New on FinoLib · Volume II: Capital Markets
I have just published a new article on FinoLib, my free finance education library: Bond Convexity and the Yield Curve: Spot and Forward Rates. It is part of Volume II: Capital Markets. Below is a summary of the topic in plain, expert language, so you can follow the idea even before you open the full article.
3.3 Convexity — The Curve Beneath Duration
The idea in plain words
Duration gives a straight-line estimate of how a bond’s price changes, but real price changes bend away from the line. Convexity measures that bend. For ordinary bonds the bend works in your favor: prices rise more when yields fall than they drop when yields rise by the same amount. It matters most for long bonds and big moves. Callable bonds and mortgage-backed securities can have the opposite, negative convexity, which works against you.
Key points
■Convexity grows roughly with the square of maturity: 9.11 for the 3-year Part 3 bond, 165.6 for a 20-year 7% bond.
■For the 20-year bond at −200 bp, duration alone predicts +21.36%; the exact gain is +25.10%.
■Calls and mortgage prepayments cap gains in rallies and extend losses in sell-offs: negative convexity.
A closer look
Duration is a useful approximation, but it is a straight-line (linear) estimate of a relationship that is actually curved. Convexity corrects for this curvature: it captures the fact that a bond’s price rises slightly more than duration alone would predict when yields fall, and falls slightly less than duration alone would predict when yields rise. In other words, positive convexity is a small but real bonus baked into most ordinary bonds — good news happens a bit more than expected, bad news a bit less.
Decision rule
Add the convexity term when ½ × C × Δy² exceeds your tolerance, say 0.10% of price, which happens beyond Δy = √(2 × 0.001 ÷ C): about 148 bp for the Part 3 bond (C = 9.11) but only 35 bp for the 20-year 7% bond (C = 165.6). For callables and MBS, use effective convexity from an option model. For day-to-day DV01 hedging of small moves, ignore convexity.
The costliest mistake
Sizing a large-move hedge on duration alone. For $100 million of the 20-year 7% bond, duration predicts a 21.36% gain if yields fall 200 bp; the exact gain is 25.10%. The $3.74 million gap is loss a duration-sized short hedge fails to offset. Avoid it: for long bonds and big scenarios, reprice at the stressed yield.
Recap
Convexity corrects duration’s straight-line estimate and grows roughly with the square of maturity, so it matters most for long bonds and large moves (165.6 for a 20-year 7% bond versus 9.11 for the Part 3 bond). Calls and mortgage prepayments create negative convexity, and duration hedges fail when the curve twists, spreads widen, moves are large or collateral calls demand cash.
3.4 The Yield Curve, Mathematically — Spot Rates and Forward Rates
The idea in plain words
A spot rate is the yield on a single payment due on one future date. A forward rate is the rate for borrowing between two future dates, and today’s spot curve already locks it in. You can work out spot rates from the prices of coupon bonds, step by step, in a method called bootstrapping. Forward rates then act as break-even rates. They also include a term premium, so they are not pure forecasts of where rates will go.
Key points
■Par yields of 4.50%, 4.80% and 5.00% bootstrap to spot rates of 4.500%, 4.807% and 5.015%.
■Forward rate = (1 + longer spot)^n ÷ (1 + shorter spot)^m − 1, annualized over the gap.
■Nominal yield minus TIPS real yield is breakeven inflation: 2.36% for 10 years on October 1, 2026.
A closer look
Volume I, Part 2 introduced the yield curve as a plot of yields across different maturities. Underneath that plot are two related but distinct concepts. A spot rate is the yield on a bond that pays no coupons and matures at a single future date (a zero-coupon bond) — the “pure” interest rate for that specific maturity. A forward rate is the interest rate implied for borrowing between two future dates, mathematically extracted by comparing two spot rates of different maturities.
Decision rule
Treat forward rates as breakevens, not forecasts. Extend maturity only if you expect yields at your horizon to rise by less than the flip point (yield advantage ÷ duration at the horizon). If your view matches the forward curve, extending adds volatility for little more than the term premium. Ignore the rule for liabilities matched by date.
The costliest mistake
Treating extra yield as extra return. Buying the 10-year for its 0.80-point pickup over the bill, then seeing yields rise 100 bp, turns 5.24% into −1.57%: 6.01 points behind the bill, $601,000 on $10 million. Avoid it: compute the flip point before you extend.
What the full article covers
In the full article I take you through these sections:
■3.3 Convexity — The Curve Beneath Duration
■3.4 The Yield Curve, Mathematically — Spot Rates and Forward Rates
It includes worked examples, a look under the hood at the mechanics, a clear decision rule, the one mistake that costs people the most, a recap, a reveal-on-click quiz and the official sources behind every figure.
Volume II: Capital Markets
Read the full article on FinoLib
The complete article, Bond Convexity and the Yield Curve: Spot and Forward Rates, with every worked example, diagram, quiz and source, is free to read in Volume II, Capital Markets on finolib.com.
Explore the Finance Education Series: Vol. II — Capital Markets · Vol. I — Global Economy · Vol. III — Finance Encyclopedia · Vol. IV — Finance Careers · Vol. V — Personal Finance