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 solved iteratively from price; there's no closed form for coupon bonds. The financial calculator's TVM solver does the iteration on exam day (BA II Plus, HP 12C, HP 10B II. The only calculators GARP allows).

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.

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 the present value of its cash flows, assuming reinvestment at YTM.
• 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 3,103 words, 21 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

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