Bond Price–Yield Convexity Visualizer
A bond is a promise: pay fixed coupons on schedule, then repay the face value at maturity. Because those cash flows are fixed, the bond's price is the only thing that can change when the market's required return — the yield to maturity (YTM) — changes. Adjust the sliders to see how price, coupons, and maturity interact.
5.00%
5.00%
10 yrs
Current Bond Price
$1,000.00
At Par
Cash-Flow Breakdown
Coupon per period
$25.00
Number of periods (n)
20
PV of coupon payments
—
PV of face value
—
When the coupon rate equals the required return, each coupon exactly compensates investors for the time value of money — so the bond prices at face value (at par).
Pricing formula
$$P \;=\; \underbrace{\frac{C}{r}\!\left(1 - \frac{1}{(1+r)^n}\right)}_{\text{PV of coupons}} \;+\; \underbrace{\frac{F}{(1+r)^n}}_{\text{PV of face value}}$$ where \(C = F \times c/m\) is the coupon per period, \(r = y/m\) is the periodic required return, \(n = T \times m\) is the total number of periods, \(y\) is the annual YTM, \(m\) is coupons per year, and \(F\) is the face value.
$$P \;=\; \underbrace{\frac{C}{r}\!\left(1 - \frac{1}{(1+r)^n}\right)}_{\text{PV of coupons}} \;+\; \underbrace{\frac{F}{(1+r)^n}}_{\text{PV of face value}}$$ where \(C = F \times c/m\) is the coupon per period, \(r = y/m\) is the periodic required return, \(n = T \times m\) is the total number of periods, \(y\) is the annual YTM, \(m\) is coupons per year, and \(F\) is the face value.
Key Concepts
- Prices and yields move in opposite directions. Raise the YTM slider — the price falls. All future cash flows are discounted more heavily, lowering their present value. Lower the YTM — the price rises. This inverse relationship is fundamental to fixed income.
- The relationship is curved (convex), not linear. The price-yield curve bows toward the lower-left. For equal-sized yield increases and decreases, the price gain from a yield drop is larger than the price loss from a yield rise. This asymmetry is called convexity and is always good for bondholders.
- Longer maturities mean greater price sensitivity. More payments lie far in the future, and distant payments are discounted over more periods — so they respond more to yield changes. Compare a 2-year bond with a 25-year bond at the same coupon and YTM.
- Lower coupons mean greater price sensitivity. A low-coupon bond delivers more of its value through the face-value repayment at maturity, the single most distant cash flow. A zero-coupon bond is the extreme case: the entire value is that one payment.
- Coupon rate = YTM → price = face value. Set the coupon rate and YTM to the same value. The bond trades at par. Move them apart: higher YTM → discount bond (price < face); lower YTM → premium bond (price > face).