A bond with a $100 par value has a 5.25% annual coupons and is due to mature at

Question

A bond with a $100 par value has a 5.25% annual coupons and is due to mature at the end of 16 years. The bond will be redeemed at maturity for an amount equal to its par value less a service charge. The service charge is equal to 25% of the excess (if any) of the par value over the purchase price. A prospective purchaser offers a price that will produce a yield equivalent to a 7% anual effective interest rate, taking into account the deduction of the service charge. It is noted that (1.07)^16=3. In which of the following ranges does the price lie?

A. < $65, B. > $65 but < $71, C. > $71 but < $77, D. > $77 but < $83, E. > $83

Please, no excel spreadsheets.

Explanation / Answer

Calculation of Bond Price using YTM formula :

YTM = (C + ((F-P)/n))) / ((F+P)/2)

YTM = yield equivalent = 7% = 0.07

C = Annual Coupon amount = 100*5.25% = 5.25

F = maturity value = 100- (25% *(100-P)) =(100-25 + 0.25 P) = (75 + 0.25 P)

P = Price of the bond

n = number of years = 16

Hence ,

0.07 = (5.25 + (((75 + 0.25 P) -P)/16))) / (((75 + 0.25 P) +P)/2)

0.07 = (5.25 + ((75 - 0.75 P)/16))) / ((75 + 1.25 P)/2)

0.07 * ((75 + 1.25 P)/2) = (5.25 + ((75 - 0.75 P)/16)))

0.07 * ((75 + 1.25 P)/2) = (5.25 + ((75 - 0.75 P)/16)))

(2.625 + 0.04375 P ) = (84 + 75 – 0.75 P) / 16

16* (2.625 + 0.04375 P ) = (84 + 75 – 0.75 P)

16* (2.625 + 0.04375 P ) = (84 + 75 – 0.75 P)

42 + 0.7 P = 159 - 0.75 P

1.45 P = 159 -42

P = 80.69

Hence the Price should be equal to $80.69

So the correct answer is “:D. > $77 but < $83

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