modified-duration-and-price-sensitivity — yield duration and convexity

Duration errors cost real money. In 2022, the Bloomberg U.S. Aggregate Bond Index carried a duration of 6.0 years versus its long-term average of 4.97 years (Hartford Funds, 2025). When yields rose 236 basis points that year, the index lost -13.01%. Investors who assumed "bonds are safe" without checking duration got blindsided. The practical antidote: know exactly how to translate yield changes into price changes—and when simple duration math breaks down.

What Modified Duration Actually Tells You

Modified duration is your percentage price sensitivity per 1% yield change. A bond with modified duration of 5.0 will drop approximately 5% if yields rise 100 basis points (and gain approximately 5% if yields fall 100 bps).

The formula:

Modified Duration = Macaulay Duration / (1 + y)

Where y is the yield to maturity (expressed as a decimal, matching the compounding frequency).

The point is: Macaulay duration measures weighted-average time to receive cash flows (in years). Modified duration converts that time metric into a price sensitivity metric. They're related but serve different purposes.

Causal chain:

Macaulay duration (time-weighted cash flows) → Modified duration (price sensitivity) → Dollar duration/DV01 (dollar risk per bp)

The Core Calculation (Step by Step)

You own a 5-year annual-pay bond with a 5% coupon, $1,000 face value, trading at $1,100 (a premium bond). Yield to maturity is 2.82%.

Step 1: Establish Macaulay Duration

Macaulay duration for this bond is 4.571 years (weighted-average time to receive cash flows, discounted at 2.82%).

Step 2: Convert to Modified Duration

ModDur = MacD / (1 + y)

4.571 / (1 + 0.0282) = 4.445

Step 3: Apply to a Rate Scenario

If yields rise by 100 basis points (from 2.82% to 3.82%):

Estimated price change = -ModDur x Change in yield

-4.445 x 1.00% = -4.445%

Dollar impact: $1,100 x -4.445% = -$48.90

New estimated price: approximately $1,051

The practical point: Modified duration gives you a linear approximation. For small yield changes (under 50 bps), this estimate is usually accurate within a few basis points. For larger moves, you need the convexity adjustment.

When Duration-Only Estimates Fail (The Convexity Gap)

Duration assumes a straight-line relationship between price and yield. The actual relationship is curved (convex for option-free bonds). This gap widens with larger yield moves.

Example with convexity adjustment:

Bond price: $100
Modified duration: 7.02
Convexity: 64.93
Yield change: +45 bps (0.0045)

Duration-only estimate: -7.02 x 0.0045 = -3.16%

Convexity adjustment: 0.5 x 64.93 x (0.0045)^2 = +0.066%

Total estimated price change: -3.16% + 0.066% = -3.09%

New estimated price: $96.91

Why this matters: The convexity adjustment is always positive for option-free bonds (cushions both declines and amplifies gains). Ignoring convexity on a 100 bps move for a 20-year bond with duration 12 and convexity 180 can mean estimation errors of 0.9% or more.

Real-World Stress Test: The 2022 Rate Shock

The 2022-2023 Fed hiking cycle provides a brutal case study:

The setup:

Fed funds rate: 0.00-0.25% (start) to 5.25-5.50% (end)
Total increase: +525 bps over 16 months
10-year Treasury yield: 1.52% (end 2021) to 3.88% (end 2022)
10-year yield change: +236 bps

The damage:

Bloomberg U.S. Aggregate: -13.01% (2022)
Bloomberg Municipal Index: -8.8% (2022)
Aggregate peak-to-trough: -16.73% (July 2020 to October 2022)

Duration math check:

If the Aggregate Index had duration of 6.0 years and yields rose 236 bps:

Estimated loss = -6.0 x 2.36% = -14.16%

Actual loss was -13.01%. The difference (approximately 1.15%) reflects positive convexity cushioning the decline—exactly as the formula predicts.

The durable lesson: Duration math works. The problem in 2022 was that investors held portfolios with historically elevated duration (6.0 years vs. 4.97-year average) without recognizing the amplified downside.

Duration Across Bond Types (Reference Ranges)

Short-term:

Treasury bills: < 1 year
Short-term corporates: 1-3 years

Intermediate:

Agency MBS: ~5.18 years (though this fluctuates with prepayments)
Investment-grade corporates: ~8 years
Bloomberg Aggregate: 6.0 years (current), 4.97 years (historical average)

Long-term:

30-year Treasuries: 15-20+ years
Zero-coupon 30-year: 30 years (duration equals maturity for zeros)

The test: Can you state the approximate duration of your bond portfolio without looking it up? If not, you're flying blind on rate risk.

DV01: Converting Duration to Dollars

DV01 (dollar value of one basis point) translates percentage sensitivity into dollar terms:

DV01 = Modified Duration x Market Value x 0.0001

Example:

Portfolio value: $10,000,000 Modified duration: 6.5 years

DV01 = 6.5 x $10,000,000 x 0.0001 = $6,500

Interpretation: Your portfolio gains or loses $6,500 for each 1 basis point move in yields. A 50 bps move means $325,000 at risk.

Typical DV01 values per $1 million par (CME Group, 2024):

2-year Treasury: $185
5-year Treasury: $450
10-year Treasury: $850
30-year Treasury: $2,131

The practical point: Use DV01 for dollar-based risk budgeting (how much can you afford to lose?). Use duration for percentage-based comparisons across different-sized positions.

Detection Signals: You're Likely Misusing Duration If...

You assume a "6% duration" means the bond matures in 6 years (duration and maturity are related but not identical)
You apply modified duration to callable bonds without checking effective duration (the call option caps price gains)
You stress-test for 200 bps moves using duration alone and wonder why actual losses differ
You compare DV01s across different-sized positions without normalizing (doubling position size doubles DV01, but duration is unchanged)
You treat fund "average duration" as precise when the underlying holdings span a wide range

Callable Bonds: When Modified Duration Fails

Modified duration assumes fixed cash flows. But callable bonds have uncertain cash flows—the issuer might call early when rates drop.

Effective duration handles this:

EffDur = (PV_down - PV_up) / (2 x rate_change x PV_base)

Example: 4-year 6% callable bond

PV at current rates: $100.00
PV if rates fall 50 bps: $103.10
PV if rates rise 50 bps: $95.80

EffDur = (103.10 - 95.80) / (2 x 0.005 x 100) = 7.30

Interpretation: A 100 bp rate increase reduces value by approximately 7.30%. But note: the price gain for a 50 bp decline ($3.10) is less than the loss for a 50 bp rise ($4.20). That asymmetry signals negative convexity from the embedded call option.

The point is: For MBS, callable munis, or callable corporates, always use effective duration. Modified duration will overstate your expected gains when rates fall (because the call caps your upside).

Checklist: Using Duration Correctly

Essential (high ROI)

Know your portfolio duration before entering positions (check fund fact sheets or calculate weighted average)
Use duration x yield change for quick price estimates on moves under 50 bps
Add convexity adjustment for moves of 100 bps or more
Use effective duration for callables including MBS, callable munis, and callable corporates

High-impact refinements

Convert to DV01 for dollar risk budgeting and hedge ratio calculations
Stress-test for 100, 200, and 300 bps scenarios before assuming "bonds are safe"
Check key rate durations if your portfolio has significant curve exposure (barbell vs. bullet)

Your Next Step

Pull up your fixed-income holdings (including bond funds and ETFs). For each position:

Find the stated duration in the fund fact sheet or bond details
Calculate your portfolio-weighted duration (duration x weight for each position, then sum)
Run a stress scenario: What happens if yields rise 100 bps? 200 bps?

Interpretation:

Portfolio duration under 3 years: Modest rate sensitivity (your equity allocation is likely the bigger risk)
Duration 4-6 years: Moderate; a 100 bps rise means 4-6% decline
Duration above 7 years: Elevated; stress-test carefully for rate shock scenarios

Action: If your duration exceeds your risk tolerance for rate moves, consider shortening through Treasury bills, short-term bond funds, or duration hedges.

Related: Macaulay Duration Calculation Walkthrough | Convexity Concept and Calculation | Effective Duration for Callable Bonds | Stress Testing Portfolios for Rate Shocks

Source: CFA Institute, Yield-Based Bond Convexity and Portfolio Properties (2025). Hartford Funds, Duration of the Bloomberg US Aggregate Bond Index (2025). PIMCO, Understanding Duration (2024). CME Group, Trading the Treasury Yield Curve (2024).

Modified Duration and Price Sensitivity