To calculate how a higher interest rate affects a bond, reprice every remaining coupon and the principal at the new 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. For a larger move, add convexity.
For example, a $1,000 five-year bond paying a 4% annual coupon is worth $1,000 when its yield is 4%. If the yield rises to 5%, its calculated price falls to $956.71: a $43.29 decline, or 4.33%. The exact result is smaller than a five-year maturity rule because the investor receives coupon payments before maturity.
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Why higher yields push bond prices lower
A fixed-rate bond promises cash flows that were set when it was issued. When market yields rise, newly issued bonds offer more attractive returns. The older bond must therefore sell at a lower price so that its fixed coupons and principal produce the new market yield for a buyer.
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The calculation below assumes a fixed-coupon bond whose promised cash flows do not change. It also treats the yield as an annual rate compounded at the bond’s coupon frequency. A callable, puttable, mortgage-backed or otherwise cash-flow-sensitive bond needs a different analysis because a rate change can alter the cash flows themselves.
Information you need before calculating the change
Collect these inputs for the bond and keep their units consistent:
- Face value (F): the principal repaid at maturity, often $1,000.
- Annual coupon dollars (C): the coupon rate multiplied by face value. A 4% coupon on $1,000 pays $40 per year.
- Coupon frequency (m): the number of coupon payments each year, such as 1 for annual or 2 for semiannual payments.
- Remaining coupon periods (N): the number of payments still due, not simply the number of calendar years.
- Current annual yield (y): the yield used to calculate the starting value.
- Shocked annual yield: the yield after the assumed rate increase.
A one-percentage-point increase is 100 basis points, or 0.0100 when used in a formula that requires a decimal. A 50-basis-point increase is 0.0050.
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Exact method: reprice every remaining cash flow
The most direct method discounts each remaining coupon and the final principal at the shocked yield. For a fixed-coupon bond, use:
P = Σ[t=1..N] (C/m)/(1 + y/m)^t + F/(1 + y/m)^N
Here, P is the bond’s present value, C/m is the coupon paid each period, y/m is the yield per coupon period and N is the number of remaining periods.
Step-by-step calculation
- Calculate the current price. Use the current yield in the formula and retain the resulting price as
P_current. - Convert annual inputs. Divide the annual coupon by
m, divide the annual yield bym, and count every remaining coupon period. - Calculate the shocked price. Replace the current yield with the higher yield and discount every remaining coupon and the principal.
- Find the dollar change. Compute
P_shocked − P_current. A negative number is a loss. - Find the percentage change. Compute
(P_shocked/P_current − 1) × 100.
This method is exact for the stated fixed cash flows and yield convention. It does not approximate the price-yield curve with a straight line. If you have a yield for each maturity rather than one single yield, discount each cash flow with the appropriate point on that curve.
Clean price, full price and accrued interest
The formula values the cash flows at the valuation date under the stated timing assumptions. A trade between coupon dates requires a distinction between the quoted clean price and the full, or dirty, price. The full price includes accrued interest owed to the seller for the portion of the coupon period that has elapsed. Add accrued interest using the applicable day-count convention when comparing the model value with a transaction amount. Otherwise, a change in accrued interest can be mistaken for a change caused by market yields.
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Worked example: a five-year, 4% bond after a one-point rise
Assume:
- Face value: $1,000
- Annual coupon rate: 4%
- Annual coupon: $40
- Coupon frequency: once per year
- Remaining maturity: five years, so five coupon periods
- Current yield: 4%
- Shocked yield: 5%
1. Current price
Because the 4% coupon rate equals the 4% current yield, the present value of the five $40 coupons and the $1,000 principal is $1,000. Thus, P_current = $1,000.
2. Price at the higher yield
At a 5% yield, discount the five coupons and the principal at 5%:
P_shocked = 40/(1.05)^1 + 40/(1.05)^2 + 40/(1.05)^3 + 40/(1.05)^4 + 40/(1.05)^5 + 1,000/(1.05)^5
The result is P_shocked = $956.71.
3. Measure the impact
- Dollar change:
$956.71 − $1,000 = −$43.29. - Percentage change:
($956.71/$1,000 − 1) × 100 = −4.33%.
The bond loses 4.33% in this immediate mark-to-market calculation. The loss is not the same as a realized loss unless the bond is sold at that price. If held to maturity and the issuer pays as promised, the principal repayment remains $1,000, although the investor still bears interim market-price risk and reinvestment risk.
Modified duration: the fast estimate
Modified duration gives the first-order percentage sensitivity of a bond’s price to a yield change. Use:
ΔP/P ≈ −D* × Δy
D* is modified duration in years and Δy is the annual yield change expressed as a decimal. To estimate the dollar move, multiply the percentage estimate by the current price:
ΔP ≈ −P × D* × Δy
How to read a duration number
If a bond has modified duration of 6 and its yield rises by one percentage point, the estimate is:
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ΔP/P ≈ −6 × 0.0100 = −0.0600
That implies an approximately 6% decline. At a $1,000 price, the estimated dollar loss is about $60 before convexity. A yield decrease of one percentage point would produce an approximately 6% gain under the same duration-only approximation.
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Duration is a sensitivity, not a maturity date and not a guarantee. A five-year bond normally has duration below five years because some value is received through coupons before maturity. A bond fund’s published duration works the same way as a portfolio-level estimate: FINRA’s examples describe a fund with 10-year duration declining about 10% for a one-percentage-point rise, while a fund with two-year duration would decline about 2% for that move.
The shortcut is most useful for relatively small yield changes when cash flows are fixed and the relevant yield curve move can reasonably be treated as parallel. It is less reliable for large shocks, long maturities or securities whose cash flows can change.
Convexity: improving the estimate for larger moves
Bond prices and yields have a curved relationship. Modified duration draws a straight-line approximation at the current yield, while convexity captures part of the curve. Add the convexity term as follows:
ΔP/P ≈ −D* × Δy + 0.5 × C* × (Δy)^2
C* is modified convexity. The yield change must again be in decimal form.
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For a normally positively convex bond, the correction generally means:
- When yields fall, the duration-only estimate understates the price gain.
- When yields rise, the duration-only estimate overstates the price loss.
The correction grows with the square of the yield change, so it matters much more for a large shock than for a small one. It also tends to matter more for longer-maturity bonds. Convexity is not a substitute for exact repricing when the cash flows are known; it is a useful refinement when you need a quick estimate and have a convexity measure available.
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Which calculation should you use?
| Method | Best use | Accuracy and effort | Cash-flow assumption |
|---|---|---|---|
| Exact present-value repricing | A specific bond and a stated shocked yield | Most accurate for the specified cash flows; requires discounting every payment | Cash flows are known and unchanged |
| Modified-duration shortcut | Fast estimate for a relatively small yield move | Low calculation effort; first-order approximation | Fixed cash flows and an approximately parallel yield shift |
| Duration plus convexity | Larger yield moves, longer maturities or a precision cross-check | More accurate than duration alone; still an approximation | Fixed cash flows with measured duration and convexity |
| Effective duration and scenario pricing | Callable, puttable, mortgage-backed or otherwise option-sensitive bonds | Designed for changing cash flows; requires rate scenarios or an appropriate model | Cash flows may change as rates move |
A practical workflow for your own bond
- Define the scenario. State whether the rate shock is 25, 50 or 100 basis points, whether it applies to one yield or the whole curve, and the valuation date.
- Gather the bond terms. Record face value, coupon rate, coupon frequency, remaining maturity, current yield and shocked yield.
- Convert to period values. Use
C/mfor each coupon andy/mfor each discount-period yield. Count the remaining periods precisely. - Reprice at the current yield. This establishes a consistent starting value under the same assumptions.
- Reprice at the shocked yield. Discount every remaining coupon and the principal, then sum the present values.
- Calculate both impacts. Report the dollar change and the percentage change relative to the original price.
- Cross-check with duration. Multiply modified duration by the decimal yield change and apply the negative sign.
- Add convexity when appropriate. Use the second-order formula for a larger move, long maturity or when the approximation needs improvement.
- Adjust for transaction conventions. If the trade is between coupon dates, separate clean price from full price and add accrued interest under the relevant day-count rule.
- Escalate option-sensitive bonds. Use effective duration, effective convexity or scenario pricing when rates can change the timing or amount of cash flows.
How maturity, coupon and duration change the result
Longer maturity
With other terms held equal, a longer-maturity bond returns more of its value later. Those distant cash flows are discounted more heavily when yields rise, so the percentage price decline is generally larger.
Lower coupon
A lower coupon postpones more of the bond’s value until maturity. That generally increases duration and makes the percentage price more sensitive to a yield change than a higher-coupon bond with otherwise similar terms.
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Higher duration
Duration provides a direct comparison across bonds and funds. A duration of 8 implies an approximately 8% price decline for a one-percentage-point yield rise, before convexity and assuming the duration remains a useful measure for the scenario. A duration of 2 implies approximately 2% under the same conditions.
Rate-shock size
A 10-basis-point move and a 200-basis-point move should not be treated as equally reliable duration-only exercises. The larger the move, the more the price-yield curve matters and the more useful convexity or exact repricing becomes.
Illustrative duration comparisons
FINRA Investor Education Foundation gives two portfolio-level illustrations: a bond fund with 10-year duration would decrease about 10% for a one-percentage-point rise in rates, while a two-year-duration fund would decrease about 2%. These are approximations, not promises of performance.
FINRA also cites a Fitch example involving a two-percentage-point rate rise. A medium investment-grade corporate bond with duration 8.4 corresponded to an approximately 15% loss, while a similar bond with duration 14.5 corresponded to an approximately 26% loss. The example shows why duration differences can produce materially different outcomes, but it does not replace repricing the particular security, spread and cash-flow schedule you own.
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Callable bonds
An issuer may redeem a callable bond when rates fall, removing some high-coupon cash flows just when a simple fixed-cash-flow model would predict the largest price gain. Use effective duration and scenario pricing rather than assuming the payment schedule is fixed.
Puttable bonds
A holder’s put right can change expected cash flows when rates or credit conditions move. Effective measures account for the option’s effect; modified duration based on an unchanging schedule may not.
Mortgage-backed securities
Prepayments can accelerate or slow as rates change, altering both the timing and amount of cash flows. Effective duration, effective convexity and scenario analysis are appropriate instead of a single fixed schedule.
Nonparallel yield-curve moves
The one-yield formula applies one shocked yield to all periods. In practice, short-, intermediate- and long-term rates can move by different amounts. Reprice each cash flow using the corresponding shocked curve when the shape of the curve changes materially. A single duration number can hide that exposure.
Between-coupon valuation dates
Ignoring accrued interest can produce an apparent price change that is really the normal transfer of coupon income between buyer and seller. Compare like with like: clean prices against clean prices, or full prices against full prices, using the correct day-count convention.
Common calculation mistakes
- Using 1 instead of 0.0100 for a one-point move. Duration formulas require a decimal yield change. Entering
1instead of0.01multiplies the estimate by 100. - Using maturity as duration. Maturity is when principal is repaid; duration reflects the timing of all cash flows and is usually shorter for a coupon bond.
- Applying the sign incorrectly. A positive yield change has a negative first-order price effect. The duration formula includes the minus sign.
- Discounting the principal but forgetting coupons. Exact repricing requires every remaining coupon plus the final principal.
- Mixing annual and periodic rates. If coupons are semiannual, use the per-period coupon, the yield divided by two and the correct number of semiannual periods.
- Treating an estimate as an exact result. Duration is first-order; convexity is second-order. Use full present-value repricing when the cash flows are known and the answer matters.
- Ignoring embedded options. A changing payment schedule invalidates the fixed-cash-flow assumption behind ordinary modified duration.
- Confusing mark-to-market loss with a default loss. A lower market price reflects the yield required by new buyers. It becomes a realized trading loss only if you sell at that price, although credit deterioration and default are separate risks.
What the result means for an investor
The calculated percentage is the estimated change in market value for the stated rate scenario. It is most relevant if you may sell, if your account is marked to market, or if you are comparing interest-rate risk across bonds and funds. Holding a plain fixed-coupon bond to maturity can return the stated principal if the issuer performs, but the market value can still fluctuate before then. Coupons received before maturity also mean that a bond’s sensitivity is governed by duration rather than by the calendar maturity alone.
Use the exact present-value result when you know the cash-flow schedule and need a defensible number. Use modified duration for a rapid answer to a small shock, and add convexity for a larger move. If rates can change the security’s cash flows, move to effective measures and scenario pricing.
Quick Recap
Key takeaways
- Higher yields generally mean lower prices for existing fixed-rate bonds.
- Exact impact is
P_shocked − P_current, where each price is the present value of all remaining coupons and principal at its respective yield. - Modified duration estimates the percentage move:
ΔP/P ≈ −D* × Δy. - Convexity refines that estimate:
ΔP/P ≈ −D* × Δy + 0.5 × C* × (Δy)^2. - Longer maturities, lower coupons and higher duration generally indicate greater percentage sensitivity, all else equal.
- Callable, puttable, mortgage-backed and other cash-flow-sensitive bonds require effective duration, effective convexity or scenario repricing.
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