Price–yield curvature, strips and duration intuition
◈ 10 cardsTabulate price against yield to see the inverse, convex relation — a fall in yield helps more than a rise hurts — then rank bonds by sensitivity: longer maturity and lower coupon move most, and the strip is the extreme.
The curve, not the line
Price Prairie Grid's 5.2 % ten-year at five yields, =-PV(B1/2,20,26,1000) filled down a column of yields in B:
| Yield | Price | Change from 6 % |
|---|---|---|
| 4 % | 1,098.11 | +16.76 % |
| 5 % | 1,015.59 | +7.99 % |
| 6 % | 940.49 | — |
| 7 % | 872.09 | −7.27 % |
| 8 % | 809.74 | −13.90 % |
Price and yield move inversely — that much was Lesson 10.3. The new fact is the asymmetry: a one-point fall in yield adds 7.99 %, a one-point rise takes only 7.27 %. Plot the five points and the line bows toward the origin — it is convex. Each successive one-point rise costs fewer dollars (82.52, 75.10, 68.40, 62.35) because each is a smaller fraction of a smaller price and discounts flows already heavily discounted. Convexity is good for the holder: gains outrun losses for the same move.
Maturity and coupon — the duration intuition
How much a bond moves for a given yield change depends on how far out its dollars sit. A one-point rise from 6 % to 7 % on three 5.2 % bonds:
| Term | Price at 6 % | Price at 7 % | Change |
|---|---|---|---|
| 3 years | 978.33 | 952.04 | −2.69 % |
| 10 years | 940.49 | 872.09 | −7.27 % |
| 20 years | 907.54 | 807.80 | −10.99 % |
Longer maturity, bigger move. The same logic runs on the coupon: a lower coupon puts more of the bond's value in the distant par and less in the near coupons, so it moves more for its term. The extreme is the strip (Lesson 9.4) — no coupons, every dollar at the end. The ten-year strip at 6 %: =-PV(0.03,20,0,1000) = 553.68; at 4.5 %: =-PV(0.0225,20,0,1000) = 640.82 — a +15.74 % move, against +12.27 % for the coupon bond over the same 1.5-point fall. In AFM 121 duration is this ranking: longer maturity and lower coupon mean higher duration mean more interest-rate risk. Rank four bonds cold: 20-year strip > 20-year 8 % > 3-year strip? No — the 3-year strip has 3 years of duration and the 20-year coupon bond has more than that; 20-year strip > 20-year 8 % coupon > 3-year strip > 3-year 8 % coupon.
The two-row table
When rates rise a bondholder has two risks pulling opposite ways (Lesson 9.6): the price falls, but every coupon is now reinvested at a higher rate. Rates fall: price up, reinvestment down. Duration is where the two balance; for a holder whose horizon equals the bond's duration, the two roughly cancel.
Recompute
Fill the 8 % row for the three terms: 3-year 926.61, 10-year 809.74, 20-year 722.90 — the same yield, deepening down the maturity axis. Rank by sensitivity, cold: long before short, low coupon before high, strip at the top.