Yield Duration vs. Curve Duration for Fixed Income

So a bond's duration is a measure of how its price changes when there's a yield change. Fine. But which yield? This could mean the yield to maturity on that bond, or it could mean a shift in spot rates for all maturities.

Suppose you're looking at how the price changes given a change in just the bond's yield to maturity. What does that sound like?

Yield duration is the price sensitivity of the bond's own yield, and curve duration is the price sensitivity to the entire yield curve.

For example, consider this simple, 5-year bond with annual coupons, with a price of about 88.82, priced with its 7.785% yield to maturity: $$\displaystyle 88.82 \approx \frac{5}{1.07785} + \frac{5}{1.07785^2} + \frac{5}{1.07785^3} + \frac{5}{1.07785^4} + \frac{105}{1.07785^5} $$ Add 100 basis points to the yield, so that the bond's yield to maturity is 8.785%. What is the new bond price?

Correct!

Not quite. It's 85.20.

$$\displaystyle 85.20 \approx \frac{5}{1.08785} + \frac{5}{1.08785^2} + \frac{5}{1.08785^3} + \frac{5}{1.08785^4} + \frac{105}{1.08785^5} $$ The price change is: $$\displaystyle \frac{85.20}{88.82} - 1 \approx -0.04076 = 4.076 \% $$ That's fair. The yield duration is about 4. Of course for a more accurate measure of effective yield duration you'd want to look at small yield changes in both directions; this is just for a simple illustration.

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That's right! And you won't find a huge difference here:

That wouldn't work. You want to think of each rate changing much like the yield to maturity changed.

No, this can be done in a very similar way.

$$\displaystyle 85.21 \approx \frac{5}{1.05} + \frac{5}{1.06^2} + \frac{5}{1.07^3} + \frac{5}{1.08^4} + \frac{105}{1.09^5} $$ A price drop closer to 4.06% instead of 4.08%. A yield duration is likely more appropriate when you're looking at a corporate bond, where the yield may be more affected by activities of the company. It's the most common duration measure, and best for assessing interest rate risk.

Now suppose that same bond was priced at 88.82 with an underlying yield curve of 4%, 5%, 6%, 7%, and 8% for each of the five years. $$\displaystyle 88.82 \approx \frac{5}{1.04} + \frac{5}{1.05^2} + \frac{5}{1.06^3} + \frac{5}{1.07^4} + \frac{105}{1.08^5} $$ How might you estimate the curve duration in a similar way?

Yes!

No, this is yield duration.

85.2

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Add 100 bp to each rate

Yield duration

Curve duration

Other responses

Split the 100 bp change among the various rates

It's impossible to do something similar with a rate curve like this

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