Modified Duration
Modified duration as a measure of price sensitivity to yield changes
From Time to Sensitivity: What is Modified Duration?
While Macaulay Duration measures the weighted-average time of cash flows, Modified Duration ($D_{mod}$) converts this concept into what most market participants mean when they say "duration": price sensitivity.
Modified Duration measures the estimated percentage change in a bond's price for a 1% (100 basis point) change in its yield. It is the single most important metric for quantifying interest rate risk.
Mathematically, it is the first derivative of the price-yield function, a direct measure of the slope of the price-yield curve at a given point.
Calculation Formula
Modified Duration is easily derived from Macaulay Duration ($D_{mac}$). The formula adjusts $D_{mac}$ by the bond's yield per period.
For JGBs with semi-annual compounding (where $k=2$):
\[D_{mod} = \frac{D_{mac \text{ (years)}}}{1 + (y/k)} = \frac{D_{mac \text{ (years)}}}{1 + (y/2)}\]Where:
- $D_{mac \text{ (years)}}$ = Macaulay Duration in years
- $y$ = Yield to Maturity (YTM)
- $k$ = Compounding periods per year (2 for JGBs)
The Price Approximation Formula
The reason $D_{mod}$ is so powerful is its use in approximating price changes. The formula is:
\[\%\Delta P \approx -D_{mod} \times \Delta y\]Where:
- $\%\Delta P$ = The percentage change in the bond's price
- $-D_{mod}$ = The negative of the Modified Duration
- $\Delta y$ = The change in yield (e.g., +0.0010 for a 10 bps rise)
The negative sign is critical: it reflects the inverse relationship between price and yield. If yields rise, prices fall, and vice versa.
Worked JGB Example
Let's continue with our 3-year JGB from the previous section.
- Macaulay Duration ($D_{mac}$): 2.9631 years
- Yield to Maturity (y): 0.8% (or 0.008)
- Compounding (k): 2
Step 1: Calculate Modified Duration
$$D_{mod} = \frac{2.9631}{1 + (0.008 / 2)}$$ $$D_{mod} = \frac{2.9631}{1.004} = \textbf{2.9513}$$
Interpretation: This bond has a Modified Duration of 2.9513. This means that for a 1% (100 bps) change in yield, the bond's price will change by approximately 2.9513% in the opposite direction.
Step 2: Use $D_{mod}$ to Estimate Price Change
Let's estimate what happens if the BOJ hikes rates and this bond's yield rises 20 basis points (from 0.8% to 1.0%).
- $\Delta y$ = +0.0020 (or 20 bps)
- $D_{mod}$ = 2.9513
$$\%\Delta P \approx -2.9513 \times (+0.0020)$$ $$\%\Delta P \approx -0.00590 \text{ or } \textbf{-0.590%}$$
If the bond's original price was ¥1,005,903.53, the estimated price drop would be:
$$\text{Price Drop} \approx -0.00590 \times ¥1,005,903.53 = \textbf{-¥5,935}$$
This gives a portfolio manager an instant, powerful estimate of their risk.
The Limitation of Duration
Modified Duration is a linear approximation of a curved (convex) relationship. This means:
- It is very accurate for small yield changes (like 1-2 bps).
- It becomes inaccurate for large yield changes (like 100 bps).
Specifically, duration always *underestimates* the price gain from a yield drop and *overestimates* the price loss from a yield rise. This "error" is a positive attribute for the bondholder, and it is measured by Convexity, which is covered in two sections.
Advanced Topic: Key Rate Duration
While Modified Duration measures a bond's sensitivity to a parallel shift in the entire yield curve (all maturities move by the same amount), real-world yield curve movements are rarely parallel. The curve can:
- Steepen: Long-term rates rise more than short-term rates
- Flatten: Short-term rates rise more than long-term rates
- Twist: Some maturities rise while others fall
- Butterfly: The middle of the curve moves differently than the wings
For portfolios containing bonds of multiple maturities, a single Modified Duration number is insufficient to capture these complex risks. This is where Key Rate Duration (also called Partial Duration or Grid Point Sensitivity) becomes essential.
What is Key Rate Duration?
Key Rate Duration measures how a bond's (or portfolio's) price changes when the yield at a specific maturity point changes by 1%, while yields at other maturities remain constant.
Instead of a single duration number, you get a vector of durations at predefined "key rates" or "grid points" along the yield curve.
Standard JGB Grid Points
Japanese institutional investors and risk managers commonly use these grid points:
- 2-Year
- 5-Year
- 10-Year
- 20-Year
- 30-Year
- 40-Year (for institutions with ultra-long liabilities)
Each key rate duration answers the question: "If the yield at this specific maturity changes by 1%, how much does my bond/portfolio value change?"
Mathematical Concept
For a single bond, the sum of all key rate durations approximately equals the bond's Modified Duration:
\[\text{Modified Duration} \approx \sum_{i} \text{KRD}_i\]Where $\text{KRD}_i$ is the Key Rate Duration at grid point $i$ (2Y, 5Y, 10Y, etc.).
For a portfolio, key rate durations reveal which parts of the yield curve pose the greatest risk.
Example: 10-Year JGB Key Rate Exposures
Consider a 10-year JGB with Modified Duration of 9.5 years. Its key rate duration profile might look like:
| Grid Point | Key Rate Duration | Interpretation |
|---|---|---|
| 2Y | 0.2 | Minimal sensitivity to 2Y rate changes |
| 5Y | 1.8 | Some sensitivity (nearby maturity) |
| 10Y | 7.0 | Highest sensitivity (bond's own maturity) |
| 20Y | 0.5 | Minimal sensitivity |
| 30Y | 0.0 | Nearly zero sensitivity |
| Total | 9.5 | Sum ≈ Modified Duration |
This shows that the bond is most sensitive to changes in 10-year rates (KRD = 7.0), with smaller sensitivities to neighboring maturities.
Portfolio Example: JGB Portfolio with Mixed Maturities
Now consider a ¥100 billion portfolio containing:
- ¥30 billion in 5-year JGBs
- ¥50 billion in 10-year JGBs
- ¥20 billion in 30-year JGBs
The portfolio's aggregate key rate durations might be:
| Grid Point | Portfolio KRD | DV01 (¥ per 1bp) |
|---|---|---|
| 2Y | 0.5 | ¥5 million |
| 5Y | 3.2 | ¥32 million |
| 10Y | 5.8 | ¥58 million |
| 20Y | 2.1 | ¥21 million |
| 30Y | 4.5 | ¥45 million |
| Total | 16.1 | ¥161 million |
Interpretation:
- If only 10Y yields rise by 10bp, the portfolio loses: ¥58m × 10 = ¥580 million
- If only 30Y yields rise by 10bp, the portfolio loses: ¥45m × 10 = ¥450 million
- If all yields rise by 10bp (parallel shift), the portfolio loses: ¥161m × 10 = ¥1.61 billion
Why Key Rate Duration Matters for JGB Investors
1. Curve Steepening/Flattening Risk
During BOJ policy transitions (e.g., YCC exit in 2024), the JGB yield curve experienced significant steepening:
- 2Y yields rose sharply (from -0.10% to +0.25%)
- 10Y yields rose moderately (from 0.60% to 0.90%)
- 30Y yields barely moved (stayed around 1.80%)
A portfolio with high 2Y KRD suffered losses, while portfolios concentrated in 30Y KRD were relatively unaffected. Modified Duration alone couldn't capture this differential impact.
2. Targeted Hedging
If a portfolio manager identifies that most risk comes from 10Y exposure (high 10Y KRD), they can:
- Sell 10Y JGB futures to hedge that specific grid point
- Leave other grid points unhedged if they view them as less risky
- Implement curve trades to reduce specific KRD exposures (see Chapter 4.3)
3. Butterfly Trade Construction
Butterfly trades (see Chapter 4.4) are specifically designed to create targeted key rate exposures:
- Positive KRD at wing maturities (e.g., 5Y and 20Y)
- Negative KRD at body maturity (e.g., 10Y)
- Near-zero KRD at other grid points
This precision is impossible without key rate duration analysis.
Relationship to Grid Point Sensitivity (GPS)
In Japanese institutional practice, Key Rate Duration is often called Grid Point Sensitivity (GPS) or グリッドポイント感応度. These terms are essentially synonymous:
- Key Rate Duration = Percentage change in value for 1% yield change at a grid point
- Grid Point DV01 = Yen change in value for 1bp yield change at a grid point
GPS will be covered in greater detail in Section 3.5.
Limitations and Practical Considerations
1. Modeling Assumptions
Key rate duration assumes that when one grid point moves, others stay constant. In reality:
- Nearby maturities often move together (correlation)
- Changes propagate through the curve
- More sophisticated models use principal component analysis (PCA) to capture these effects
2. Computational Complexity
Calculating key rate durations requires:
- Repricing each bond multiple times (once per grid point)
- Yield curve interpolation between grid points
- More sophisticated risk systems than simple duration calculations
3. Grid Point Selection
The choice of grid points is somewhat arbitrary. Some institutions use:
- Sparse grids: 2Y, 5Y, 10Y, 20Y, 30Y (faster, less precise)
- Dense grids: 1Y, 2Y, 3Y, 4Y, 5Y, 7Y, 10Y, 15Y, 20Y, 25Y, 30Y, 40Y (slower, more precise)
Key Takeaways
- Single duration numbers are insufficient for portfolios with exposure across multiple maturities
- Key Rate Duration decomposes risk into sensitivities at specific yield curve points
- JGB institutional practice uses grid points at 2Y, 5Y, 10Y, 20Y, 30Y, and sometimes 40Y
- KRD enables targeted hedging and precise curve trade construction
- Essential for understanding non-parallel curve movements, which are the norm in JGB markets
Understanding key rate duration is critical for professional JGB portfolio management and forms the foundation for sophisticated strategies like butterfly trades and duration-neutral curve positioning.