A number of statistics can be used to describe the sensitivity of bond price to a change in the yield. The standard measures of Macaulay's duration, modified duration, convexity, and PVBP (present value of a basis point), are presented. All of these measures rely on either the first derivative of bond price with respect to yield or the second derivative of bond price with respect to yield.

1. Macaulay Duration

Bond duration can be determined as:

where:

P = dirty price (clean price plus accrued interest) of the bond per 100 units (PPH) face value.

y = the required annual nominal redemption yield, expressed as a decimal.

h = number of coupon periods in a year.

Duration is defined as the price elasticity of the bond; that is, duration describes the percentage change in the price for a given percentage change in yield.

Note that because bond price is a negative function of yield, the first derivative of price with respect to yield will also be negative, and the negative sign is therefore used in the above equation to ensure that the duration is always expressed as a positive number.

2. Modified Duration

Modified duration is the percentage change in bond price for a given change in yield, and is calculated as:

As with Macaulay duration, modified duration is always expressed as a positive number even though the relationship between yield and prices is negative.

3. Convexity

Convexity is used to approximate the percentage change in the bond price that is not explained by the duration measure (resulting from the curvature in the bond price and yield relationship). Bond convexity is defined as follows:

4. PVBP

The price value of a basis point change for every $100 face value can be determined as:

As with the duration statistics, PVBP is always expressed as a positive number.

See Also

Discrete Risk Statistic Calculations