INTEREST RATE SENSITIVITY An investor purchased the following 5 bonds. Each bond

Question

INTEREST RATE SENSITIVITY An investor purchased the following 5 bonds. Each bond had a par value of $1,000 and an 10% yield to maturity on the purchase day. Immediately after the investor purchased them, interest rates fell, and each then had a new YTM of 6%. what is the percentage change in price for each bond after the decline in interest rates? Fil in the following table. Round your answers to the nearest cent or to two decimal places. Enter all amounts as pasitive numbers. Price @ 10% Price @ 6% Percentage Change 10-year, 10% annual coupon 10-year zero 5 year zero 30-year zero 5100 perpetuity

Explanation / Answer

YTM = [C+( F-P)/N]/(F+P)/2 C= COUPON RATE F=FACE VALUE P= PRICE N= NUMBER OF YEARS PRICE @10% N YTM C F FIND P 10 YEARS , 10% ANNUAL COUPON 10 0.1 100 1000 0.1 = [100+( 1000-P)/10]/(1000+P)/2 0.1 = [(1000+1000-P)/10]/500+0.5P 0.1 = [(2000-P)/10]/500+0.5P 0.1(500+0.5P) = 200-0.1P 50+0.05P = 200 – 0.1P 0.05P +0.1P = 200-50 0.15P = 150 P=150/0.15 P=1000 10 YEARS , ZERO 10 0.1 0 1000 0.1 = [0+( 1000-P)/10]/(1000+P)/2 0.1(500+0.5P) = 100-0.1P 50+0.05P = 100 – 0.1P 0.05P +0.1P = 100-50 0.15P = 50 P=50/0.15 P=333.33 5 YEARS , ZERO 5 0.1 0 1000 0.1 = [0+( 1000-P)/5]/(1000+P)/2 0.1(500+0.5P) = 200-0.2P 50+0.05P = 200 – 0.2P 0.05P +0.2P = 200-50 0.25P = 150 P=150/0.25 P=600 30 YEARS , ZERO 30 0.1 0 1000 0.1 = [0+( 1000-P)/30]/(1000+P)/2 0.1(500+0.5P) = 33.33-0.03P 50+0.05P = 33.33 – 0.03P 0.05P +0.03P = 33.33-50 0.08P = -16.67 P=-16.67/0.08 P=-208.38 WE TAKE $1 AS POSITIVE NUMBER TO BE TAKEN $100 PERPETUITY INDEFINITE 0.1 100 1000 PV of perpetuity = C/YTM 100/0.1= 1000 PRICE @6% N YTM C F FIND P 10 YEARS , 10% ANNUAL COUPON 10 0.06 100 1000 0.06 = [100+( 1000-P)/10]/(1000+P)/2 0.06 = [(1000+1000-P)/10]/500+0.5P 0.06 = [(2000-P)/10]/500+0.5P 0.06(500+0.5P) = 200-0.1P 30+0.03P = 200 – 0.1P 0.03P +0.1P = 200-30 0.13P = 170 P=170/0.13 P=1307.69 10 YEARS , ZERO 10 0.06 0 1000 0.06 = [0+( 1000-P)/10]/(1000+P)/2 0.06(500+0.5P) = 100-0.1P 30+0.03P = 100 – 0.1P 0.03P +0.1P = 100-30 0.13P = 70 P=70/0.13 P=538.46 5 YEARS , ZERO 5 0.06 0 1000 0.06 = [0+( 1000-P)/5]/(1000+P)/2 0.06(500+0.5P) = 200-0.2P 30+0.03P = 200 – 0.2P 0.03P +0.2P = 200-30 0.23P = 170 P=170/0.23 P=739.13 30 YEARS , ZERO 30 0.06 0 1000 0.06 = [0+( 1000-P)/30]/(1000+P)/2 0.06(500+0.5P) = 33.33-0.03P 30+0.03P = 33.33 – 0.03P 0.03P +0.03P = 33.33-30 0.06P = 3.33 P=3.33/0.06 P=55.5 $100 PERPETUITY INDEFINITE 0.06 100 1000 PV of perpetuity = C/YTM 100/0.06= 1666.67 $ $ PRICE @10% PRICE @ 6% DIFFERENCE % CHANGE =(DIFFERENCE / OLD PRICE )*100 10 YEARS , 10% ANNUAL COUPON 1000 1307.69 307.69 30.769 10 YEARS , ZERO 333.33 538.46 205.13 61.5396153962 5 YEARS , ZERO 600 739.13 139.13 23.1883333333 30 YEARS , ZERO 1 55.5 54.5 5450 $100 PERPETUITY 1000 1666.67 666.67 66.667

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