Why Do Longer Bonds Fall More When Yields Rise?

01 · The setup

One purchase decision, two repayment dates

A treasury team needs a place for funds it may use later. It compares a 2-year bond with a 10-year bond. Each has face value USD 1,000.00, a 4.00% fixed annual coupon paid twice a year, and an initial nominal annual yield of 4.00%. Both therefore cost USD 1,000.00. The longer bond does not offer a larger coupon in this deliberately controlled comparison.

Before choosing, the team asks a practical question: if investors demand a yield of 5.00% immediately after purchase, which bond would lose more market value? A fixed coupon is a contractual cash payment; it does not fix the price at which another investor will buy the bond. This is the distinction the two repayment dates bring into focus.

02 · Shared assumptions

Hold the cash flows fixed before changing the yield

These are fictional, option-free bonds with identical credit quality. Valuation is immediately after a coupon date, with no elapsed time or accrued interest. The model uses a flat yield for each bond and changes that yield by the same 100.00 basis points. It excludes default, taxes, trading costs and exchange-rate effects. No central-bank decision or market forecast is implied.

Each half-year coupon is USD 20.00. The short bond has fewer payments to discount; the long bond leaves more cash flows, including principal repayment, further away. A new higher required return reduces the present value of those unchanged payments. The price must adjust until their discounted value matches what a buyer will pay.

03 · Compare the numbers

The longer wait produces the larger price change

At the new yield, the short bond is worth USD 981.19, a change of -1.88%. The long bond is worth USD 922.05, a change of -7.79%. Both still promise the same coupon per payment. The difference is the timing of the cash flows exposed to the higher discount rate.

The result is not a rule that maturity alone determines risk. Coupon size, yield level and embedded options also matter. A high-coupon long bond receives more of its value earlier than a comparable low-coupon bond. Here those other features are deliberately held constant so the effect of a later principal payment can be seen.

The shock changes a market yield, not a central-bank policy rate directly. Markets can price an expected policy move before the announcement; a particular bond's yield also includes compensation for its own risks. Knowing that a central bank has raised a rate is therefore insufficient to calculate either of these price changes.

04 · Understand the model

Duration measures sensitivity; convexity improves the estimate

The modified durations are 1.90 for the short bond and 8.18 for the long bond. Modified duration gives a local percentage-price approximation for a small change in the quoted annual yield. It is not the maturity date, nor a promise that a particular loss will occur. A one-percentage-point yield rise is a decimal change of 0.01, not 1.

For the long bond, duration alone estimates a new price of USD 918.24. Adding its convexity of 78.90 gives USD 922.19, closer to the full discounted-cash-flow price of USD 922.05. Convexity describes curvature: the price-yield relationship bends rather than following the tangent line forever.

The correction here is positive for either sign of the yield change because this plain fixed-cash-flow bond has positive convexity. It remains an approximation. Try a much larger change below and compare the estimates with exact repricing; do not turn duration into a substitute for the cash-flow model.

05 · Decision conditions

Match the bond to the time the money is needed

A team certain it can hold an option-free bond until maturity faces a different decision from one that must sell to meet a payment next year. A market-price decline does not by itself change the promised principal, but it matters if the bond must be sold, marked to market or used as collateral. Holding to maturity also leaves credit risk, inflation risk and the opportunity cost of locking in a rate.

The useful conclusion is therefore conditional: in this controlled case, the longer bond is more sensitive to the same yield shock. Real yield curves can twist instead of shifting together; credit spreads and liquidity can change separately. Callable bonds can behave differently and require appropriate option-sensitive measures. Use the sensitivity tool to vary assumptions, then ask whether the holding period and cash-flow dates fit the actual need.

Finally, this is an immediate price comparison with no intervening coupon. It is not an investment return over a year. A return calculation would also need the holding period, income received and what happens to that income. That is the next question addressed by the companion bond total-return case.