EF 3320 Lecture 11(2)

# The bond price with 91 interest equals 88798 the

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The bond price with 9.1% interest equals \$887.98. The difference of \$9.28 is quite close to \$9.36 predicted by duration formula.

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Duration is a Local Concept Suppose that in the last example the YTM changed to 10%. What’s the price change predicted by the duration formula? P = - 11.37 × 897.26 × 0.01/1.09 = - \$93.59. What’s the price change predicted by the annuity formula? P = 811.46 - 897.26 = - \$85.80.
Duration is a Local Concept Now the divergence is more substantial. This points to an important limitation of the duration formula. Duration is a local concept, and its value changes as the YTM changes . Why?

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Convexity The relationship between bond prices and yields is not linear. Duration rule is a good approximation for only small changes in bond yields. Bonds with greater convexity have more curvature in the price-yield relationship.
Bond Price Convexity: 30-Year Maturity, 8% Coupon; Initial YTM = 8%

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Convexity Measures how much a bond’s price-yield curve deviates from a straight line Second derivative of price with respect to yield divided by bond price Allows us to improve the duration approximation for bond price changes = = + + + = = + + + = T t t t T t t t t t y CF y P y P P t t y CF y y P 1 2 2 2 2 1 2 2 2 2 ) ( ) 1 ( ) 1 ( 1 1 Convexity ) ( ) 1 ( ) 1 ( 1
Convexity Recall approximation using only duration: New bond price The predicted percentage price change accounting for convexity: New bond price y D P P × - = * ( 29 × × + × - = 2 * ) ( Convexity 2 1 y y D P P [ ] y D P P × - × + = ) ( * [ ] × × × + × - × + = 2 * ) ( Convexity 2 1 ) ( y P y D P P

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Numerical Example with Convexity Consider the bond in Figure 16.3, with a 30-year 8% coupon bond selling at par value (\$1,000), to yield 8%. We can find that the modified duration is 11.26, and the convexity is 212.4. If the yield increases from 8% to 10%, the bond price will fall to \$811.46, a decline of 18.85%. The duration rule indicates that After correcting for convexity % 52 . 22 02 . 0 26 . 11 * - = × - = × - = y D P P ( 29 % 27 . 18 or 1827 . 0 ) 02 . 0 ( 4 . 212 2 1 02 . 0 26 . 11 ) ( Convexity 2 1 2 2 * - - = × × + × - = × × + × - = y y D P P
Numerical Example with Convexity What if yields fall by 2%? If yields decrease instantaneously from 8% to 6%, what’s the percentage price change of this bond? Note that predicted change is NOT SYMMETRIC . ( 29 % 76 . 26 or 0.2676 ) 02 . 0 ( 4 . 212 2 1 02 . 0 26 . 11 ) ( Convexity 2 1 2 2 * = × × + × = × × + × - = y y D P P

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Convexity of Two Bonds
Investors Like Convexity Bonds with greater curvature gain more in price when yields fall than they lose when yields rise. The more volatile interest rates, the more attractive this asymmetry.

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