Scenario Comparison
Bond Price Sensitivity
Move yield to maturity and compare exact bond repricing with duration and duration–convexity estimates.
Start with a fixed-coupon bond
Most promised coupons and principal are already fixed. If new bonds offer a higher required return, an older bond generally needs a lower price to compete. When required yields fall, its price generally rises.
A small example
A bond with face value 1,000, a 5% coupon, ten years remaining and semiannual coupons is worth about 1,081.76 at a 4% base YTM. At a 5% scenario YTM, exact repricing gives 1,000. The roughly 81.76 drop comes from discounting the same cash flows at a higher yield.
Bond assumptions
Enter bond assumptions and build the analysis to explore the price–yield relationship.
Keep the three rates distinct
The coupon rate sets the bond's promised annual coupon relative to face value and does not change with the slider. YTM is the nominal annual yield that makes the present value of all scheduled payments equal the price. Base YTM determines today's base price; scenario YTM is only for repricing. Market rates can affect required YTM, but the terms are not identical for every bond.
FORMULA
Three ways to look at the same bond
First reprice the bond at the scenario YTM, then compare two quick estimates against that result.
Exact repricing
Put the new yield into the full bond-pricing formula and discount each future cash flow again. This gives the actual result used for comparison.
Duration estimate
Treat the price–yield curve near today's yield as a straight line and use its slope to estimate the price move quickly.
Duration + convexity
The true price–yield curve bends, so convexity corrects some of the curvature that duration's straight line misses.
Then see the short formulas
Duration estimateΔP/P ≈ −Dmod × Δy
Duration + convexityΔP/P ≈ −Dmod × Δy + ½ × C × (Δy)²
50 bps = 0.50 percentage points = 0.005 in decimal form. Use 0.005 for Δy in the formulas.
Show full calculation
- Coupon per period
- Face value × coupon rate ÷ m, where m is coupon payments per year.
- Periods and periodic yield
- N = years to maturity × m; periodic yield = YTM ÷ m.
- Full bond price
- P = Σ[coupon per period ÷ (1+YTM/m)^t] + face value ÷ (1+YTM/m)^N, discounting periods 1 through N.
- Macaulay duration
- Multiply each payment's time in years by its present value, sum those amounts, and divide by the base price.
- Modified duration
- Dmod = Macaulay duration ÷ (1+base YTM/m). Δy = scenario YTM − base YTM in decimal form.
- Convexity scaling
- C is the second-order sensitivity of the base bond's cash flows. It includes the m² scaling and works with a decimal change in nominal annual YTM.
Scope and common mistakes
This model assumes fixed coupons, no embedded options, scheduled payments, and valuation on a coupon date. Accrued interest, default, taxes, fees, liquidity, and call features are excluded. Repricing holds other assumptions constant; it is not a market-price forecast. A second-order approximation can still miss large changes.
FAQ
Why does a higher YTM usually lower price?
The same coupons and principal have lower present values at a higher discount rate.
Can I use a negative YTM?
Yes, if 1+YTM/m is positive and the resulting finite price passes the existing pricing core's limits.
Is convexity always more accurate?
No. It adds second-order curvature and often helps for moderate moves, but higher-order errors remain.
Is this the settlement price?
No. The fixed-cash-flow model assumes valuation on a coupon date and excludes accrued interest.
Related bond tools
Further reading
Bond pricing, premium and discount bonds, and accrued interest · Coupon rate, current yield, and YTM · Duration and convexity