Bond Yield and Duration: Why Bond Prices Fall When Interest Rates Rise

Bond Pricing as Present Value of Future Cash Flows

A bond's price is nothing more than the present value of all the cash it will pay you in the future. Each coupon payment and the final repayment of face value (also called par value) are discounted back to today using the market's required yield, or yield to maturity (YTM). The formula sums the discounted value of every coupon plus the discounted face value: Price equals the sum, for each period t, of Coupon divided by (1 plus yield) raised to the power t, plus Face Value divided by (1 plus yield) raised to the power of the total number of periods. The key insight is that the coupon payments and face value themselves never change once the bond is issued — they are fixed by contract. What changes is the discount rate used to value them, and that discount rate is set by the broader market for similar debt. If newly issued bonds start offering a higher yield than your existing bond's coupon rate, your bond becomes less attractive at its old price, so its market price must drop until its effective yield matches the new market rate. This present-value framework is exactly the same one used to value any stream of future cash flows, from mortgages to corporate projects, and it is the mathematical foundation for everything else in bond analysis.

The Inverse Relationship Between Price and Yield

Because a bond's price is calculated by discounting fixed cash flows at the market yield, price and yield are mathematically locked into an inverse relationship: as the discount rate in the denominator increases, the present value of every future payment decreases, so the price falls. Conversely, when yields drop, those same fixed cash flows become relatively more valuable compared to what new bonds are offering, so the price rises. This relationship is not a market quirk or a psychological pattern — it is a direct consequence of discounting. Think of it intuitively: if you own a bond paying a fixed 4 percent coupon and suddenly new bonds of similar risk are issued paying 5 percent, nobody will pay you full face value for your lower-paying bond. Its price must fall until its total return (coupon income plus price appreciation to par at maturity) matches the 5 percent now available elsewhere. The reverse holds if rates fall: your 4 percent bond becomes more attractive than newly issued 3 percent bonds, so investors bid its price up above par. This inverse price-yield relationship is the single most important concept in fixed-income investing, and it explains why bond funds can lose value even though every underlying bond eventually pays back its stated principal.

Duration: Measuring Interest Rate Sensitivity

Duration measures how sensitive a bond's price is to changes in yield, and it comes in two closely related forms. Macaulay duration is the weighted-average time (in years) it takes to receive all of a bond's cash flows, where each payment's weight is the proportion of the bond's total present value it represents. A bond with larger, earlier cash flows (high coupons) has a shorter duration than a bond with the same maturity but smaller coupons, because more of its value arrives sooner. A zero-coupon bond's Macaulay duration equals its maturity exactly, since all value arrives in a single final payment. Modified duration adjusts Macaulay duration by dividing it by (1 plus yield per period), converting it into a direct sensitivity measure: the approximate percentage change in price for a one percentage point (100 basis point) change in yield. The relationship is expressed as: percentage change in price is approximately equal to negative modified duration multiplied by the change in yield. This makes duration an incredibly useful shorthand — a bond with a modified duration of 8 will lose roughly 8 percent of its value if yields rise by 1 percentage point, and gain roughly 8 percent if yields fall by 1 percentage point. Longer maturities, lower coupons, and lower yields all push duration higher, meaning greater price volatility for a given rate move.

Convexity: The Second-Order Correction

Duration assumes the price-yield relationship is a straight line, but it is actually curved — specifically, it is convex, bowing below the tangent line that duration draws. This means duration's linear estimate always overstates the price decline when yields rise and understates the price gain when yields fall. Convexity captures this curvature as a second-order term, much like acceleration refines a velocity-only estimate of motion. The more complete approximation becomes: percentage change in price is approximately equal to negative modified duration multiplied by the change in yield, plus one-half multiplied by convexity multiplied by the change in yield squared. Because the convexity term is added regardless of whether yields rise or fall (since the change in yield is squared, it's always positive), convexity always works in the bondholder's favor — it softens losses when rates rise and amplifies gains when rates fall. Bonds with greater convexity (typically those with longer maturities, lower coupons, and more dispersed cash flows) are more desirable, all else equal, because they offer a more favorable asymmetry between upside and downside price moves. Portfolio managers actively manage both duration and convexity together, since two bonds can share identical duration yet behave quite differently during large rate swings.

A Worked Example: Repricing a 10-Year Bond

Consider a bond with a $1,000 face value, a 4 percent annual coupon ($40 per year), and 10 years to maturity. If the market yield is exactly 4 percent, the bond's price equals its face value: $1,000, since the coupon rate matches the required return. Now suppose the market yield rises to 5 percent. Repricing the bond means discounting all eleven cash flows (ten $40 coupons plus the $1,000 principal) at 5 percent: the present value of the coupon stream is $308.87 and the present value of the principal is $613.91, giving a new price of about $922.78 — a drop of 7.72 percent. How well does duration predict this? This bond's Macaulay duration at the original 4 percent yield works out to approximately 8.44 years, giving a modified duration of 8.44 divided by 1.04, or about 8.11. The duration approximation estimates the price change as negative 8.11 multiplied by 0.01 (the 1 percentage point yield increase), predicting a drop of about 8.11 percent, or a price near $918.90. The actual drop of 7.72 percent is smaller than duration's linear prediction — exactly as convexity theory predicts, since the true price-yield curve bows above the straight-line estimate, cushioning the loss. Adding a convexity correction term brings the estimate much closer to the true $922.78 price, illustrating why professional bond desks always use both measures together rather than relying on duration alone.

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