2007 FRM Learning Outcome

• LO 23.3: Calculate and interpret the effective duration of a security, given a change in yield and the resulting change in price.

Duration is a measure of price sensitivity

For FRM candidates, there are two ways to view duration:

• As a measure of a bond price sensitivity to interest rates. That is, if interest rates change by 10 basis points, how much does the bond price change approximately? This is the essential risk perspective. The 'approximately' modifier refers to the second view...
• As a first derivative. This would not be helpful except it connects to an important theme: we often use first derivatives as linear approximations. As the slope of a tangent line, the line is inaccurate except locally.When the actual dynamic is nonlinear (e.g., the bond price/yield curve, the plot of call option value to stock price or strike price) the first derivative is just an approximation; e.g., the delta of option only works "locally" for small changes.

If you keep in mind this limitation of the first-order derivative, it comes in handy. For example, convexity (in the case of bonds) and the Taylor expansion (in the case of stock options) are meant to help fix the gap between the nonlinear reality and the linear approximation (e.g., duration, delta).

 

Formula for duration

The learning outcome asks for effective duration. Duration refers to either modified or effective duration. Modified does not assume that a yield change will change the underlying bond cash flows; effective duration incorporates the realism (the feedback loop, if you will) that yield changes impact the underlying cash flows. In the case of the simple bond (illustrated below), it does not matter. It matters for bonds with embedded derivatives. Both are different from the Macaulay duration. The Macaulay duration is the duration implied when somebody says "the bond's duration is four years." Thinking of duration as time, however, it not as useful to risk managers. Better to view duration as sensitivity.

The formula for (effective) duration is given by:

 

 

An example

This example is further demonstrated in the EditGrid spreadsheet at the end of this article (to upload to Excel or other format, select File > Save As..).

Here are the bond assumptions (left-hand column of spreadsheet below):

• Bond Par: $1,000
• Coupon: 4% (2% every six months)
• Years to maturity: 10 years (twenty semiannual periods)
• Yield to maturity (YTM, or just 'yield'): 6%

We only need to decide how much to shock the yield. We will use 20 basis points (equal to 0.2%). Given the shock in 20 bps, the numerator is the difference of the two re-priced bonds (i.e., one priced at 5.8% and another at 6.2%). The result is about a duration of about 7.93:

 

EditGrid spreadsheet

There are only two columns. The left are bond assumptions. The right column computes the (effective) duration but requires the shock value (20 basis points) as an input.

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