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# Assume interest is paid semiannually paid 0045 0045 n

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Unformatted text preview: lly. paid 0.045 0.045 n (1+YTM)n - 1 YTM2 0.045 + 11+YTM YTM (1+YTM) 0.045 (1+YTM)n - 1 YTM +1 +1 0.045 0.045 n (1+0.04)n - 1 0.042 0.045 + 11+0.04 0.04 (1+0.04) 0.045 (1+0.04)n - 1 0.04 +1 0.045 0.045 20 (1+0.04)20 - 1 0.042 0.045 + 11+0.04 0.04 (1+0.04) 0.045 20 (1+0.04)20 - 1 0.04 +1 +1 =13.82 six-month periods =13.82 = 6.91 years 6.91 Another Duration Example Another Find the duration of a 7% 15-year bond Find sold to yield 9%. Assume interest is paid annually. annually. 0.07 0.07 15 (1+0.09)15 - 1 0.092 0.07 + 11+0.09 0.09 (1+0.09) 0.07 15 (1+0.09)15 - 1 0.09 +1 +1 = 9.24 years 9.24 Modified duration calculated as Macaulay duration / (1 + YTM) Relationship between duration and bond price changes: approx. pct. pct. point change in = - modified duration × change in YTM bond price For this bond, if its yield to maturity For rises to 11 percent, its price will fall by rises fall about about 9.24/1.09 × 2, or 16.95 percent. 9.24/1.09 Convexity The above formula suggests that the relationship between bond price and yield to maturity is linear. Actually, the relationship is convex to the origin. This implies that regardless of which way interest rates move, the percentage change in bond price is more favorable than the above formula would suggest. YTM 7% 9% 11% Price* Price* Approx. Actual 980.96 1000.00 838.79 838.79 69...
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