Duration

Duration tells us how sensitive a bond’s price is to changes in yield. There are actually several “flavors” or duration, including dollar duration, modified duration, and Macaulay duration. DV01 is a closely related statistic. The concept of duration can be applied to other fixed income products (interest rate swaps, bond futures, etc.).

Dollar Duration

Dollar duration is the negative of the first derivative of a bond’s price with respect to its yield. If a bond with a notional, N, and an annual coupon payment, c, has a yield, y, per annum on an annual basis, then the price, P, of the bond is

Note, P is the dirty price or economic value of the bond, and not the market quoted or clean price. The dollar duration is then,

Because the price of a plain vanilla bond goes up when the yield goes down, and vice versa, flipping the sign of the derivative makes dollar duration positive for a long bond position.

The precise formula for dollar duration will vary depending on how we define yield. For example, if, rather than quoting yield on an annual basis, we quote yield on a continuous basis, then the price of a bond would be

The corresponding formula for dollar duration would then be

Dollar duration is a popular risk measure for portfolios of fixed income securities. The dollar duration of a portfolio is just the sum of the dollar duration of the various instruments in the portfolio. If the dollar duration of a bond position is $30 million and the dollar duration of a bond future position is $20 million, then the combination of the two positions would have a dollar duration of $50 million.

If a portfolio has a dollar duration of $50 million, then a 1% increase in yields will result in a change in value of approximately −$50 million x 1% = −$500,000. The result is approximate because the price formula is not linear. The approximation will generally be more accurate the smaller the change in the yield.

Modified Duration

Dollar duration is great for portfolios, but because it varies with the size of our position it is less useful for comparing one bond to another. Rather than the dollar change per bond, what we would like is the rate of change or the change per dollar of a bond. To do this, we divide dollar duration by P,

Macaulay Duration

When bond traders talk about the duration of a bond, they are usually talking about Macaulay duration. Macaulay duration is the weighted average time to maturity of the cash flows of a bond, where each cash is weighted by its present value discounted using the yield of the bond. It turns out that this quantity is very closely related modified duration. When yield is quoted on an annual basis, we simply need to multiply by (1 + y) to get the Macaulay duration. When yield is quoted on a continuous basis, Macaulay duration and modified duration are, in fact, equal.

For yield quoted on an annual basis, we have,

And for yield quoted on a continuous basis,

A zero-coupon bond with T years to maturity has a Macaulay duration of T. If, instead, the bond pays a coupon, then the bond will have a Macaulay duration that is less than T (the Macaulay duration is a weighted average of the return of notional at time T, and coupon payments made at time less than or equal to T).

DV01

DV01 (pronounced dee-vee-oh-one) is the dollar value of a basis point. This quantity is sometimes referred to as PV01 (present value of a basis point) or BP01 (the clearly redundant, basis-point=oh-one). DV01 tells us how much the value of a bond will change if its yield increases or decreases by a basis point. We can calculate DV01 from dollar duration,  

As with the other duration measures, the change in value calculated in this fashion is only a linear approximation, but, because a basis point is so small, it is usually a very good approximation.

Another way to calculated DV01 is to fully reprice the bond for both an up and down move of one basis point, and then to average the price change,

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