So far in this reading, we focused on how to use duration and convexity to measure the price change given a change in YTM. Now, we will look at what causes the YTM to change.
Example 19: Calculating modified duration
Consider a 4-year, 9% coupon paying semi-annual bond with a YTM of 9%. The duration of the bond is 6.89 periods. Calculate the modified duration.
Solution:
= 6.593 periods or 3.297 years.For a one percent change in the annual YTM, the percentage change in the bond price is 3.297%.
Example 20: Calculating the change in the credit spread on a corporate bond
The (flat) price on a fixed-rate corporate bond falls one day from 96.55 to 95.40 per 100 of par value because of poor earnings and an unexpected ratings downgrade of the issuer. The (annual) modified duration for the bond is 5.32. What is the estimated change in the credit spread on the corporate bond, assuming benchmark yields are unchanged?
Solution:
Given that the price falls from 96.55 to 95.40, the percentage price decrease is 1.191%.
-1.191% ≈ -5.32 × ∆Yield,
∆Yield=0.2239%
Given an annual modified duration of 5.32, the change in the yield-to-maturity is 22.39 bps.
The approach used in this reading to estimate duration and convexity with mathematical formulas is called analytical duration. This approach implicitly assumes that benchmark yields and credit spreads are uncorrelated with one another.
However, in practice, changes in benchmark yields and credit spreads are often correlated. So, many fixed income analysts use an alternate approach – Empirical duration. This approach uses statistical methods and historical bond prices to derive the price-yield relationship for specific bonds or bond portfolios.
For a government bond with little or no credit risk, the analytical and empirical duration would be similar because bond prices are largely driven by changes in the benchmark yield.
However, for a high-yield bonds with significant credit risk, the analytical and empirical duration will be different. In a market stress scenario, many investors switch to high quality government bonds due to which their yields (i.e., benchmark yields) fall. But at the same time the credit spreads on high-yield bonds will widen (i.e., credit spreads and benchmark yields are negatively correlated). The wider credit spreads will fully or partially offset the decline in government benchmark yields. Thus, the empirical duration for high yield bonds will be lower than their analytical duration.
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