26 9 2 1 2 07 1 0050 68 2 2 1 0050 2 2 p dp p dp cx r

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26 . 9 2 1 2 07 . 1 0050 . 0 * 68 . 2 2 1 * ) 0050 (. ) ( 2 2 P dP P dP CX R MD P dP R Price = Price* dP P = $1079.94 * -.012831 = $-13.86 New Price = P 0 + Price = $1079.94 + $-13.86 = $1066.08 30
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F520_Bond Convexity improves on our solution, especially for larger price changes. N I PV PMT FV 3*2 7/2 CPT= 1079.93 .10*1000/2 =50 1000 3*2 7.5/2 CPT= 1066.06 .10*1000/2 =50 1000 3*2 10/2 CPT= 1000.00 .10*1000/2 =50 1000 On page 24, we had predicted 1065.96 for 50 bp increase(off 10 cents), with our new measure of convexity we are only 2 cents off. Expected Price Change assuming +300 bp: % 3513 . 7 073513 . 004167 . 07768 . 26 . 9 2 1 2 07 . 1 03 . 0 * 68 . 2 ) 03 (. 2 P dP P dP w/ convexity $1079.94 * -.073513 = $-79.39 $1079.94 + $-79.39 = $1000.55 w/o convexity $1079.94 * -.07768 = $-83.90 $1079.94 + $-83.90 = $996.04 over-estimates decrease. And this is with a very small convexity measure. 31
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F520_Bond The duration measure, which we have been calculating, assumes a flat yield curve. The appropriate duration can be calculated by discounting the coupons and principal value of the bond by the discount rates or yields on appropriate maturity zero coupon bonds. In an upward sloping yield curve, the adjusted duration measure will be smaller, since more discounted cash flows are discounted at higher rates. Our models have assumed no default risk. An adjustment can be made by calculating the expected cash flow in the duration measure. The detailed duration calculation is only for fixed rate securities, but can be adjusted. (The approximation will work for other securities if properly done.) Floating rate securities have a duration equal to the time period between the now and when the interest rate is readjusted. Restructuring the balance sheet in order to reduce duration gap is time consuming and can be expensive. Immunization based on duration requires costly continuous portfolio re-balancing to ensure that the investment duration exactly matches the investment horizon. 32
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F520_Bond 5% coupon bond with 20 years to maturity (calculated in textbook reading). Annual SemiAnnual Coupon 5% 2.50% Yield 9% 4.50% Maturity 20 40 Par 100 100 Pmt # Rec'd PV PV*t PV*t*(t+1) 1 2.5 2.3923 2.3923 4.7847 2 2.5 2.2893 4.5786 13.7359 3 2.5 2.1907 6.5722 26.2889 4 2.5 2.0964 8.3856 41.9281 5 2.5 2.0061 10.0306 60.1838 6 2.5 1.9197 11.5184 80.6291 7 2.5 1.8371 12.8595 102.8760 8 2.5 1.7580 14.0637 126.5733 9 2.5 1.6823 15.1403 151.4035 10 2.5 1.6098 16.0982 177.0801 11 2.5 1.5405 16.9455 203.3456 12 2.5 1.4742 17.6899 229.9689 13 2.5 1.4107 18.3388 256.7436 14 2.5 1.3499 18.8991 283.4858 15 2.5 1.2918 19.3770 310.0323 16 2.5 1.2362 19.7788 336.2391 17 2.5 1.1829 20.1100 361.9799 18 2.5 1.1320 20.3760 387.1443 19 2.5 1.0833 20.5818 411.6367 20 2.5 1.0366 20.7321 435.3750 21 2.5 0.9920 20.8313 458.2895 22 2.5 0.9493 20.8835 480.3216 23 2.5 0.9084 20.8926 501.4232 24 2.5 0.8693 20.8622 521.5552 25 2.5 0.8318 20.7957 540.6872 26 2.5 0.7960 20.6962 558.7964 27 2.5 0.7617 20.5667 575.8667 28 2.5 0.7289 20.4099 591.8885 29 2.5 0.6975 20.2286 606.8577 30 2.5 0.6675 20.0250 620.7750 31 2.5 0.6388 19.8014 633.6460 32 2.5 0.6112 19.5600 645.4798 33 2.5 0.5849 19.3026 656.2892 34 2.5 0.5597 19.0312 666.0903 35 2.5 0.5356 18.7473 674.9015 36 2.5 0.5126 18.4525 682.7438 37 2.5 0.4905 18.1484 689.6402 38 2.5 0.4694 17.8363 695.6154 39 2.5 0.4492 17.5174 700.6954 40 102.5 17.6227 704.9077 28,901.2147 63.1968 1,373.9652 44,404.2118 643.4226 semi-annual total CX 21.7410 321.7113 semi-annual price CX 10.8705 Macaulay 160.8556 total CX 10.4024 Modified duration convexity yearly CX = semi-annual/2 2 33
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F520_Bond Cont (demonstration of textbook problem) Without Convexity: 2% increase in rates % 8 . 20 208 . 02 . * 4 . 10 * P dP P dP R MD P dP With Total Convexity: 2% increase in rates % 58 . 17 1758 . 0322 . 208 .
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