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.
Price-Sensitivity Estimate
The first-order price estimate is:
If a bond has modified duration of 6.0, a 1.00% yield increase implies roughly a 6.0% price decline before convexity adjustment. A 0.50% yield increase implies roughly a 3.0% price decline.
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.
Practical Example
Suppose a bond has:
- market value of
$1,000,000 - modified duration of
4.5 - yield shock of
+0.50%
The estimated price effect is:
On $1,000,000, that is about a $22,500 price decline before convexity, spread changes, tax effects, and liquidity costs.
What To Verify
Before relying on modified duration, verify:
- yield convention, compounding convention, and settlement date
- clean price versus dirty price treatment
- coupon rate, maturity date, payment frequency, and day-count convention
- whether the security has call, put, prepayment, sinking-fund, or amortization features
- whether the duration is calculated at the security level, portfolio level, or benchmark level
- whether the yield shock is parallel or tied to a specific curve point
- whether convexity materially changes the estimate for the size of the rate move
Modified duration should be tied to a calculation date and a pricing source. A stale duration number can be materially wrong after yield, price, or cash-flow assumptions change.
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.
These sources are convention checks. A security-specific duration decision still needs the bond record, pricing model, yield convention, and portfolio objective.
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.