The second half's payoff needs one precise tool: duration, the measure of how much a bond's price moves when its yield moves. Get the intuition first, then the formula.
Why do long bonds move more?
The intuition. A bond is a stream of fixed future cash flows (coupons plus the face value at maturity). Its price is the present value of that stream at the current market yield. When the yield rises, every one of those future cash flows is discounted harder, so the price falls. And here is the crux: the cash flows further in the future are discounted over more years, so they lose proportionally more value when the rate rises. A bond with more of its value in distant cash flows (a longer maturity, or a lower coupon) therefore falls more for a given yield rise. That sensitivity is what "duration" measures. This is the same idea as the terminal-value point in interest rates and asset prices and the growth-stock point in disinflation vs deflation: one principle (distant cash flows are rate-sensitive) wearing three costumes.
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What is the formula for bond duration?
The formula. Macaulay duration is the weighted-average time to receive the bond's cash flows, each weighted by its share of the bond's present value, measured in years. A zero-coupon bond pays only at maturity, so its Macaulay duration equals its maturity; a coupon bond pays some cash early, so its duration is less than its maturity. What you actually use for price moves is modified duration:
modified duration = Macaulay duration / (1 + y/k)
(where y is the annual yield and k the coupons per year), and it delivers the one equation you will use forever:
% change in bond price ≈ − modified duration × change in yield
So a bond with modified duration 8 loses about 8% for every +1% (100 bp) rise in its yield (and gains ~8% for a 100 bp fall). Longer-maturity and lower-coupon bonds have higher duration and therefore bigger price swings, the precise, quantified reason "long bonds move more." A 30-year bond might have a duration near 18–20 and move nearly ten times as much as a 2-year (duration ~1.9) for the same yield change, which is why the long end is where the action (and the risk) is, and why an insurer or pension fund matching 30-year liabilities lives and dies by the long end while a bank funding itself overnight barely feels it.
What is convexity?
One honest refinement: convexity. The duration equation is a linear approximation to a relationship that is actually curved (convex): the true price–yield line bows, so duration overstates the loss on a big rate rise and understates the gain on a big rate fall, a small, favourable error for the bondholder. For the modest moves you will usually analyze, duration is close enough; for large moves you add a convexity correction (+ ½ × convexity × (Δy)²). Worked Example 3 shows the duration prediction and the exact repricing side by side, and the gap between them is the convexity, so you will see the concept as a number rather than a word. The practical takeaway for the module: duration is the bridge between a rate move and a price move, it is larger for longer-dated and lower-coupon bonds, and it is the exact mechanical cousin of the "equity duration" idea that makes growth stocks the long bonds of the stock market, which is where interest rates and asset prices takes it.