Bond convexity and duration: Macaulay, modified, DV01 and the formulas

Bond duration measures how far a bond’s price moves when yields change; bond convexity measures how that sensitivity changes as yields move. On 4 November 2026 the US 10-year Treasury yield closed at 4.90%, down 5 basis points on the day. A 10-year bond with a 4.90% coupon had a modified duration of 7.83, so its price rose about 0.39%: $3,922 on a $1 million position.

What is bond duration?

Duration has two readings that share one number. As time, Macaulay duration is the average wait until a bond pays you back, each payment weighted by its present value. A 10-year par bond at 4.90% pays 20 half-yearly coupons and its principal, and the weighted average lands at 8.02 years, before maturity because the coupons arrive early. As risk, modified duration turns that time into a price sensitivity: about 7.83% of price for a one-percentage-point move in yield.

The Macaulay duration formula

The modified duration formula

Modified duration is the figure risk systems usually mean by "duration". It grows with maturity and shrinks as the coupon or the yield rises. On the US curve of 4 November 2026:

Why zero-coupon bonds have the longest duration

Coupons cut duration because part of the value comes back early. A 30-year zero-coupon bond at 5.28% has a modified duration of 29.23, nearly twice the 14.97 of the 30-year par bond, which is why long zero-coupon bonds swing the most when yields change.

DV01: the dollar value of a basis point

DV01, also called PV01 or the dollar value of a basis point, is how traders size hedges and compare positions across maturities. $1 million of 2-year notes carries $189 of DV01, so it takes about $7.9 million of them to match the rate risk of $1 million of 30-year bonds. The basis point itself is covered in basis points. Across countries the 10-year looks alike:

What is bond convexity?

Duration draws a straight line through a curved relationship. The price-yield curve of a plain bond bends upward: as yields fall its duration lengthens and gains accelerate, and as yields rise duration shortens and losses slow. Convexity measures that bend. For a 30-year par bond at the 4 November yield of 5.28%:

The convexity formula

For a plain bond convexity is positive: duration underestimates the gain when yields fall and overestimates the loss when they rise, by about ½ × convexity × Δy² either way. Longer maturities and lower coupons have more of it, and it grows roughly with the square of duration: 4.6 for the 2-year, 74.1 for the 10-year, 335.3 for the 30-year.

Is higher convexity better?

At the same duration, more convexity means a better result from any large parallel move, up or down. A barbell shows it: 54.6% of the money in the 2-year and 45.4% in the 30-year has the 10-year’s duration of 7.83 with twice its convexity.

Convexity usually has a price: positions with more of it tend to yield less, and it only pays when yields move a long way. The last column shows the other catch. Same duration, but the barbell made $888 less on 4 November, because the two ends of the curve fell less than the middle. Duration treats every yield as moving together, and that day they did not.

Negative convexity and effective duration

Some bonds bend the other way. A callable bond can be repaid early by its issuer, which becomes likely when yields fall, so its price stalls near the call price. A mortgage-backed security behaves the same way, because homeowners refinance when rates drop. When early repayment is likely, both have negative convexity: duration shortens as yields fall and lengthens as they rise, the opposite of a plain bond.

Their cash flows change with yields, so the Macaulay formula no longer applies. Effective duration and effective convexity reprice the bond with a model after a yield shift each way:

Most bond funds publish an effective duration for the same reason. It is the figure to compare with the numbers on this page, not the fund’s average maturity.

Key rate duration: when the curve does not move in parallel

Key rate durations split a bond’s duration across points on the curve, so each tenor’s own move can be applied to it. For a par bond at a curve point, most of it sits at its own maturity; the barbell above has its exposure at 2 and 30 years, where the curve fell 3 and 4 basis points against 5 at 10 years. Summed, key rate durations come back to about the ordinary duration, which is why one number is enough for a parallel shift and not for a day like this one. The yield curve inversion guide covers how the curve’s shape moves.

Duration and convexity calculator

Calculate duration, convexity and DV01 in Python

duration_convexity.py reads the latest yields from the bonds API, one request per tenor, prices a par bond at each, and prints its two durations, convexity, DV01 and the day’s profit or loss on $1 million. It needs Python 3.9 or newer, the standard library and a TICKERLAYER_API_KEY environment variable.

"""Duration, convexity and DV01 of par bonds at today's yields. Python 3.9+, standard library only."""
import json
import os
import sys
import urllib.request

API = "https://api.tickerlayer.com"
KEY = os.environ.get("TICKERLAYER_API_KEY") or sys.exit("Set TICKERLAYER_API_KEY first.")
TENORS = {"US:2Y": 2, "US:5Y": 5, "US:10Y": 10, "US:30Y": 30}
FREQ = 2  # Treasury coupons are paid twice a year
FACE = 1_000_000


def snapshot(symbol):
    req = urllib.request.Request(f"{API}/bond/snapshot/{symbol}", headers={"x-api-key": KEY})
    with urllib.request.urlopen(req, timeout=30) as resp:
        return json.load(resp)


def cash_flows(coupon, years):
    """(time in years, cash flow per 100 of face) for a bullet bond priced on a coupon date."""
    n = round(years * FREQ)
    return [(k / FREQ, coupon / FREQ + (100 if k == n else 0)) for k in range(1, n + 1)]


def price(coupon, years, ytm):
    y = ytm / 100 / FREQ
    return sum(cf / (1 + y) ** (t * FREQ) for t, cf in cash_flows(coupon, years))


def measures(coupon, years, ytm):
    """Price, Macaulay and modified duration in years, convexity in years squared."""
    y = ytm / 100 / FREQ
    pv = weighted = curve = 0.0
    for t, cf in cash_flows(coupon, years):
        k = t * FREQ
        pv += cf / (1 + y) ** k
        weighted += t * cf / (1 + y) ** k
        curve += cf * k * (k + 1) / (1 + y) ** (k + 2)
    macaulay = weighted / pv
    return pv, macaulay, macaulay / (1 + y), curve / pv / FREQ**2


print(f"{'tenor':<7}{'yield':>7}{'Macaulay':>10}{'modified':>10}{'convexity':>11}{'DV01 per 1M':>13}{'day':>8}{'P&L per 1M':>12}")
for symbol, years in TENORS.items():
    s = snapshot(symbol)
    ytm, move = s["rate"], s["change_bps"]
    p, mac, mod, conv = measures(ytm, years, ytm)  # a par bond: coupon equal to the yield
    dv01 = mod * p / 100 * FACE / 10_000
    pnl = (p / price(ytm, years, s["prev_rate"]) - 1) * FACE  # the same bond, repriced from yesterday's yield to today's
    print(f"{symbol:<7}{ytm:>6.2f}%{mac:>10.2f}{mod:>10.2f}{conv:>11.1f}{dv01:>13,.0f}{round(move, 1):>+6}bp{pnl:>+12,.0f}")
print(f"yields dated {s['date']}; change against {s['prev_date']}")

tenor    yield  Macaulay  modified  convexity  DV01 per 1M     day  P&L per 1M
US:2Y    4.58%      1.93      1.89        4.6          189    -3bp        +567
US:5Y    4.69%      4.51      4.41       22.9          441    -5bp      +2,208
US:10Y   4.90%      8.02      7.83       74.1          783    -5bp      +3,922
US:30Y   5.28%     15.37     14.97      335.3        1,497    -4bp      +5,998
yields dated 2026-11-04; change against 2026-11-03

Swap the symbols in TENORS for UK:10Y or DE:30Y to price other curves, and set FREQ = 1 for German and French government bonds, which pay one coupon a year. The yields are daily observations, each dated in the response.

Interest rate risk: what duration leaves out