Convexity

Two option-free bonds priced to the same yield both carry a modified duration of 7.0, but Bond P has materially higher convexity than Bond Q. Yields then move sharply - down 150 basis points in one scenario, up 150 basis points in another. What does the convexity difference imply about the two bonds?

1. A.Convexity is relevant only to callable bonds and mortgage-backed securities, so it has no effect on two option-free bondsThose securities are where NEGATIVE convexity appears, which is a genuine and important point - but every bond has convexity. Option-free bonds simply have positive convexity.
2. B.Because their durations are identical, the two bonds will experience the same percentage price change regardless of how large the yield move isThat is true only in the limit of very small yield changes. At 150 basis points the linear duration estimate has drifted materially from the true price, and convexity is what accounts for the gap.
3. C.Bond P outperforms when yields fall but underperforms when yields rise, since convexity is a directional betOnly half right. The curvature works in the investor favor on both sides for an option-free bond, which is why investors pay up for convexity.
4. D.Bond P outperforms in both scenarios, gaining more than duration predicts when yields fall and losing less than duration predicts when yields riseCorrect. Positive convexity improves the outcome on both sides, and the advantage widens as the yield move gets larger.

Why: Duration is a straight-line estimate of a price-yield relationship that is actually curved. Convexity measures that curvature. For an option-free bond convexity is positive, which means the true price rises MORE than duration predicts when yields fall, and falls LESS than duration predicts when yields rise. Higher convexity therefore helps in both directions, and the benefit grows with the size of the yield move.

Market interest rates decline sharply. Compared with an otherwise similar NON-callable bond, a CALLABLE bond's price will:

1. A.Match the non-callable exactlyWrong. The embedded call option guarantees divergence.
2. B.Rise more, because callables always carry higher couponsWrong-but-tempting. Higher coupons compensate for call risk - they do not create superior price appreciation when calls loom.
3. C.Fall, because falling rates hurt all bondsWrong. Falling rates RAISE bond prices; direction is not the issue.
4. D.Rise less, compressing near the call price as redemption becomes likelyCorrect. Negative convexity caps the callable's rally.

Why: Falling rates make the issuer's call option valuable, capping the callable bond's appreciation near the call price while the non-callable rises freely; call risk is the mirror of the investor's reinvestment risk. Citation: callable bond convexity analysis. Takeaway: rate rallies bypass callable holders - the upside is compressed.

Wexler Securities compares a noncallable 6 percent corporate bond with a current-coupon agency mortgage-backed pass-through of similar stated duration. Market interest rates then fall by 100 basis points, and the mortgage-backed security appreciates noticeably LESS than the corporate bond. The BEST explanation is:

1. A.The mortgage-backed security pays monthly rather than semiannually, which mathematically eliminates its price sensitivity to rates.Wrong. More frequent payments modestly shorten duration but never eliminate price sensitivity. Payment frequency is not the driver here.
2. B.The mortgage-backed security carries higher credit risk, and its credit spread widened as rates fell.Wrong. An agency pass-through carries minimal credit risk, and nothing in the facts suggests spread widening. The effect described is a convexity effect.
3. C.The mortgage-backed security exhibits negative convexity - falling rates accelerate prepayments, shortening its effective life and capping price appreciation.Correct. The homeowners' prepayment option truncates the upside of a pass-through in a rate rally.
4. D.The mortgage-backed security has a longer duration, so it responds less to a change in interest rates.Wrong on two counts: durations were stated as similar, and a LONGER duration means GREATER price sensitivity, not less.

Why: A noncallable bond exhibits positive convexity: as rates fall its price rises at an accelerating rate. A mortgage-backed pass-through behaves differently because the homeowners behind it hold what amounts to a free prepayment option. When rates fall, refinancing accelerates, principal comes back early, and the security's effective life shortens exactly when a longer life would have been most valuable. That is NEGATIVE convexity, and it caps price appreciation in a rally - while doing nothing to protect the holder in a selloff, when slower prepayments extend the security instead.