Explain how price moves against yield and why bonds trade at a premium or discount
The idea
Price and yield move inversely, because price is the present value of fixed cash flows discounted at the yield. When the coupon rate exceeds the required yield the bond trades at a premium; when it is below, at a discount; when equal, at par. The relationship is convex rather than linear: a fall in yield raises the price by more than an equal rise lowers it, and that curvature increases with maturity and decreases with coupon. Convexity is a benefit to the holder, which is why a duration-based estimate always understates the price of an option-free bond.
A practice question
A bond's coupon rate is below its required yield. How does it trade?
- At a premiumWhy not: That requires the coupon to *exceed* the required yield, giving investors a reason to bid above par.
- At a discount — correct
- At parWhy not: Par pricing requires the two rates to be equal. A shortfall has to be made up in the price.
The coupon does not deliver the return investors require, so the price must fall below par for the total return to compensate.
A harder one
Which bond has the greater curvature in its price-yield relationship?
- A long-maturity, low-coupon bond — correct
- A short-maturity, high-coupon bondWhy not: Both features pull the other way — they return value early, which reduces convexity as well as duration.
- Curvature is the same for all option-free bondsWhy not: It varies with maturity and coupon, for the same reason duration does: how far out the cash flows sit.
Convexity rises with maturity and falls with coupon, because both determine how far out the cash flows sit.
The hardest tier
Why does a duration-based price estimate always understate the true price of an option-free bond?
- Because duration assumes yields cannot fall below zeroWhy not: Duration makes no assumption about the level of yields. The understatement occurs for rises as well as falls.
- Because duration excludes accrued interestWhy not: Accrued interest is a separate adjustment between clean and dirty price, and it does not produce a one-sided error.
- Because duration is a linear approximation to a convex curve, which lies above its tangent — correct
The straight line touches the curve at the current yield and falls below it everywhere else. The error grows with the size of the yield change, which is why convexity is added for large moves.
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