Percentage price change approximation, duration

In order to approximate the percentage price change for a given change in yield and a given duration, we employ the following formula ... 

For example, consider once again the UK gilt principal strip whose duration we just computed is 17.636. The approximate percentage price change for a 10 basis point increase in the required yield (i.e.. Ay = +0.001) is... 

EXHIBIT 4.20 Application of Duration to Approximate the Percentage Price Change... 

We demonstrated this property earlier using a UK gilt principal strip priced to yield 4.435% with a duration of 17.631. For a 10 basis point change in yield, the estimate was accurate for both an increase and decrease in yield. However, for a 200 basis point change in yield the approximate percentage price change was off considerably. 

If the yield decreases from 4.435% to 5.435%, the convexity adjustment to the approximate percentage price change based on duration would also be 1.65%. 

As stated above, Macaulay duration equals modified duration multiplied by (1+r). The first two components of the right-hand side of (2.13) taken together are therefore equivalent to modified duration, and equation (2.13) expresses the approximate percentage change in price as modified duration multiplied by the change in yield. 

The nominal spread is the traditional spread measnre. The nominal spread is simply the difference between the yield on a nongovernment issue and the yield on a comparable maturity government. When the spread is taken to be the nominal spread, spread duration indicates the approximate percentage change in price for a 100 basis point change in the nominal spread holding the government yield constant. 

Finally, the option-adjusted spread (OAS) is the constant spread that, when added to all the rates on the interest rate tree, will make the theoretical value equal to the market price. Spread duration based on OAS can be interpreted as the approximate percentage change in price of a nongovernment for a 100 basis point change in the OAS, holding the government rate constant. 

If the expression for Macaulay durarion, (2.5), is substituted into equation (2.4), which calculates the approximate percentage change in price, (2.7) is obtained. This is the definition of modified duration. 

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