Bond Yields, Duration, Convexity, and DV01

A fixed-income desk holds $500 million face of a 10-year Treasury and needs to hedge against rate moves. The trader pulls duration off the screen, multiplies, and sells $80 million of 2-year futures. Two weeks later rates rise 50 basis points. The hedge captures most of the loss but leaves an $800,000 residual. The duration number was right; convexity captured the rest, and the trader who knows that doesn't get a margin call.

YTM is the discount rate that, applied uniformly across all cash flows, equates the bond price to the sum of present values:

$$P = \sum_{i=1}^{n} \frac{C_i}{(1 + y/k)^{k t_i}}$$

YTM is generally solved numerically from price. A financial calculator's TVM solver performs that calculation; simple one- or two-period cases can also be solved algebraically.

YTM has two interpretations. Mathematical: the IRR of the bond's cash flows at the current price. Investment: the realized return if held to maturity AND all coupons reinvested at YTM AND no default. The reinvestment-at-YTM assumption is restrictive; realized returns deviate from YTM when reinvestment rates differ.

Common mistakes

• Confusing Macaulay and modified duration. Macaulay is in years (weighted-average time); modified is the price-sensitivity coefficient (in years per unit yield). The numbers differ by a factor of $(1 + y/k)$. Trap: a question asks for "the duration that gives % price change". The answer is modified, not Macaulay.
• Forgetting the convexity adjustment for large moves. Linear duration is accurate for small yield changes but biased for moves beyond 50 bp. For a 200 bp move, ignoring convexity leaves 1-2% on the table.
• Treating callable-bond duration as analytical duration. A callable bond's effective duration is shorter than its analytical duration. And gets shorter as rates approach the call threshold. Using analytical duration to hedge a callable book overstates the hedge size.

Bottom line

• Yield to maturity (YTM) is a bond's IRR: the single discount rate equating price to promised cash flows. Realizing that compound return requires reinvestment at YTM and no default through maturity.
• Macaulay duration is the weighted-average time to cash flows; modified duration = Macaulay / (1 + y/k) gives the % price change per 1% yield change.
• DV01 (dollar value of an 01) = price change per 1 bp: $\text{DV01} \approx \text{Modified D} \times P / 10{,}000$; hedge by matching DV01s.
• Convexity is the second-order term: $\Delta P / P \approx -D_{\text{mod}} \Delta y + \frac{1}{2} C (\Delta y)^2$; always positive for option-free bonds and grows quadratically with the yield move.

Exam shortcut

When a question gives modified duration and yield change, multiply (with a negative sign) for the percentage price change. For moves bigger than 50 bp, ALWAYS add the convexity term. Duration-only gets the sign right but the magnitude wrong. For DV01, divide modified-duration price-sensitivity by 10,000. Memory aid: "Macaulay measures time, modified measures price, DV01 measures dollars." Three measures, three units.

The full lesson (about 6,046 words, 40 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

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