Positive convexity
Also written Positively convex bond · Buying convexity · Convexity buying
The favourable asymmetry of an option-free bond: when yields fall its price rises more than duration predicts, and when yields rise it falls less — so buying convexity is an active fixed income strategy.
In plain language
Duration tells you how much a bond's price moves for a small change in yield. It does that with a straight line.
The real relationship between price and yield is not a straight line. It bends. Convexity is the name for that bend, and it is a second-order effect that only matters for large yield moves.
When a bond has positive convexity, the bend works in the holder's favour, both ways:
- yields fall — the price rises more than duration said it would;
- yields rise — the price falls less than duration said it would.
That is a one-sided gift, and it has a consequence worth money. Take two bonds with the same duration. If one has more convexity than the other, it is expected to earn more whenever interest rates move at all — in either direction.
Which is why managers deliberately go out and buy it.
How it works
The workbook's statement (Chapter 19, section 19.3.1, under Buying Convexity). For larger movements in yields, the yield-price movement deviates from a linear relationship. Convexity is a second order effect which captures bond price movements for large changes in yields. If a bond has positive convexity then:
- the bond price increases more if interest rates decline than the duration estimate would suggest;
- the bond price decrease is less when interest rates rise than the price decrease estimated by duration.
The return consequence, in the workbook's own words. The expected return of a bond with positive convexity will be higher than the return of an identical duration, lower convexity bond when interest rates change. That is the whole case for treating convexity as something to buy rather than merely to measure.
How a manager changes it. A portfolio's convexity can be changed by:
- shifting the maturity or duration distribution of the bonds in the portfolio;
- adding or removing bonds with the desired convexity properties; or
- using derivatives.
Where it fits in yield curve strategy. Convexity plays a big role in managing interest rate risk when a yield curve movement is expected in terms of its level, slope or curvature. The chapter's three S's of the yield curve name the shape of the curve as the one decided by curvature and measured through convexity — the other two being the shift (a parallel move) and the slope (steepness or flatness).
Buying convexity sits among the active interest-rate strategies. Section 19.3.1 lists it alongside a barbell of short and long maturities, zero coupon bonds with bullet payments, floaters, maturity extension and roll down.
No figure is attached. The passage gives no convexity number, no basis-point illustration and no worked example. It states the two directional relationships and the equal-duration return comparison, and no more.
A worked example
Illustrative figures applying the workbook's two relationships. A manager compares two bonds, both with a modified duration of 7.0 and both priced at Rs 100. Bond H has high convexity; Bond L has low convexity. She has Rs 10,00,00,000 to place.
Yields move 200 basis points, which is large enough for the second-order effect to show.
| Duration estimate | Bond H (high convexity) | Bond L (low convexity) | |
|---|---|---|---|
| Yields fall 2% | +14.00% | +14.90% | +14.25% |
| Yields rise 2% | −14.00% | −13.20% | −13.75% |
In rupees on Rs 10 crore:
- Yields fall. Bond H gains Rs 1,49,00,000; Bond L gains Rs 1,42,50,000. Bond H is Rs 6,50,000 better.
- Yields rise. Bond H loses Rs 1,32,00,000; Bond L loses Rs 1,37,50,000. Bond H is Rs 5,50,000 better again.
The manager did not have to be right about the direction of rates. She only had to be right that rates would move. That is the workbook's point in a sentence: at identical duration, the higher-convexity bond is expected to return more whenever interest rates change.
If instead rates barely move — say 10 basis points — the two bonds behave almost identically, because convexity is a second-order effect and needs a large move to matter. Any premium she paid for Bond H is then wasted.
Why NISM asks about it
Chapter 19 (Fixed Income Portfolio Management Strategies), section 19.3.1, covers positive convexity under Buying Convexity as one of the active interest-rate strategies.
The chapter's sample question 2 asks it in almost exactly these words: which of these is/are CORRECT when a fixed income instrument has positive convexity? — with the correct pair being the price rise is more than the duration estimate when interest rates decline and the price fall is less than the duration estimate when interest rates rise. Learn those two directions as a pair; the distractors simply swap them.
Expect also a question on how a portfolio's convexity can be changed, and on which of the three S's of the yield curve convexity measures (the shape).
Common exam traps
- Positive convexity helps in both directions. More gain on a fall and less loss on a rise. The wrong answers in the sample question give you one right leg and one wrong one.
- It is a second-order effect and needs a large yield move. For small moves, duration alone is close enough and convexity earns nothing.
- Equal duration is the comparison the workbook makes. The higher-convexity bond wins only against a bond of the same duration; comparing bonds of different durations tells you nothing about convexity.
- The bond does not have to be right about direction. Convexity pays on movement, not on a view.
- Convexity is the measure; positive convexity is the property. This page is about the property and the strategy of buying it; the measure, its formula and its role as the correction to duration are on the convexity page.
- Convexity measures the shape of the yield curve, not its shift or its slope. All three are tested together in the three S's.
Check yourself
1.Which statements are CORRECT for a fixed income instrument with positive convexity? (i) Price rise is more than the duration estimate when rates decline. (ii) Price rise is less than the duration estimate when rates decline. (iii) Price fall is more than the duration estimate when rates rise. (iv) Price fall is less than the duration estimate when rates rise.
- a)I and III only
- b)II and IV only
- c)II and III only
- d)I and IV only
Show the answer
Answer: (d) I and IV only
With positive convexity, the price rises more than duration predicts when rates fall (I), and falls less than duration predicts when rates rise (IV).
This is why, for identical duration, the higher-convexity bond has the higher expected return when rates change. Every other option includes at least one reversed statement.
Where this is taught
Free preparation for NISM Series XXI-BRelated terms
- Modified DurationMacaulay's duration divided by (1 + yield) — the percentage by which a bond's price moves for a one percentage point change in interest rates, and so the standard measure of interest rate risk.
- Zero coupon bondA bond that pays no coupons: it is issued below face value and repays the face value at maturity, so its duration equals its maturity.
- DurationA measure of how sensitive a bond's price is to changes in interest rates — the higher the duration, the larger the price swing for a given change in rates.
- ConvexityThe curvature of the price-yield relationship — the correction duration misses, because duration is a straight line and the true relationship bends.
- Macaulay durationThe weighted average time, in years, to receive a bond's cash flows, each weighted by the present value of that cash flow — the bond's effective payback period.
- Barbell strategyA bond portfolio concentrated at the short and long ends of duration with little in the middle, used when rates could move sharply in either direction.
- Roll downAn active fixed income strategy that profits from the passage of time: on an upward-sloping yield curve a bond's remaining maturity shortens, its yield falls, and its price rises.
- Three S's of the yield curveThe workbook's own name for the three ways a yield curve can move — Shift, Slope and Shape — that a fixed income manager watches when taking an interest-rate view.