Markets02
Bonds and yields
A is a loan you can trade, and the only hard idea in it is that the payments are fixed while the required return is not. When required returns change, the only thing that can adjust is the price.
Price from yield
Bond price
| 3% | 5% | 6% | 9% | |
|---|---|---|---|---|
| Price | 1,056.57 | 1,000.00 | 973.27 | 898.75 |
At a 5% it prices exactly at par, because the and the required return are the same number. That is the anchor: premium above, discount below, always.
Yield from price
Going the other way has no closed form. There is no rearrangement of the pricing formula that isolates , so every calculator solves it numerically. The implementation behind the tool below uses bisection, because price falls monotonically as yield rises, so a bracket cannot fail. Newton's method is faster and occasionally wanders off, and a bond calculator that sometimes diverges is worse than one that is slow.
Duration, which is a sensitivity rather than a time
Price change from duration
Macaulay is the weighted average time to receive the cash flows, which is why it is quoted in years. Modified duration converts it into the percentage price move per point of yield. For the three-year bond above, Macaulay is 2.857 years and modified is 2.696.
Duration is below maturity for any coupon bond, because some cash arrives before the end. For a zero-coupon bond they are identical, since all the cash arrives at once.
Convexity, the part duration misses
| Duration alone | With convexity | Exact | |
|---|---|---|---|
| Yields rise 1% | negative 26.24 | negative 25.75 | negative 25.76 |
| Yields fall 1% | 26.24 | 26.72 | 26.73 |
Duration overstates the loss and understates the gain, by the same amount both ways, because it draws a straight line through a curve. Adding the term closes most of the gap. On a three-year bond the difference is cents. On a thirty-year bond it is percentage points, and a portfolio hedged on duration alone is systematically mis-hedged.
Price it yourself
From the money track
Why bond funds fall when rates rise
The same mechanism, from the side of someone holding a bond fund in a retirement account.
Test yourself
01Rates rise 1%. Which falls more, a 2-year bond or a 20-year bond?
The 20-year, by a lot. Its duration is roughly 13.1 years against about 2.0 for the short one, so it loses roughly ten times as much for the same yield move. The follow-up is why anyone owns long bonds, and the answer is that the same sensitivity works in reverse when rates fall, which is the entire trade.
02A bond trades above par. What does that tell you?
Its coupon is above the current market yield for that credit and maturity, so investors pay a premium to receive above-market income. The premium erodes as maturity approaches, because the bond repays exactly par at the end. That amortization is a real cost that a yield calculation captures and a coupon yield does not.
03Why does convexity favor the bondholder?
Because the price-yield curve bends upward. Duration says the price change is linear in yield, and it is not: gains from falling yields are larger than losses from equal rises. On the three-year bond here, a 1% rise costs $25.76 while duration alone predicts $26.24. The difference is the curvature, and it is always in the holder's favor for a plain bond.
A tutor that knows this lesson. It asks before it explains, and it will not tell you what to do with your own money.
Every figure in this lesson that names a company comes from Target Corporation's Form 10-K for Fiscal 2025, the year ended 2026-01-31, filed 2026-03-11. Educational material, not investment advice.