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Learn / Risk & ALM

Duration and convexity

Duration is a first guess at how a price changes when yields change. Convexity is the correction, because the price-yield relationship is curved. Together they tell an asset-liability team which side of the book moves more when rates move.

The classroom bond is priced at 100. Its modified duration is 7. Its convexity is 60. Every figure on this page uses that bond, or the $100 million book with durations 6 and 12.

01

Several durations, one idea

Macaulay duration is the weighted average time until the bond’s cash flows arrive, measured in years. A cash flow due next year pulls the average in. A cash flow due in thirty years pulls it out. Modified duration converts that average into a price sensitivity: about how many percent the price changes for a 1 percentage point change in yield.

Effective duration is the same idea measured by shifting the curve and repricing, which matters when the bond has a call, a prepayment, or some other choice that changes the cash flows. Key-rate duration splits the sensitivity across maturities, so you can see the 2-year piece separately from the 10-year piece.

Duration is the tangent. Convexity is the bow.Bond price falls as yield rises. A straight tangent is the duration estimate. The curve bends away from that line, and that bend is convexity.Duration is the tangent. Convexity is the bow.Teaching sketchPriceYieldDuration, the straight guessConvexity, the bend
Classroom bond at price 100, modified duration 7, convexity 60. Up 100 basis points is about −6.7 points. Down 100 basis points is about +7.3 points. Convexity adds on both sides.

02

The arithmetic of one move

The classroom formula, as a fraction of price, is minus duration times the yield change, plus one-half times convexity times the yield change squared. For a 1 percentage point rise, that is −7 × 0.01 + 0.5 × 60 × 0.01 × 0.01 = −0.070 + 0.003 = −0.067. On a price of 100, that is about −6.7 points. The straight duration guess was −7 points. Convexity gave back 0.3 points.

For a 1 percentage point decline, the duration piece flips sign and the convexity piece does not, because the yield change is squared. The change is about +7.0 + 0.3 = +7.3 points. The price rises a bit more than duration alone suggested, and it falls a bit less than duration alone suggested. That is the bow in the picture.

03

Dollar duration, DV01, and PV01

Percent of price is useful. Dollars are what a book manages. Dollar duration here is the value of one basis point. On $100 million with modified duration 7, one basis point is $100 million × 7 × 0.0001 = $70,000. This page uses DV01 and PV01 as that same dollar: the present-value change for a one basis point move. Other desks split those two names more finely. Here they are one classroom dollar.

Key-rate duration spends that dollar across the curve. If half of the $70,000 sits on the 2-year and half sits on the 10-year, a twist that lifts only the 10-year moves about $35,000 per basis point, not the full $70,000. A parallel shift moves both. A twist does not.

04

Assets and liabilities on the same ruler

Put the insurance book on this ruler. Assets are $100 million with duration 6, so they move about $60,000 per basis point. Liabilities are $100 million with duration 12, so they move about $120,000 per basis point. The gap is $60,000 per basis point. A parallel rise marks the liabilities down by more dollars than the assets.

Who moves more per basis pointAsset dollar duration is $60,000 per basis point. Liability dollar duration is $120,000 per basis point. The gap is $60,000.Who moves more per basis pointTeaching sketchAssets$60,000 per bpLiabilities$120,000 per bpGap to close with a receive-fixed swap: $60,000 per basis point
Both sides are $100 million. Assets have duration 6. Liabilities have duration 12. The longer liabilities move about twice as much when the curve shifts in parallel.

Closing that parallel gap is the job of the interest-rate hedging page. A twist is what remains after a parallel hedge, because the 2-year and the 30-year no longer share one move.