Introduction

Bond investors quickly encounter a central rule: when market yields rise, existing fixed-rate bond prices generally fall, and when yields fall, prices generally rise. The reason is the mathematics of present value.

The Inverse Relationship

One of the most important relationships in fixed income is that bond prices and market yields generally move in opposite directions. When required yields rise, the present value of a bond’s fixed future cash flows falls. When required yields fall, present value rises.

This is not a market convention; it follows directly from discounting.

Why the Relationship Works

Imagine a bond with a 4% coupon. If newly issued bonds of similar maturity and credit quality offer 6%, investors will not normally pay full face value for the older 4% bond. Its price must fall until the return available at the lower price becomes competitive.

If market yields instead fall to 2%, the 4% coupon becomes attractive and investors may pay more than face value.

A Numerical Example

Consider a five-year, $1,000 bond paying a 5% annual coupon. At a 5% required yield, it trades approximately at par. If the required yield rises, discounting the same $50 coupons and $1,000 principal at the higher rate produces a lower price.

The cash flows did not change. The discount rate did. This distinction is central to understanding mark-to-market losses in bond portfolios.

The Role of Duration

Not all bonds respond equally to a one-percentage-point change in yields. Duration summarizes first-order price sensitivity. A bond with modified duration of 7 would be expected to lose roughly 7% for a 1 percentage-point parallel increase in yield, before accounting for convexity.

Long maturities and low coupons generally increase duration, while shorter maturities and larger coupons generally reduce it.

Yield Changes of Different Magnitudes

The price-yield relationship is curved rather than perfectly linear. Therefore, the estimated gain from a large fall in yields is not exactly the mirror image of the estimated loss from an equally large rise. Convexity captures this curvature.

For small yield changes, duration often provides a useful approximation. For larger moves, convexity becomes increasingly relevant.

Real-World Implications

Bond investors can suffer mark-to-market losses even when the issuer continues making every promised payment. Rising yields can reduce the market value of existing fixed-rate bonds, particularly long-duration securities.

Conversely, falling yields can create substantial capital gains. This is why fixed income is not synonymous with fixed price.

Key Takeaways

Bond prices and yields move inversely because future cash flows are discounted at the market-required rate. The magnitude of the price response depends on duration and convexity as well as the size and nature of the yield move.

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