CFA Level I: Fixed Income Risk and Return

Understanding how to measure and manage interest rate risk is the cornerstone of fixed income portfolio management. As a future analyst or portfolio manager, you must master the tools that quantify how bond prices move with changing yields, allowing you to construct portfolios that align with specific risk tolerances and market views. This knowledge is not just academic; it’s directly applied in immunization strategies, asset-liability management, and tactical trading decisions, forming a critical part of the CFA and MBA curriculum.

Bond Price Sensitivity and the Concept of Duration

At its core, duration is a measure of a bond's sensitivity to changes in interest rates. It is expressed as a weighted average term to maturity of the bond's cash flows. The most fundamental measure is Macaulay duration, calculated as the present-value-weighted average time until each cash flow is received. For a bond, the formula is:

$$MacDur=\frac{{\sum }_{t=1}^n\frac{t\times C}{(1+y{)}^t}+\frac{n\times F}{(1+y{)}^n}}{P}$$

Where $t$ is the time period, $C$ is the coupon payment, $y$ is the yield per period, $F$ is the face value, $n$ is the total number of periods, and $P$ is the bond's price. While insightful, Macaulay duration is not directly used for estimating price changes.

For that, we use modified duration, which provides a linear approximation of the percentage price change for a given change in yield. It is derived from Macaulay duration:

$$ModDur=\frac{MacDur}{(1+\frac{y}{m})}$$

where $m$ is the number of compounding periods per year. The estimated percentage price change is then approximately $–ModDur\times \Delta y$. For example, a bond with a modified duration of 5.0 will see its price fall by approximately 5% for a 1% (100 basis point) increase in yield.

However, modified duration assumes a bond's cash flows do not change with yield, which is only true for option-free bonds. For bonds with embedded options, such as callable or putable bonds, we use effective duration. This measures sensitivity by calculating the price change for a small parallel shift in the benchmark yield curve, accounting for how the option's value changes:

$$EffDur=\frac{P_{-}-P_{+}}{2\times P_0\times \Delta y}$$

Here, $P_{-}$ and $P_{+}$ are the bond prices if the yield falls or rises by $\Delta y$, and $P_0$ is the original price.

From Percentage to Dollar Risk: Money Duration and PVBP

While modified duration gives a percentage change, practitioners often need to know the dollar impact. Money duration (or dollar duration) provides this, calculated as:

$$MoneyDur=ModDur\times P_0$$

The change in price in currency units is approximately $–MoneyDur\times \Delta y$. A closely related and very practical measure is the price value of a basis point (PVBP), also known as the dollar value of a 01 (DV01). This estimates the change in a bond's price from a one basis point (0.01%) change in yield. It can be derived from money duration:

$$PVBP=\frac{MoneyDur\times 0.0001}{1}\approx \frac{P_{-}-P_{+}}{2}$$

Portfolio managers use PVBP to aggregate interest rate risk across different securities and to hedge exposures precisely, as it translates duration into a direct dollar risk figure.

The Limitations of Duration and the Role of Convexity

A critical shortcoming of duration is that it assumes a linear relationship between yield and price, which is not accurate. In reality, the relationship is curved, or convex. This means that for a given change in yield, the duration-based estimate will underestimate the actual price increase when yields fall and underestimate the actual price decrease when yields rise. This beneficial curvature is measured by convexity.

Convexity is the second derivative of the price-yield function. For an option-free bond, the formula for convexity (in periods) is:

$$Convexity=\frac{{\sum }_{t=1}^n\frac{t(t+1)C}{(1+y{)}^{t+2}}+\frac{n(n+1)F}{(1+y{)}^{n+2}}}{P_0}$$

To make it usable, we often convert to an annual figure and use it to improve our price change approximation. The combined duration and convexity approximation for percentage price change is far more accurate:

$$\%\Delta P\approx (-ModDur\times \Delta y)+(\frac{1}{2}\times Convexity\times (\Delta y{)}^2)$$

The convexity adjustment term, $\frac{1}{2}\times Convexity\times (\Delta y{)}^2$, is always positive for an option-free bond, correcting the duration error. Bonds with higher convexity are more desirable in volatile markets, as they gain more when yields fall and lose less when yields rise, all else being equal.

Advanced Duration Measures and the Integration of Credit Risk

For complex portfolios, a single duration number may be insufficient. Key rate duration measures sensitivity to changes in specific points (or "key rates") on the yield curve, such as the 2-year, 5-year, or 10-year spot rates. This allows a portfolio manager to understand exposure to yield curve reshaping (non-parallel shifts) and to structure targeted hedges. The sum of key rate durations typically equals the effective duration.

Finally, you must understand the interaction of default risk and interest rate risk. These are the two primary risks of a corporate bond. Importantly, they are not independent. During periods of economic stress, credit spreads (the yield premium over a government bond) tend to widen as default risk increases. Simultaneously, central banks may cut policy rates, causing government yields to fall. This creates a complex dynamic: the bond's price may be pushed down by the widening spread but propped up by falling benchmark rates. Effective duration, which uses the benchmark curve, may fail to capture the full price decline driven by credit deterioration. A comprehensive risk measurement requires analyzing spread duration (sensitivity to changes in the credit spread) in conjunction with interest rate duration.

Common Pitfalls

1. Using Modified Duration for Bonds with Embedded Options: This is a fundamental error. Modified duration is invalid for callable bonds because their cash flows change when interest rates move. Failing to use effective duration will lead to significant misestimation of interest rate risk.
2. Ignoring Convexity for Large Yield Changes: For yield moves exceeding 50-100 basis points, the linear estimate from duration becomes increasingly inaccurate. Neglecting the positive convexity adjustment will cause you to systematically underestimate bond price increases and overestimate price decreases.
3. Assuming a Single Duration Measures All Curve Risk: Treating a portfolio's overall effective duration as sufficient ignores yield curve risk. A portfolio with the same effective duration as a benchmark can have very different key rate duration exposures, leading to unexpected losses if the curve steepens or flattens.
4. Overlooking the Credit-Rate Risk Interaction: Analyzing a corporate bond's interest rate risk in isolation from its credit risk is misleading. In a "flight-to-quality" scenario, the two risks can push the bond's price in opposite directions, and a model that doesn't account for spread change will fail.

Summary

• Duration is the primary measure of interest rate sensitivity. Modified duration estimates percentage price changes for option-free bonds, while effective duration must be used for bonds with embedded options.
• Money duration and the price value of a basis point (PVBP) translate percentage sensitivity into concrete dollar risk, which is essential for hedging and portfolio management.
• Convexity measures the curvature in the price-yield relationship. Using a duration and convexity approximation provides a far more accurate price change estimate than duration alone, especially for larger yield movements.
• Key rate duration decomposes risk exposure to specific maturities on the yield curve, enabling management of non-parallel shift risk.
• In corporate bonds, default risk and interest rate risk interact. A full risk assessment requires analyzing both benchmark yield changes (via duration) and credit spread changes (via spread duration).