A 100 par value 6 percent bond with semi-annual coupons is purchased at 110 to yield a nominal rate of 4 percent convertible semi-annually. A similar 3 percent bond with semi-annual coupons is purchased at P to provide the buyer with the same yield. Calculate P.

Options:

(A) 90 (B) 95 (C) 100 (D) 105 (E) 110

(A) 90

Explanation:

par value = $100

coupon rate 6%, semiannual

market price $110

YTM = 4%

similar coupon:

par value = $100

coupon rate 3%, semiannual

market price $???

YTM = 4%

first of all, since the market rate is higher than the coupon rate, the bond will be sold at a discount, therefore, options C, D and E can be eliminated.

that leaves us with options A ($90) and B ($95)

now we can use the YTM formula to find n for both options:

YTM = [coupon + [(face value - market value)/n]} / [(face value + market value)/2]

0.04 = 3 + [(100 - 90)/n]} / [(100 + 90)/2]

0.04 = (3 + 10/n) / 95

3.8 = (3n + 10) / n

3.8n = 3n + 10

0.8n = 10

n = 10/.8 = 12.5 years

0.04 = 3 + [(100 - 95)/n]} / [(100 + 90)/2]

0.04 = (3 + 5/n) / 95

3.8 = (3n + 5) / n

3.8n = 3n + 5

0.8n = 5

n = 5/.8 = 6.25 years

now we must replace n in the YTM formula for the first bond:

bond price $90

YTM = 3 + [(100 - 110)/12.5]} / [(100 + 110)/2]

YTM = 2.2 / 105 = 2.09% X 2 = 4.18% ≈ 4%

bond price $95

YTM = 3 + [(100 - 110)/6.25]} / [(100 + 110)/2]

YTM = 1.4 / 105 = 1.33% X 2 = 2.67% WRONG

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