Modified Duration
Modified duration estimates the percentage price change of a fixed-income security for a small change in yield.
Modified duration estimates how much a bond’s price should change for a small change in yield. It is one of the most practical fixed-income risk measures because it converts a duration number into an approximate price-sensitivity number.
The sign is important: when yields rise, fixed-rate bond prices usually fall. When yields fall, fixed-rate bond prices usually rise.
Core Formula
Modified duration is commonly written as:
With periodic compounding, a more general version is:
Where \(y\) is yield and \(n\) is the number of compounding periods per year.
Worked Example: Modified Duration Price Estimate
The first-order price estimate is:
Continue the three-year bond example from Macaulay Duration. The bond has a 2.861-year Macaulay duration, 4% yield with annual compounding, and price of $1,027.75.
For a 50-basis-point yield increase, (\Delta y = 0.005):
Applying that estimate to the current price gives an estimated new price of about $1,013.61. Repricing the fixed cash flows exactly at a 4.5% yield gives about $1,013.74. The small difference comes from curvature that the first-order duration estimate omits.
The close result does not make modified duration exact. A larger yield move would generally increase the gap, and the estimate is not appropriate if calls, puts, or prepayments cause expected cash flows to change.
Why It Matters
Modified duration matters because it gives portfolio managers, traders, and risk teams a fast estimate of interest-rate exposure.
It helps answer:
- how much a bond price may move for a small yield change
- which bond has more rate sensitivity
- how much duration a portfolio carries versus a benchmark
- how large a Treasury or futures hedge might need to be
- whether a yield pickup is enough to compensate for extra duration risk
It is a linear approximation. It is useful for small yield moves, but it is not a full valuation model.
Modified Duration vs. Nearby Measures
| Measure | What it answers | Best use | Main limitation |
|---|---|---|---|
| Macaulay Duration | What is the weighted-average time to receive cash flows? | Timing and immunization concepts | Not directly a price-change estimate |
| Modified Duration | How much does price change for a small yield change? | Plain fixed-rate bond rate-risk estimates | Assumes cash flows stay fixed |
| Effective Duration | How sensitive is price when cash flows may change? | Callable, putable, and prepayable bonds | Depends on model assumptions |
| Convexity | How much curvature is missing from the duration estimate? | Larger rate moves and curved price-yield analysis | More complex than a first-pass duration estimate |
For a plain fixed-rate bullet bond, modified duration is usually a good first-pass risk measure. For a callable bond or mortgage-backed security, effective duration is usually more relevant because expected cash flows can change when rates move.
Public Source Checks
Useful public references include:
- FINRA Brush Up on Bonds: Interest Rate Changes and Duration: Investor education on duration as a measure of bond price sensitivity to interest-rate changes.
- Investor.gov Bond Funds and Income Funds: SEC investor education noting duration, volatility, interest-rate risk, credit risk, and prepayment risk in bond funds.
- MSRB Evaluating a Municipal Bond’s Interest Rate Risk: Municipal-bond education on duration and interest-rate-risk evaluation.
- OCC Interest Rate Risk Comptroller’s Handbook: Defines modified duration as an approximate price response that assumes expected cash flows do not change.
These sources are convention checks. A security-specific duration decision still needs the bond record, pricing model, yield convention, and portfolio objective.
This example is educational and is not an investment recommendation or interest-rate forecast.
When Modified Duration Misleads
Modified duration can mislead when:
- a large yield shock is treated as if the price-yield curve were perfectly linear
- a callable or prepayable bond is modeled as if cash flows were fixed
- credit spreads move at the same time as Treasury yields
- yield changes are nonparallel across the curve
- the duration was calculated from stale price or yield data
- liquidity, taxes, and transaction costs dominate the theoretical price estimate
- a portfolio duration figure hides concentrated key-rate exposure
Use modified duration as a first-pass estimate. Then check convexity, effective duration, key-rate duration, spread risk, liquidity, and the actual cash-flow structure.
Related Terms
- Duration: The broader bond price-sensitivity concept.
- Macaulay Duration: The timing measure used to derive modified duration.
- Effective Duration: Duration measure preferred when expected cash flows can change.
- Convexity: The curvature adjustment beyond linear duration.
- Key Rate Duration: Duration broken into specific curve points.
- Interest-Rate Risk: The broader risk modified duration helps quantify.