Duration and Convexity: Why Some Bonds Fall Much Further Than Others

Anna and Ben both buy a government bond on the same day. Both bonds pay a 4% coupon and are priced to yield 4%. The only difference: Anna's matures in 2 years, Ben's in 30.

The next day, yields jump from 4% to 5%. Anna's bond falls by about 1.9%. Ben's falls by about 15.4%.

Same issuer, same coupon, same move in rates. Anna barely notices. Ben loses about 15% of his money. The number that explains the gap is duration. Convexity explains why duration's answer is slightly off. Both come up again and again on the CFA and FRM exams.

First, why bond prices fall when yields rise

A bond's coupons are fixed. If new bonds start paying 5% and yours pays 4%, nobody will buy yours at the old price. Its price has to drop until a new buyer earns a 5% yield on it, the same as on new bonds.

In maths terms, a bond's price is the present value of its future payments, and a higher yield means discounting those payments more heavily. Payments far in the future are hit hardest, which is why Ben's 30-year bond suffers so much more than Anna's.

Duration: how sensitive a bond is to yields

You will meet two kinds of duration first (a third, effective duration, comes later), and the exam expects you to keep them apart.

Macaulay duration is the weighted average time until you receive the bond's payments, measured in years. Each payment is weighted by its share of the bond's present value. A zero-coupon bond pays everything at the end, so its Macaulay duration equals its maturity. A coupon bond pays some money earlier, so its duration is shorter than its maturity.

Modified duration turns that into price sensitivity:

It tells you the approximate percentage change in price for a 1 percentage point change in yield:

Ben's bond has a modified duration of about 17.3, so duration says a 1 percentage point rise in yield costs him roughly 17%. Anna's has about 1.9, so she loses roughly 1.9%. Ben's real loss, 15.4%, is smaller than the estimate; convexity, below, explains why.

A worked example

Take a 5-year bond with a 5% annual coupon, priced at 100 to yield 5%.

• Macaulay duration: 4.55 years.
• Modified duration: 4.55 ÷ 1.05 = 4.33.

Now let the yield rise by 1%, from 5% to 6%. Duration says the price should fall by about 4.33%, to 95.67. If you actually discount the cash flows at 6%, the price is 95.79, a fall of 4.21%.

Close, but not exact. Duration overstated the loss by 0.12 percentage points. Run the same test with yields falling to 4%: duration predicts a gain of 4.33%, and the real gain is 4.45%. This time duration understated the gain.

In both directions the bond did a little better than duration said. That is convexity at work.

Convexity: why the straight line misses

If you plot a bond's price against its yield, you get a curve that bends upwards, not a straight line. Duration is the straight line that touches the curve at today's yield. For a small move it is very close. For a bigger move the curve pulls away from the line. For an ordinary bond, it always pulls away on the bondholder's side: smaller losses when yields rise, bigger gains when they fall.

Convexity measures how much the curve bends. Add it to the duration estimate and you get:

The example bond has a convexity of about 23.9. For a 1% rise in yield, written as 0.01:

• Duration effect: −4.33 × 0.01 = −4.33%
• Convexity effect: ½ × 23.9 × 0.01² = +0.12%
• Total: −4.21%, almost exactly the real price change of −4.21%.

Notice that the convexity term is squared, so it is positive whether yields rise or fall. For an ordinary bond with no options attached, convexity always helps you.

Try it yourself

Change the maturity, the coupon and the yield, then move yields up or down. Watch the straight duration line drift away from the real price as the move gets bigger.

Duration in money: DV01

Risk managers often want the answer in currency, not percent. DV01 (also called PVBP, the price value of a basis point) is how much the position's value changes for a one basis point (0.01 percentage point) move in yield:

For $1 million of the example bond: 4.33 × $1,000,000 × 0.0001 ≈ $433. Every basis point that yields rise, the position loses about $433. It is quoted as a positive number. On the CFA exam the related idea is money duration, modified duration × the bond's full price, and PVBP ≈ money duration × 0.0001. Hedgers use DV01 to size the trade on the other side: match the DV01s and small rate moves cancel out.

What makes duration longer or shorter

• Longer maturity, longer duration. Ben's 30-year bond versus Anna's 2-year. (One rare exception: very long bonds trading far below par, where duration can start to shrink.)
• Lower coupon, longer duration. Less money arrives early, so the average wait is longer. A zero-coupon bond has the longest duration for its maturity.
• Lower yield, longer duration. Far-off payments are discounted less, so they carry more weight.

When convexity works against you

The exception is callable bonds and mortgage-backed securities. When rates fall, the issuer can repay early (or homeowners refinance), so the price cannot rise much. Their price curve flattens or bends the other way at low yields. That is called negative convexity.

Because their cash flows change when rates move, modified duration, which assumes fixed cash flows, no longer works. Analysts use effective duration and effective convexity instead, worked out by repricing the bond with yields moved up and down.

Where it shows up on the exams

• CFA Level I (Fixed Income, 11–14%): Macaulay, modified and effective duration, convexity, the price change estimate with both terms, and money duration and PVBP.
• CFA Level II: effective duration and convexity for bonds with embedded options.
• CFA Level III: changing and hedging portfolio duration with swaps and futures (core), plus yield curve and liability-driven strategies in the Portfolio Management pathway.
• FRM Part I (Valuation and Risk Models, 30%): DV01, duration and convexity, and hedging interest rate risk with them.

The three mistakes that cost marks

1. Using the yield change as a whole number. A 1% change is 0.01 in the formula. Plug in 1 and the duration term comes out 100 times too big and the convexity term 10,000 times too big.

2. Using Macaulay duration to estimate a price change. Price sensitivity needs modified (or effective) duration. Macaulay is a time measure.

3. Forgetting the signs. The duration term is negative for a rise in yields. The convexity term is positive in both directions for an ordinary bond, so don't flip it when yields fall.

Back to Anna and Ben

Neither bond was safer in the sense of default: they were the same issuer. Ben simply held far more interest rate risk, and duration would have told him in one number before he bought. If you remember one thing, make it this: duration is the size of the bet you are making on interest rates.

Related reading: firms change their duration with swaps, explained in Interest Rate Swaps Explained. To turn DV01 into a loss you might see on a bad day, see Value at Risk.

Want to check you have it? The fixed income questions in the JephAi Questions Bank include duration, convexity and PVBP calculations, each with a worked explanation.