To measure exactly how a higher interest rate changes a bond’s value, reprice every remaining coupon and the principal at the new yield. Subtract the original price from that shocked price for the dollar change, then divide by the original price for the percentage change. For a fast estimate, multiply modified duration by the change in yield; add convexity when the move is large or the bond is long-dated.
The key relationship is inverse: when market yields rise, the price of an existing fixed-rate bond generally falls. The size of the fall depends mainly on the bond’s remaining cash flows, maturity, coupon, duration, convexity and any embedded options.
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Why higher yields push existing bond prices down
A fixed-coupon bond promises cash flows that were set when it was issued. If newly issued bonds begin offering higher yields, an older bond with a lower coupon is less attractive unless its market price declines enough to give a buyer the new required yield. This is why the FINRA Investor Education Foundation described the core maxim in 2017 as: “when interest rates rise, bond prices fall, and vice versa.”
The effect applies to the market value before maturity. If you hold an otherwise non-defaulting bond to its maturity date, you generally receive its stated principal, but that does not remove interim mark-to-market risk. Selling earlier can produce a gain or loss because the bond must be sold at the price investors require at that time.
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Two bonds exposed to the same rate increase can have very different price changes. Longer remaining maturities and lower coupons generally create greater percentage sensitivity because more of the bond’s value arrives farther in the future. Duration summarizes that sensitivity; convexity captures the curvature that duration alone misses.
Collect the inputs before calculating
For a fixed-coupon bond whose cash flows will not change, gather these values:
- Face value (F): the principal repaid at maturity, such as $1,000.
- Annual coupon dollars (C): coupon rate multiplied by face value. A 4% coupon on $1,000 pays $40 per year.
- Coupon frequency (m): the number of payments each year, such as 1 for annual or 2 for semiannual payments.
- Remaining coupon periods (N): the number of payments still due, not the original term at issuance.
- Current annual yield (y): the yield used to calculate the bond’s current price.
- Shocked annual yield: the new yield after the assumed rate increase.
Express yields as decimals in calculations. A 1-percentage-point increase is 100 basis points, or 0.0100, not 1.00 in the formula. Keep the coupon frequency consistent with the yield convention: divide both the annual coupon and annual yield by the same number of periods.
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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 (dirty) price is:
P = Σ[t=1..N] (C/m)/(1 + y/m)t + F/(1 + y/m)N
The first term discounts each coupon payment. The second discounts the principal repayment at the final period. Calculate the price twice:
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- Current price: use the current yield.
- Shocked price: use the higher yield.
Then calculate:
Dollar change = Pshocked − Pcurrent
Percentage change = (Pshocked / Pcurrent − 1) × 100
This approach is the most accurate choice when the bond’s future cash flows are known. It directly implements the valuation principle that a bond’s price is the present value of its promised cash flows discounted at the required yield.
Clean price versus full price
The formula above gives the full price, which includes interest accrued since the last coupon date. Market quotations often show a clean price that excludes accrued interest. If the transaction occurs between coupon dates, calculate accrued interest using the applicable day-count convention and add it to the clean price to obtain the full amount exchanged. Use the same price basis when comparing the current and shocked values; otherwise, accrued interest can be mistaken for a rate-driven gain or loss.
Step-by-step exact calculation
- List the bond terms. Record face value, coupon rate and dollars, payment frequency, remaining maturity, current yield and the shocked yield.
- Convert to per-period figures. Set the coupon payment to
C/m, the discount rate toy/m, and the period count toN. - Discount each coupon at the shocked yield. For period t, divide
C/mby(1 + yshock/m)t. - Discount principal. Divide F by
(1 + yshock/m)N. - Sum the present values. The result is the new full price.
- Repeat at the original yield if the current market price is not already known, producing a consistent comparison.
- Subtract and divide. Compute the dollar and percentage changes using the two formulas above.
- Check the result with duration. A duration estimate should point in the same direction and be close for a small yield move.
- Add convexity for a larger move. Use the second-order estimate described below, or rely on the exact repricing when precision matters.
Worked example: a five-year, 4% bond after a one-point rise
Assume a $1,000 face-value bond with five years remaining, a 4% annual coupon paid once per year, and a yield that rises from 4% to 5%.
| Input or result | Value |
|---|---|
| Face value | $1,000 |
| Annual coupon | $40 |
| Remaining annual payments | 5 |
| Current yield | 4% |
| Shocked yield | 5% |
| Price at 4% yield | $1,000.00 |
| Price at 5% yield | $956.71 |
| Dollar change | −$43.29 |
| Percentage change | −4.33% |
The current price is exactly par because the 4% coupon rate equals the 4% yield. At a 5% yield, discounting all five $40 coupons and the $1,000 principal produces a price of $956.71. The decline is smaller than a simple five-year-maturity rule would suggest because the investor receives coupons before maturity. The bond’s relevant duration is below five years, which is why duration and convexity—not maturity alone—are the appropriate sensitivity measures.
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Duration: the fast first-order estimate
Modified duration measures the approximate percentage change in price for a one-unit change in annual yield. The shortcut is:
ΔP/P ≈ −D* × Δy
D* is modified duration in years, and Δy is the yield change expressed as a decimal. To convert the percentage estimate into dollars:
ΔP ≈ −P × D* × Δy
Interpreting a duration of five or six
If a bond has modified duration of 5, a 1-percentage-point (0.0100) rise in yield implies an approximate 5% price decline, before convexity and assuming the yield change applies in parallel to the relevant curve. A bond priced at $1,000 with duration 6 would have an estimated change of:
−$1,000 × 6 × 0.0100 = −$60
That is approximately −6%, taking the price to about $940. It is an estimate, not a guarantee: the actual result depends on the full cash-flow calculation and the bond’s convexity.
What duration comparisons show
FINRA’s 2017 investor education material describes a bond fund with 10-year duration as losing about 10% for a 1-percentage-point rate rise, while a two-year-duration fund would lose about 2% for the same move. It also cites a Fitch example in which a 2-point 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. Those figures illustrate the sensitivity relationship; they are not universal results for every fund or bond.
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Duration is most useful for quick screening, portfolio comparison and small, approximately parallel yield shifts. FINRA summarizes the interpretation simply: “The higher the duration number, the greater the sensitivity to interest rates changes.”
Convexity: improving the estimate for larger moves
Bond prices and yields do not form a straight line. The duration shortcut is the tangent-line approximation at the current yield. Convexity measures the curvature of that price-yield relationship. Include modified convexity when the yield move is large, the maturity is long, or a closer estimate is important:
ΔP/P ≈ −D* × Δy + 0.5 × C* × (Δy)2
C* is modified convexity. Because the squared yield change is always positive, positive convexity generally means the duration-only method understates the price gain when yields fall and overstates the price loss when yields rise. The correction grows as the yield move and the bond’s convexity increase.
For a small change, the convexity term may be negligible. For a multi-point move or a long-maturity bond, omitting it can materially distort the estimate. Exact present-value repricing remains preferable when all cash flows are known; convexity is a practical second-order cross-check when you have duration and convexity measures but are not rebuilding every cash flow.
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| Method | Accuracy | Work required | Best use | Key limitation |
|---|---|---|---|---|
| Exact present-value repricing | Highest when cash flows are fixed and inputs are correct | Discount and sum every remaining cash flow | Specific bond quotes, trade decisions and precise scenario analysis | Requires complete cash-flow schedule; clean/dirty treatment must be consistent |
| Modified-duration shortcut | First-order approximation | Multiply duration by the decimal yield change | Small moves, rapid comparisons and portfolio-level screening | Misses curvature and can be inaccurate for large moves |
| Duration plus convexity | Better second-order approximation | Add the convexity term to the duration result | Larger moves, long maturities and precision checks | Still assumes the relevant cash flows and sensitivity measures remain appropriate |
| Effective duration/convexity or scenario pricing | Designed for changing cash flows | Revalue under rate scenarios | Callable, puttable, mortgage-backed and other option-embedded bonds | Requires an interest-rate model or reliable scenario valuations |
All three fixed-cash-flow approaches generally assume a parallel shift: the yields used to discount each period move together. If short- and long-term rates move by different amounts, use a key-rate or scenario analysis rather than one single duration number; the formulas presented here do not establish a specific nonparallel method or result.
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Bonds whose cash flows can change
Standard modified duration assumes the amount and timing of future cash flows are known. That assumption can fail for callable bonds, puttable bonds, mortgage-backed securities and other instruments with embedded options. A rate increase can change the likelihood or timing of a call, prepayment or exercise, so the bond’s cash flows change at the same time as its discount rate.
For these securities, use effective duration and effective convexity, which are based on revaluation under rate scenarios that allow cash flows to change. Scenario pricing is preferable when a single duration number cannot represent the option behavior. Do not apply the fixed-cash-flow formula blindly and present the result as exact for an option-embedded bond.
How coupon, maturity and timing affect the result
Coupon size
Lower-coupon bonds return less cash before maturity, leaving more value exposed to distant discounting. Holding other features constant, their percentage price response to a yield change is generally larger than that of higher-coupon bonds.
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A longer remaining term usually increases duration because principal is discounted over more periods. The effect is based on remaining maturity, not the bond’s original term. A bond with two years left can have very different sensitivity from the same issue when it had ten years remaining.
Coupon frequency
Payment frequency changes the timing of cash flows. Convert both coupon and yield to per-period values before discounting; mixing an annual yield with semiannual periods produces a wrong price and an unreliable rate impact.
Settlement date
A trade between coupon dates requires accrued interest and the applicable day-count convention. Comparing a clean quoted price with a full price can create an apparent change that is unrelated to the yield shock.
Common mistakes and how to correct them
- Using 1 instead of 0.0100 for a one-point move: convert percentage points to decimal yield units before multiplying by duration.
- Using maturity as duration: coupons arrive before maturity, so calculate or obtain modified duration rather than assuming the two are equal.
- Discounting only the principal: include every remaining coupon; omitting them exaggerates the price decline for coupon-paying bonds.
- Applying a duration estimate to a large shock without qualification: add convexity or perform exact repricing.
- Ignoring coupon frequency: divide both annual coupon and annual yield by the payment frequency and use the correct number of periods.
- Mixing clean and dirty prices: add accrued interest consistently when the valuation date falls between coupon dates.
- Assuming all bonds have fixed cash flows: use effective measures or scenario pricing for securities with embedded options.
- Interpreting a maturity payment as protection from interim losses: principal repayment at maturity does not prevent a market-value loss if you sell earlier.
- Treating a duration example as a promise: duration describes approximate sensitivity under stated assumptions; it is not a guaranteed return.
A practical decision process
- Need the price for one identifiable fixed-coupon bond? Use exact present-value repricing with the shocked yield.
- Need a rapid answer for a small, parallel move? Use modified duration and report the result as an approximation.
- Is the move large or the maturity long? Add modified convexity, or use exact repricing if the cash flows are available.
- Can rates move unevenly across maturities? Test separate curve scenarios rather than relying on one parallel-shift duration.
- Can the bond’s cash flows change? Use effective duration/convexity or scenario pricing instead of fixed-cash-flow measures.
- Will the trade settle between coupons? Keep clean price, accrued interest and full price clearly separated.
The result should always state the assumed current yield, shocked yield, yield units, price basis, remaining maturity and whether the calculation is exact or an approximation. Those details determine whether another reader can reproduce the number and understand what it means.
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