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.
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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.
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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
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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
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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
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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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