Duration, convexity and immunisation

Duration is an average time

Macaulay duration is the present-value-weighted average of the times at which cash flows arrive. Write down each payment's present value, use those as weights, average the payment times, and that is the duration. It is measured in periods, and for a single payment it is simply the time of that payment.

That interpretation makes several results obvious that look like separate formulas. A zero-coupon bond maturing in seven years has a duration of exactly seven. A perpetuity at 6 percent has a duration of one plus the rate over the rate, which is 17.667. A ten-year level annuity at 6 percent has a duration of 5.022, well short of its ten-year term, because the early payments pull the average forward.

For a bond priced at par the result is prettier still. A ten-year bond with 6 percent annual coupons priced to yield 6 percent has a price of 1000 and a Macaulay duration of 7.801692 - which is exactly the ten-year annuity-due factor at 6 percent. That identity is worth carrying, because it turns a duration calculation into a table lookup whenever the bond is at par.

Two properties follow from the averaging picture and are worth knowing before any formula. Duration falls as the coupon rate rises, because larger early payments pull the weighted average forward, and duration falls as the yield rises, because higher discounting shrinks the distant payments that were holding the average back. Both are the kind of qualitative check that catches a sign error faster than recomputation does.

Modified duration and the first-order estimate

Modified duration is Macaulay duration divided by one plus the yield per period, and it is the quantity that appears in price-change estimates. For the bond above, modified duration is 7.801692 over 1.06, which is 7.360087 - and, pleasingly, that is the ten-year annuity-immediate factor at 6 percent.

The estimate itself says that the proportional change in price is approximately minus modified duration times the change in yield. Raise the yield by one percentage point and the estimate predicts a price of 926.40 against an actual price of 929.76. The estimate is 3.37 too low.

It is always too low, and that is not an accident of these numbers. The price-yield relationship curves upward, and a straight-line estimate drawn tangent to a curve that bends upward must fall below it on both sides. Lower the yield by a point and the estimate gives 1073.60 against an actual 1077.22 - again short, this time by 3.62.

Convexity is the second-order term

Convexity is the second present-value-weighted moment of the payment times, and it plays the same role for price as acceleration does for position: it is the curvature the first-order estimate ignores. Adding half the convexity times the squared change in yield gives the second-order estimate.

For the same bond, Macaulay convexity is 70.559 and the modified convexity used in the price approximation is 69.740. At a one-point rise in yield, the two-term estimate gives 929.89 against an actual 929.76 - an error of 0.12 rather than 3.37, an improvement of nearly thirty times.

The gap widens with the size of the shock, which is the practical lesson. At a two-point rise the actual price is 865.80; duration alone predicts 852.80, an error of 13.00, while duration and convexity together predict 866.75, an error of 0.95. Duration is adequate for small moves and misleading for large ones.

Immunisation

A fund immunised against small interest rate movements is one whose assets and liabilities move together. Redington's conditions state that requirement precisely: the present values must match, the durations must match, and the convexity of the assets must exceed the convexity of the liabilities.

Take a single liability of 1,000,000 due in seven years, valued at 6 percent. Its present value is 665,057.11, its duration is 7 and its convexity is 49, because a single payment's moments are just powers of its date. Now fund it with two zero-coupon bonds maturing at three years and twelve years.

Matching present value and duration fixes both amounts uniquely: 440,052.04 due at three years and 594,766.92 due at twelve. The convexity of that pair is 69, comfortably above the liability's 49, so all three conditions hold. Shift the rate to 5 percent and the surplus is 640.64; shift it to 7 percent and the surplus is 547.43; shift it to 8 percent and the surplus is 2,027.23. In every direction the fund gains, which is what the third condition buys.

Where the model breaks

Immunisation in this form assumes a flat yield curve that moves in parallel, and neither assumption is true. A twist in the curve - short rates moving one way and long rates the other - can produce a loss on a portfolio that satisfies all three conditions, because the portfolio was only ever protected against the shifts the model contemplated.

The conditions also hold at one instant. Durations drift as time passes and as rates move, so an immunised position must be rebalanced, and the cost of rebalancing is a real drag that the theory does not price.

Full immunisation, where each liability is bracketed by asset cash flows on either side, is the stronger and more expensive answer, and it is worth knowing as the contrast case even though most questions ask for Redington.

• Redington: present values equal, durations equal, asset convexity strictly greater. Protects against small parallel shifts.
• Full immunisation: every liability flanked by asset cash flows before and after it. Protects against parallel shifts of any size.
• Cash-flow matching: assets replicate liabilities exactly. No interest rate risk at all, and usually the most expensive option.