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To measure how a rate increase affects a bond, reprice every remaining coupon and the principal at the higher yield. The difference between that shocked price and the current price is the dollar impact; divide by the current price for the percentage impact. For a quick estimate, multiply modified duration by the yield change, and add convexity when the move is large or precision matters.
What happens to a bond when interest rates rise?
Bond prices and market yields generally move in opposite directions. As the required yield rises, the fixed payments promised by an existing bond are worth less in present-value terms, so its market price falls. FINRA Investor Education Foundation summarizes the relationship as: “A maxim of bond investing is that when interest rates rise, bond prices fall, and vice versa.”
The size of the decline depends on the bond’s remaining cash flows, not just its maturity date. Lower-coupon bonds and longer-maturity bonds generally have greater percentage price sensitivity, all else equal, because more of their value comes from payments received farther in the future. A bond held to maturity still returns its contractual face value if the issuer pays as promised, but an investor who sells earlier is exposed to the market price created by the new yield.
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There are three useful levels of analysis:
- Exact present-value repricing: discount every remaining coupon and the principal at the shocked yield.
- Modified-duration estimate: use the bond’s first-order price sensitivity for a quick approximation.
- Duration plus convexity: add the curvature of the price-yield relationship for larger yield changes or longer maturities.
Exact method: reprice every remaining cash flow
For a fixed-coupon bond with face value F, annual coupon dollars C, m coupon payments per year, annual yield y, and N remaining coupon periods, the full price is:
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P = Σ[t=1..N] (C/m)/(1 + y/m)^t + F/(1 + y/m)^N
The first term discounts each coupon. The second discounts the face-value repayment. To measure the effect of a rate increase, calculate the price twice:
- Calculate
P_currentwith the current annual yield. - Calculate
P_shockedwith the higher annual yield. - Subtract the original price:
Dollar change = P_shocked − P_current. - Divide by the original price:
Percentage change = (P_shocked/P_current − 1) × 100.
Inputs you need
| Input | How it is used |
|---|---|
| Face value, F | Amount repaid at maturity |
| Annual coupon dollars, C | Total coupon cash paid in one year |
| Coupon frequency, m | Splits the annual coupon and annual yield into payment periods |
| Remaining coupon periods, N | Sets how many coupons and discount periods remain |
| Current annual yield | Produces the reference price |
| Shocked annual yield | Produces the price after rates rise |
Convert annual values to payment-period values
Divide the annual coupon by the number of payments per year: coupon per period = C/m. Divide the annual yield by the same frequency: yield per period = y/m. If a bond pays twice a year, for example, both the coupon and yield must be expressed on a semiannual basis before discounting. Keep the number of periods consistent: a five-year semiannual bond has 10 remaining coupon periods, not five.
Clean price, full price and accrued interest
The formula produces the value of the contractual cash flows. A trade between coupon dates also involves accrued interest. Market quotations commonly distinguish the clean price, which excludes accrued interest, from the full (dirty) price, which includes it. Add accrued interest using the applicable day-count convention when comparing the amount actually paid at settlement. The convention varies by instrument and market, so do not treat a clean-price change as the complete change in settlement cash.
Worked example: a five-year, 4% bond
Assume a bond has a $1,000 face value, pays a 4% annual coupon once per year, and has five years remaining. The annual coupon is $40. Suppose the yield rises from 4% to 5%.
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| Measure | Value |
|---|---|
| Face value | $1,000 |
| Annual coupon | $40 |
| Coupon frequency | Once per year |
| Remaining periods | 5 |
| Current yield | 4% |
| Higher yield | 5% |
At a 4% yield, the coupon rate equals the yield, so the current price is $1,000. At a 5% yield, discount the five $40 coupons and the $1,000 principal at 5%:
P_shocked = 40/1.05 + 40/1.05² + 40/1.05³ + 40/1.05⁴ + (40 + 1,000)/1.05⁵
The shocked price is $956.71. The impact is:
- Dollar change: $956.71 − $1,000 = −$43.29
- Percentage change: ($956.71 / $1,000 − 1) × 100 = −4.33%
The decline is smaller than a simple five-year-maturity rule would suggest because the investor receives coupons before maturity. The relevant duration is below five years; maturity alone is not the correct sensitivity measure.
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Modified duration: the fast estimate
Modified duration estimates the percentage price change for a small, parallel change in yield:
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ΔP/P ≈ −D* × Δy
Here, D* is modified duration in years and Δy is the annual yield change expressed as a decimal. One percentage point, or 100 basis points, is 0.0100. To estimate the dollar change:
ΔP ≈ −P × D* × Δy
How to read a duration number
A modified duration of five means that, for a small 1-percentage-point increase in yield, the bond’s price is expected to fall by approximately 5%, before convexity and other complications. A duration of six and a $1,000 price imply:
ΔP ≈ −$1,000 × 6 × 0.01 = −$60
The estimated new price is about $940. This is an approximation, not a promise of the exact trading price. FINRA describes duration as sensitivity to a one-percent rate change and notes that “the higher the duration number, the greater the sensitivity to interest rates changes.” CFA Institute defines modified duration as the first-derivative price sensitivity to yield.
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FINRA gives a practical illustration: a bond fund with a 10-year duration would be expected to decline about 10% for a 1-percentage-point rise in rates, while a fund with a two-year duration would decline about 2% for the same move. These are duration estimates before convexity, expenses, credit-spread changes, portfolio turnover and other fund effects.
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Convexity: improving the estimate for larger moves
Price does not change in a straight line as yield changes. The curve is generally convex, so duration alone becomes less reliable as the rate move grows. Add modified convexity with:
ΔP/P ≈ −D* × Δy + 0.5 × C* × (Δy)^2
C* is modified convexity. The first term is the duration estimate; the second term is the curvature adjustment.
What positive convexity means
For a normally positive-convexity bond, the adjustment generally means that a duration-only estimate understates the price gain when yields fall and overstates the price loss when yields rise. The difference becomes more important for larger yield changes and longer maturities. Convexity is therefore most useful when a scenario involves a substantial rate shock, a long-dated bond, or a decision that needs more precision than a first-order estimate provides.
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Convexity calculations assume the bond’s cash flows remain known while the yield changes. That assumption fails for callable, puttable, mortgage-backed and other bonds whose timing or amount of cash flow can change when rates move. Those securities require effective duration, effective convexity or full scenario pricing that allows cash flows to change.
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Choosing the right calculation
| Method | Best use | Strength | Limitation |
|---|---|---|---|
| Exact present-value repricing | Known cash flows and a specific shocked yield | Directly calculates the new price | Requires all cash-flow and timing inputs; a non-option bond assumption |
| Modified duration | Small yield changes and quick checks | Simple percentage and dollar estimate | Ignores curvature and can be inaccurate for large moves |
| Duration plus convexity | Larger moves, longer maturities and higher precision | Captures the main curvature effect | Still assumes cash flows do not change |
| Effective duration/convexity or scenario pricing | Callable, puttable, mortgage-backed or other option-sensitive bonds | Allows rate changes to alter expected cash flows | Requires an appropriate cash-flow or valuation model |
A repeatable workflow for a rate-shock calculation
- Identify the instrument. Record face value, coupon rate or annual coupon dollars, payment frequency, remaining maturity and whether the bond has embedded options.
- Determine the current yield and price basis. Use the yield corresponding to the quoted price and decide whether the comparison uses clean or full price.
- Define the shock. State the new annual yield and whether the change is a parallel shift applied to all relevant cash flows.
- Convert units. Express the coupon and yield per payment period, and convert basis points to decimals. For example, 75 basis points is
0.0075. - Reprice the remaining cash flows. Discount every coupon and the principal at the shocked periodic yield, then sum the results.
- Handle settlement timing. If the trade is between coupon dates, calculate accrued interest under the applicable day-count convention and keep clean and full prices distinct.
- Measure the result. Subtract the current price for the dollar change and divide by the current price for the percentage change.
- Cross-check with duration. Multiply modified duration by the yield change and apply the negative sign.
- Add convexity when warranted. Use the duration-plus-convexity expression for a larger move, long maturity or precision-sensitive decision.
- Use effective measures for changing cash flows. Do not rely on ordinary modified duration for securities whose exercise or prepayment behavior changes with rates.
Why two bonds react differently to the same rate increase
Remaining maturity
Cash received farther in the future is discounted for more periods. Extending the remaining maturity generally increases duration and therefore increases percentage price sensitivity, assuming comparable coupons, yields and credit characteristics.
Coupon rate
A lower coupon returns less cash early and leaves more value tied to the final principal repayment. Lower-coupon bonds generally have greater percentage sensitivity than otherwise similar higher-coupon bonds.
Current yield and coupon frequency
The exact price depends on the yield used to discount each payment and on how often payments arrive. Semiannual, quarterly and annual structures have different period counts and discount rates, so using an annual formula without converting the inputs produces a wrong price.
Yield-curve shape
The duration shortcut is normally framed as a parallel shift: every relevant yield moves by the same amount. If short-, medium- and long-term yields move by different amounts, a single duration number can miss the result. Exact repricing can instead apply the appropriate shocked rate to each cash flow or maturity point when the valuation framework supports a nonparallel scenario.
Credit spreads and other market changes
The rate-shock calculation isolates the effect of the specified yield change. In a real market, credit spreads, liquidity, inflation expectations and supply-demand conditions can move at the same time. A bond’s observed price change can therefore differ from the result of a pure Treasury-rate or benchmark-yield shock.
Interpreting large rate shocks
Duration scales the first-order effect, so a longer-duration portfolio can experience a substantial mark-to-market move even when the rate change sounds modest. FINRA cites a Fitch example in which a 2-percentage-point rate rise corresponded to an approximately 15% loss for a medium investment-grade corporate bond with duration 8.4 and an approximately 26% loss for a similar bond with duration 14.5. The figures illustrate the role of duration; actual outcomes depend on the bonds, yields, spreads and valuation date involved.
A price decline is an interim market-value result, not automatically a realized loss. If the investor sells, the current market price determines proceeds. If the investor holds a plain fixed-coupon bond to maturity and the issuer pays as promised, the contractual face-value repayment remains the maturity cash flow, although reinvestment opportunities and the market value along the way have changed.
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Common mistakes and how to avoid them
- Using 1 instead of 0.01 for a 1% move: yield changes in the formulas are decimals. One percentage point is
0.0100. - Confusing coupon rate with yield: the coupon determines the cash payment; the yield determines the discount rate. They are equal only in particular pricing situations.
- Using maturity as duration: coupons arrive before maturity, so duration is usually shorter than final maturity for a standard fixed-coupon bond.
- Mixing annual and periodic units: divide both annual coupon and annual yield by the payment frequency, and use the matching number of periods.
- Ignoring accrued interest: between coupon dates, clean price and full settlement price differ.
- Applying duration to an option-sensitive bond: callable, puttable and mortgage-backed cash flows can change as rates move; use effective measures or scenario pricing.
- Treating an estimate as a quote: duration and convexity describe sensitivity under assumptions. The tradable price can also reflect spreads, liquidity and other market factors.
- Assuming the result is a realized return: a mark-to-market loss becomes realized when the position is sold or otherwise settled at that price.
A compact decision guide
- Need the price for one fixed-coupon bond at one new yield? Use exact present-value repricing.
- Need a fast answer to “What does a duration of five mean?” Estimate a 5% fall for a 1-percentage-point yield rise, before convexity.
- Evaluating a larger shock or a long-maturity bond? Add convexity to duration.
- Analyzing a callable, puttable or mortgage-backed security? Use effective duration, effective convexity or scenario pricing that permits cash-flow changes.
- Comparing a trade between coupon dates? Separate clean price, accrued interest and full price.
Key takeaways
- Higher yields generally reduce the prices of existing fixed-coupon bonds.
- The exact impact is the new present value of every remaining coupon and the principal minus the original price.
- Modified duration gives a quick first-order percentage estimate: a duration of six implies about a 6% decline for a 1-percentage-point yield rise.
- Convexity improves the estimate for larger changes because the price-yield relationship is curved.
- Longer maturities and lower coupons generally produce greater percentage sensitivity, all else equal.
- Holding to maturity can preserve the contractual face-value repayment for a solvent issuer, but it does not remove interim market-price risk or reinvestment risk.
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