Questions: Convexity and Non-Linear Price-Yield Relationships

Question 1

A bond has modified duration of 8 and convexity of 100. Yields rise by 200 basis points. Compared to what duration alone predicts, the actual price decline will be:

• A. Larger — convexity amplifies losses when yields rise
• B. Smaller — convexity partially offsets the loss because the convexity term is always positive
• C. The same — convexity only matters when yields fall, not when they rise
• D. Larger only if the bond is a zero-coupon bond

Answer: (B)

The full formula is ΔP/P ≈ −D·Δy + (C/2)·(Δy)². The convexity term (C/2)·(Δy)² is always positive regardless of the direction of the yield move, because (Δy)² is always positive. When yields rise, this positive term partially offsets the negative duration term, making the actual price fall smaller than duration alone predicts. Convexity benefits the bondholder symmetrically in both directions.

Question 2

Two bonds have identical durations of 7. Bond A has convexity 120; Bond B has convexity 40. In a volatile rate environment with potentially large moves in either direction, which is preferable?

• A. Bond B — lower convexity means more predictable, stable price behavior
• B. Bond A — higher convexity means better price performance whether rates rise or fall
• C. They are equally attractive since duration determines rate sensitivity
• D. Bond A only if rates are expected to fall; Bond B if rates are expected to rise

Answer: (B)

Higher convexity is always preferable when holding duration constant, because it provides an asymmetric benefit: the bond gains more when yields fall than it loses when yields rise by the same amount. In volatile environments, large yield moves in either direction favor high convexity. Option D reflects the misconception that convexity helps only when rates fall — but it reduces losses equally when rates rise.

Question 3

Because the convexity correction term is always positive, a bond with positive convexity gains more when yields fall than it loses when yields rise by the same amount.

• T. True
• F. False

Answer: True

This is the key asymmetric property of positive convexity. In both directions of yield change, the (C/2)·(Δy)² term adds back to the price change — it reduces losses when yields rise and amplifies gains when yields fall. This makes price response to yield changes asymmetric: the gains exceed the losses for equal-sized moves, which is why high convexity is a desirable bond characteristic that commands a price premium.

Question 4

Convexity matters most when managing small, day-to-day interest rate fluctuations, and is less important for large rate moves.

• T. True
• F. False

Answer: False

The opposite is true. The convexity correction is proportional to (Δy)², which grows rapidly for larger moves. For a 10 basis point move, (Δy)² = 0.0001; for a 200 basis point move, (Δy)² = 0.04 — four hundred times larger. For small daily fluctuations, the duration approximation is highly accurate and convexity adds negligible correction. Convexity becomes material precisely when yields make large moves, such as during rate shocks or crises.

Question 5

Why does a callable bond exhibit negative convexity at low yields, and how does this differ from a standard non-callable bond?

Model answer: A callable bond gives the issuer the right to redeem it at par when rates fall. As yields decline, the bond's price rises toward the call price — but once yields fall far enough, the issuer will call the bond, capping price appreciation at par. The bondholder cannot benefit from further yield declines because the bond will be redeemed. Meanwhile, if yields rise, there's no such cap on losses. This produces negative convexity: price gains from falling rates are truncated while price losses from rising rates are not. A standard non-callable bond has no such cap — its price keeps rising as yields fall, giving it positive convexity with symmetric asymmetric benefit.

The call option effectively transfers upside convexity from the bondholder to the issuer, reversing the asymmetry that defines positive convexity.