Price-Yield Relationship

The price-yield relationship describes the mathematical link between a bond’s market price and its yield to maturity. As market interest rates rise, bond prices fall, and vice versa. This inverse relationship is not linear; it follows a curve that becomes steeper at lower yields and flatter at higher yields—a property called convexity that matters deeply to bond traders.

The mechanics of the inverse relationship

When a bond is issued, its coupon payment is fixed. If market yields rise after issuance, existing bonds paying the old (lower) coupon become less attractive. Buyers will pay less for them to earn a competitive yield. The price must drop until the bond’s total return—coupon plus capital gain—equals the new market yield.

Conversely, if yields fall, existing higher-coupon bonds become more valuable. New investors will bid up their prices to secure that richer income stream.

A concrete example: suppose you hold a bond paying 4 per cent annual coupon and maturing in 10 years. If market yields rise to 5 per cent, your bond’s price must decline below par so that buyers earn 5 per cent overall. If yields fall to 3 per cent, your bond’s price rises above par because it locks in a 4 per cent coupon when others offer only 3 per cent.

Why the curve is not a straight line

The relationship between price and yield is convex—not linear. This means the price change from a 1 per cent yield rise is smaller than the price change from a 1 per cent yield fall. The curve bends more steeply at low yield levels and flattens at high yield levels.

Two factors drive convexity:

Duration effect. The longer a bond’s duration—a weighted measure of its cash flow timing—the more its price swings for a given yield move. A 30-year Treasury is far more price-sensitive than a 2-year note.

Yield-decline asymmetry. When yields drop, a bond’s price can only rise so far (it approaches but never exceeds par plus accrued interest and any premium). But when yields rise, a bond’s price can fall to near zero, in theory. This asymmetry creates the curvature.

Bond traders exploit convexity. A portfolio manager holding bonds sees their value rise more than linear math predicts when yields fall and decline less than linear math predicts when yields rise. This is a free option—convexity works in the bondholder’s favour.

How price-yield curves differ across bond types

The curve’s shape varies with maturity and credit risk. A short-maturity bond exhibits steep convexity at very low yields but flattens quickly at higher yields. A long-maturity bond shows more dramatic price swings across a wide yield range.

Junk bonds and other credit-risky issues display a different profile: their price-yield curve becomes asymmetrical if default risk rises. A 5 per cent yield move may leave the bond price relatively flat or even shift the relationship entirely if the market reprices credit risk mid-calculation.

Callable bonds have embedded options that truncate the price-yield relationship at high prices. If yields fall far enough, the issuer will call the bond, capping price appreciation. This creates negative convexity for the bondholder—the curve flattens or reverses in the low-yield region.

Using the relationship in portfolio management

Fixed-income managers rely on price-yield relationships to make allocation and hedging decisions. If a manager believes yields will fall, she will extend duration—buy longer bonds that offer more price upside per yield move. If she expects rates to rise, she shortens duration.

The curve also explains duration and convexity hedging. Derivatives can neutralize both duration risk (first-order sensitivity) and convexity risk (second-order sensitivity), allowing a manager to isolate a specific market view.

Price-yield relationships also inform relative valuation. Two bonds of similar credit quality but different maturity will have different price-yield curves. By comparing their curves, traders identify which is relatively cheap and which is relatively rich, executing a bond swap to capture the mispricing.

The role of interest rate risk

The inverse relationship encodes interest rate risk. The more steeply a bond’s price falls as yields rise, the greater the interest rate risk. Investors can quantify this risk using duration, which approximates the percentage price change for a 1 per cent yield move.

A bond with a duration of 5 years will lose roughly 5 per cent in value if yields rise 1 per cent. But due to convexity, the actual loss may be slightly less than 5 per cent. This non-linearity is why both duration and convexity matter in risk measurement.

Central banks, when raising policy rates, cause the entire yield curve to shift higher. The price-yield relationship ensures that all bond investors face losses proportional to their duration—a coordinated shock that resets discount rates across the fixed-income market.

See also