Bond Fundamentals and Pricing › The Price-Yield Relationship

The relationship between bond prices and yields is fundamental to fixed income analysis. Understanding this relationship helps explain interest rate risk and informs trading strategies. Every bond investor needs to understand how yield movements translate into price changes, since this relationship determines both risks and opportunities in fixed income markets.

For the option-free bond with fixed positive cash flows used here, price decreases as its quoted yield increases within the model domain. That price-yield identity should not be read as saying every observed market-rate increase moves every bond price down: floating payments, embedded options, credit changes, and other drivers can change the comparison.

For an otherwise comparable bond paying a 5% coupon when the required nominal yield is 6%, the regular coupon-date price is below par. With yield and coupon held fixed, extending the payment schedule can change the price further. Duration measures the local percentage response to yield changes; maturity alone does not determine every dollar price-change comparison.

For an option-free bond with positive fixed payments and admissible yields, the price-yield curve is convex. Around the same base yield, a symmetric yield decline produces a larger price gain than the loss from an equal yield rise, with other inputs fixed. This does not guarantee a profitable holding period or describe bonds whose embedded options change their cash flows.

The curvature arises from the present value formula. Since price is a sum of terms like $C/(1+y)^t$, the second derivative with respect to $y$ is positive ($\frac{d^2P}{dy^2} > 0$). To understand this intuitively, consider what happens as yields change. When yields fall, each basis point decrease has a larger dollar impact because you're multiplying the cash flows by increasingly larger discount factors. The discount factors are growing at an accelerating rate as yields decline. Conversely, as yields rise, the discount factors shrink, so each additional basis point increase has a diminishing effect on price because you're taking away smaller and smaller pieces of value. This asymmetry, where gains accelerate as yields fall and losses decelerate as yields rise, is precisely what convexity measures. Convexity quantifies how much the price-yield relationship curves away from a straight line.