# Why is the duration of a bond important?

I know what duration measures, but now in the age of computers why is it useful? If the yield changes, we could just simply plug the new yield into a program, or excel or something like that, and calculate the new price of the bond.

You are right that if we exactly want to know the price of a bond after a change in the yield curve, we have to calculate it - and we can.

What we can say about duration:

• it is a linear approximation of the price change if yield change, this works rather fine with plain vanilla bonds but things get more difficult e.g. with callable bonds.
• keeping the eye on plain-vanilla bonds. If I want to compare bonds e.g. government bonds of the same country then duration is of a certain value. It tells me more about risk/return than just time to maturity. If I want to quickly characterize the bond then I could say e.g. it is a German govi with a duration of 8 years and I have a rough picture about the risk - without having calculated a single price.

Summing up: Duration should not be the end of an analysis but it is a useful start.

It is useful in risk reports because it tells a trader the interest rate risk of each bond in his portfolio. A trader then only needs to multiply the duration by the expected yield change to calculate the price change. Scenario analysis is then easier.

Hedging a bond portfolio with duration measures is common. But as these must rely on the assumption that the yield of the bond portfolio being hedged moves by the same amount as the yield of the hedging bond, it is important to choose a hedge bond with a maturity as close to that of the bond portfolio.

For standard bullet bonds (ones with fixed regular coupons) the duration can be calculated analytically by taking the derivative of the price yield formula. This makes it faster to calculate than doing two price calculations and calculating the numerical derivative.

One widely used measure is the modified duration which is equal to

Modified Duration -$\frac{1}{P} \frac{\partial P}{\partial y}$.

It calculates the percentage change in the bond price and so needs to be multiplied by the price to get the implied absolute price change sensitivity. This quantity is known as the dollar duration

Dollar Duration = $\frac{\partial P}{\partial y}$.

A similar measure called the DV01 is the change in the price of the bond for a 1 basis point increase in the bond yield.

DV01 = $P(y+1bp)-P(y)$.

Traders use ratios of the DV01 to calculate hedging amounts. It assumes implicitly the typical moves over the typical period between re-hedges are around 1bp.

More sophisticated bond traders avoid yield based risk measures and instead look at sensitivities to a zero rate which captures the term structure, i.e. measures that do not assume a constant reinvestment rate. The value of the zero rate changes depending on the payment time of each cash flow.

For large interest rate movements it becomes important to consider the curvature of the price-yield relationship. This can be done by expanding the price yield relationship to second order, which means calculating the convexity term.

Finally, some practitioners move beyond duration all together and use parametric models of the zero curve and hedge with respect to linear factors. The best known is the Nelson-Siegel model.