portfolio sensitivity to isolated yield curve movements. By constructing portfolios with different key rate exposures, managers can:

- Hedge exposure to non-parallel movements (such as a rise only in intermediate-term yields).
- Design barbell or bullet curve tilts for tactical curve bets.
- Diagnose where portfolio risk is concentrated or diversified.

> **Key Term: yield curve twist**
> A non-parallel movement in the yield curve, such as steepening, flattening, or changing curvature without shifting uniformly at all maturities.

## **Portfolio Construction: Balancing Duration, Convexity, and Key Rates**

Active managers seek portfolios with desired duration, positive convexity, and key rate exposure aligned to their views. Strategies may include:

- **Hedging liability risk:** Matching asset and liability duration to minimize surplus volatility.
- **Seeking convexity:** Preferring portfolios (often barbells) with higher convexity for uncertain rate environments.
- **Key rate overlays:** Using swaps, futures, or securities to tilt exposure at chosen points along the curve.

Key rate duration is critical for hedging portfolios against non-parallel curve shifts — essential for insurance companies, pension funds, and liability-driven investors.

### **Revision Tip**

> For the CFA exam, be able to interpret duration and convexity calculations in context, not just the formulas. Linking these measures to yield curve strategies and risk profiles is commonly tested.

## **Summary**

- Duration estimates price sensitivity to parallel shifts.
- Convexity improves accuracy by correcting for curvature, especially with large yield moves.
- Key rate duration measures price change when yields at specific maturities shift, aiding in risk management for non-parallel curve changes.
- Strategic allocation of duration and convexity, along with key rate overlays, supports precise yield curve positioning and risk control.

## **Key Point Checklist**

_This article has covered the following key knowledge points:_

- Calculating and interpreting modified duration, convexity, and key rate duration for bonds and portfolios
- Explaining how convexity improves duration-based price change estimates
- Understanding how static, dynamic, and key rate-driven yield curve strategies work
- Applying key rate duration to isolate and manage exposure to non-parallel yield curve moves
- Using duration and convexity together to structure portfolios for targeted interest rate risk and return profiles

## **Key Terms and Concepts**

- duration
- convexity
- key rate duration
- yield curve twist

Yield curve and duration strategies - Duration convexity and key rate duration