Bond duration

Duration is the average number of years an investor waits to receive a bond’s value, with each year weighted by how much of the value arrives in it. It measures how sensitive the bond’s price is to a change in interest rates.

Maturity alone cannot do that job. Two bonds can both mature in four years while one pays a large coupon each year and the other pays nothing until the end — the first gets most of its money back early, the second waits. They face very different amounts of interest rate risk, and their maturities are identical.

Duration reads two ways at once, and both are useful.

• As a period of time: how long, on average, before an investor recovers what they put in. It is the payback period of the bond’s cash flows, in present-value terms.
• As a sensitivity: the approximate percentage change in the bond’s price for every 1% change in the market interest rate.

The longer the duration, the further into the future the bond’s value is generated, and the more violently its price reacts when rates move.

Duration is the present value of each cash flow multiplied by the year it arrives in, totalled, and divided by the bond’s price.

ERS has issued 10% ZMW1,000 four-year bonds, redeemable at nominal value, with interest paid annually. The market interest rate is 6%. Four columns: the cash flow, its present value, and then that present value weighted by its year.

Two figures come out of one table. The sum of the present values is the bond’s market price, ZMW1,138.50. The sum of the weighted column, divided by that price, is the duration.

3.52 years against a maturity of 4. The gap is small because the year 4 cash flow — the last coupon plus the ZMW1,000 redemption — is 77% of the whole present value and pulls the average hard towards year 4. On a bond with a bigger coupon relative to its principal the gap would be wider.

Where the question supplies a different spot rate for each year rather than one market rate, discount each line at its own rate and the method is otherwise unchanged. On spot rates of 4%, 4.5%, 5% and 6% the same bond prices at ZMW1,145.40 with a duration of 3.51 years.

The second reading of duration is the useful one for managing risk: the bond’s price changes by roughly the duration, in percent, for every 1% change in the market interest rate — and in the opposite direction.

ERS’s bond has a duration of 3.52. Suppose the market rate falls by 200 basis points — 2%.

The estimate is ZMW0.90 out on a ZMW1,218 bond — close enough to manage a position with, and worth knowing is an estimate. Duration draws a straight line through a relationship that is actually curved, so it is at its most accurate for small movements and drifts as they get larger.

Two hundred basis points is already a fair-sized move. On a 25 basis point change the straight line and the curve are almost indistinguishable; on a 500 basis point change the gap is wide enough that the bond has to be revalued properly rather than estimated.

Note the minus sign in the formula and where it comes from. Rates and prices move in opposite directions, so a rise in rates gives a fall in price. Here rates fell, so the price rose.

Three things move duration, and knowing which way each pushes lets you rank bonds by risk without calculating anything.

The special case is the zero coupon bond. It has one cash flow, at maturity, so its duration equals its maturity exactly — a five-year zero has a duration of 5. Nothing arrives early to pull the average down, which makes zeros the most interest-rate-sensitive bonds of any given maturity.

That gives an ordering you can apply on sight. Among bonds maturing on the same date, the one with the lowest coupon has the longest duration and will move most when rates change. Among bonds with the same coupon, the one maturing latest has the longest duration.

It also explains a pattern in the market. Governments issuing thirty-year bonds are issuing the most rate-sensitive instruments in existence, which is why long-dated gilt and treasury prices swing so far on what look like small changes in policy.

Duration turns a vague worry about rates into a number that can be managed, and it is used in three ways.

Measuring a position

A treasurer holding a portfolio of bonds can calculate its duration — the average of the individual durations, weighted by market value — and read off what a rate movement would cost. A portfolio of ZMW500 million with a duration of 6 loses about ZMW30 million if rates rise by one point.

Immunisation

An institution with obligations to meet at a known future date can match the duration of its assets to the duration of those obligations. Do that and a rate change moves both sides by roughly the same proportion, leaving the position able to meet its liabilities whichever way rates go. A pension fund paying out in twelve years wants assets with a duration near 12, not bonds maturing in twelve years.

Positioning

A treasurer expecting rates to fall wants long-duration assets, to gain the most from the price rise. Expecting rates to rise, they shorten duration to lose the least. That is a deliberate bet on the direction of rates, and it should be described as one.

The limitations are worth stating alongside the uses. Duration is a linear approximation of a curved relationship, so it degrades on large movements. It assumes the whole yield curve shifts by the same amount, which is rarely what happens — short rates and long rates often move by different amounts or in different directions. And it says nothing at all about credit risk: a bond can hold its duration perfectly and still default.

Measuring exposure is the first half of the job. Doing something about it is the second, and that is where the hedging instruments come in.