TL;DR
When interest rates rise, bond prices fall. The math behind that relationship rests on four concepts every fixed-income investor needs to understand: price, yield to maturity, duration, and convexity. Master these and you can estimate how much any bond will gain or lose as rates move — without relying on a black-box model.
The Building Blocks of a Bond
A bond is a debt security — essentially an IOU. A borrower (a government or corporation) promises to pay you a fixed interest rate on a set face value, and to return that face value when the bond matures. According to the SEC’s investor education site, three numbers define every bond:
- Par value (face value) — the amount repaid at maturity, typically $1,000.
- Coupon rate — the annual interest as a percentage of par. A 4% coupon on a $1,000 bond pays $40 per year, usually split into two $20 semiannual payments.
- Maturity — the date the issuer repays the par value and the bond stops paying interest.
The Price-Yield Relationship: Why They Always Move in Opposite Directions
Bonds trade in the open market every day, and their prices fluctuate even though the coupon is fixed. Here is why: suppose you own a $1,000 bond paying 4% a year. If new bonds come to market paying 6%, no rational investor will pay you face value — they will demand a discount. Conversely, if new bonds pay only 2%, investors will happily pay a premium to lock in your 4%.
This is the iron law of bonds: price and yield move in opposite directions, always.
The exact price of a bond is the present value of every future cash flow, discounted at the current market yield:
Price = C/(1+r) + C/(1+r)² + ··· + (C+F)/(1+r)ⁿ
Where C = coupon payment per period, r = yield per period, F = face value, and n = total number of periods. When r rises, every denominator grows, so every fraction shrinks, and the price falls. Simple as that.
Worked Example
Take a $1,000 par bond with a 4% annual coupon and 10 years to maturity, paying coupons semiannually. At different market yields, its price changes dramatically:
| Yield to Maturity (YTM) | Bond Price | vs. Par ($1,000) | Status |
|---|---|---|---|
| 2% | $1,180 | +18.0% | Premium |
| 3% | $1,086 | +8.6% | Premium |
| 4% | $1,000 | 0.0% | Par (coupon rate) |
| 5% | $922 | -7.8% | Discount |
| 6% | $851 | -14.9% | Discount |
| 7% | $787 | -21.3% | Discount |
| 8% | $728 | -27.2% | Discount |
Source: Standard bond pricing formula; author’s calculation.
Three Types of Yield — and Why YTM Is the One That Matters
- Coupon yield — the annual interest payment divided by par value. On our example bond, it is always 4%, fixed forever. It ignores market price entirely.
- Current yield — annual coupon divided by the current market price. Better, but still ignores the capital gain (or loss) you will realize at maturity.
- Yield to maturity (YTM) — the total annualized return if you buy the bond today at market price, receive all coupons, and hold to maturity, assuming coupons are reinvested at the same rate. YTM is the standard yield quoted on financial terminals and is what the table above uses.
A simple rule of thumb: when a bond trades at a discount (below par), its YTM is above its coupon rate. When it trades at a premium (above par), its YTM is below its coupon rate.
Duration: The Bond’s Price-Sensitivity Speedometer
Knowing that bond prices fall when yields rise is useful. Knowing by how much is essential.
Duration is that “how much.” It measures a bond’s price sensitivity to a 1% change in interest rates. Think of it as the speedometer for your bond’s interest-rate risk.
Macaulay Duration
Macaulay Duration is the weighted average time you wait to receive all the bond’s cash flows, where each cash flow is weighted by its share of the bond’s total present value. Duration is measured in years, and it is always shorter than the bond’s maturity for coupon-paying bonds (because you receive some cash early, via coupons).
Here is a step-by-step calculation for a simpler 5-year bond — $1,000 par, 4% annual coupon, priced at par (YTM = 4%):
| Year | Cash Flow | PV @ 4% | Weight (% of Price) | Year × Weight |
|---|---|---|---|---|
| 1 | $40 | $38.46 | 3.85% | 0.0385 |
| 2 | $40 | $36.98 | 3.70% | 0.0740 |
| 3 | $40 | $35.56 | 3.56% | 0.1068 |
| 4 | $40 | $34.19 | 3.42% | 0.1368 |
| 5 | $1,040 | $854.80 | 85.48% | 4.2740 |
| Total | — | $1,000 | 100% | 4.63 years |
Macaulay Duration = 4.63 years. Even though this bond matures in 5 years, its “effective” time horizon is only 4.63 years because the earlier coupon payments pull the center of gravity forward.
Modified Duration: From Years to Price Sensitivity
Modified Duration converts Macaulay Duration into a direct price-change estimate per 1% move in yield:
Modified Duration = Macaulay Duration ÷ (1 + YTM) = 4.63 ÷ 1.04 = 4.45 years
Interpretation: if the market yield rises by 1 percentage point (say from 4% to 5%), this bond’s price falls by approximately 4.45% — or about $44.50 on a $1,000 bond.
A quick mental rule: the longer the maturity and the lower the coupon, the higher the duration, and the bigger the price swing for any given rate move. Zero-coupon bonds are the extreme case — their Macaulay Duration equals their maturity exactly, because all cash arrives in one lump at the end.
Convexity: Why Duration Is Only an Approximation
Duration is a linear approximation of a curved reality. In the price-yield chart above, notice that the line is curved, not straight — it bows outward toward the bottom-left. Duration measures the slope of that curve at a single point. Move far enough and the slope changes, making the straight-line estimate inaccurate.
Convexity measures that curvature. The fuller price-change formula adds a second-order correction term:
ΔP/P ≈ −ModDuration × Δy + ½ × Convexity × (Δy)²
For small rate moves (under 50 basis points), the convexity correction is tiny — duration alone is good enough. For larger moves, like a 200bps rate shock, convexity becomes material. Bond portfolio managers running tens of billions of dollars in treasuries model it explicitly.
Here is the important practical insight: convexity is always favorable for a straight bond. For a given duration, a more convex bond gains more when rates fall and loses less when rates rise, compared to a less convex bond. This is why investors typically pay a premium for high-convexity bonds.
The exception is negative convexity. Mortgage-backed securities (MBS) exhibit negative convexity because homeowners refinance (prepay) when rates fall, cutting off the bondholder’s upside. Callable corporate bonds share the same trait — the issuer calls the bond away precisely when you would want to keep it.
The Current Treasury Yield Landscape (May 2026)
With the 30-year Treasury yield near 5%, duration risk is front-and-center for fixed-income investors. A 30-year Treasury bond currently has a Modified Duration of roughly 17–18 years — meaning a 1% rise in long yields would drop its price by around 17-18%. The current yield snapshot illustrates the relatively flat curve from 5-year to 30-year maturities, which means investors are not being well-compensated for taking on extra duration risk at the long end.
Yahoo Finance
(^TNX, ^TYX, 13-week T-bill). Data reflects recent market close; verify before acting.
Common Mistakes to Avoid
- Confusing coupon rate with YTM. A bond with a 4% coupon trading at a discount has a YTM above 4%. The coupon is fixed; the yield is market-determined and changes every second.
- Treating “duration” as a time to maturity. A Modified Duration of 4.45 years does NOT mean the bond matures in 4.45 years. It means a 1% rate move changes the price by ~4.45%. Duration is a price-sensitivity number, not a time horizon.
- Ignoring convexity for long bonds. For 20- or 30-year treasuries, a 200bps rate move modeled with duration alone can be off by 2–4% in price terms. Always add the convexity correction for large shocks.
- Forgetting reinvestment risk in YTM. YTM assumes all coupons are reinvested at that same yield throughout the bond’s life. In practice reinvestment rates vary, so your realized total return will differ from stated YTM — often by more than you expect over a 10- or 20-year holding period.
What to Learn Next
With price, yield, duration, and convexity in your toolkit, here are the logical next steps:
- The yield curve and its shape — why the spread between the 2-year and 10-year Treasury yields is one of the most-watched recession indicators.
- Spread duration — how corporate bond credit spreads add an additional layer of risk on top of pure rate duration.
- Interest rate swaps — how institutional investors hedge duration risk without selling the underlying bonds.
- Dollar duration (DV01) — the dollar change in portfolio value for a 1 basis point (0.01%) move in yield, the standard risk measure on trading desks.
Sources
- U.S. Securities and Exchange Commission — Investor.gov: Bonds and Fixed Income
- Yahoo Finance — US Treasury Yield Rates (^TNX, ^TYX, 13-week T-bill), May 2026
Disclosure: This article was produced with AI assistance and reviewed before publication. It is for informational purposes only and is not investment advice.