You are considering investing in a security that matures in 10 years with a par value of $1,000. During the first five years, the security has an 8 percent coupon with quarterly payments (i. e., you receive $20 a quarter for the first 20 quarters). During the remaining five years the security has a 10 percent coupon with quarterly payments (i. e., you receive $25 a quarter for the second 20 quarters). After 10 years (40 quarters) you receive the par value. Another 10-year bond has an 8 percent semiannual coupon (i. e., the coupon payment is $40 every six months). This bond is selling at its par value, $1,000. This bond has the same risk as the security you are thinking of purchasing. Given this information, what should be the price of the security you are considering purchasing

$1,060.75

Explanation:

the yield to maturity of the second bond is to 4% semiannual or 8.16% effective annual rate.

so we have to calculate the quarterly interest rate that yields an effective annual rate of 8.16%:

0.0816 = (1 + i)⁴ - 1

1.0816 = (1 + i)⁴

⁴√1.0816 = ⁴√(1 + i)⁴

1.0198 = 1 + i

i = 0.019804 = 1.9804%

now we must discount the first bond using that effective interest rate:

PV of face value = $1,000 / (1 + 4%)²⁰ = $456.39

PV of first 20 coupon payments = $20 x 16.38304 (PV annuity factor, 1.9804%, 20 periods) = $327.66

now we must find the value of the last 20 coupon payments but at the end of year 5 = $25 x 16.38304 = $409.58. Then we calculate the PV = $409.58 / (1 + 4%)¹⁰ = $276.70

the bond's current market value = $456.39 + $327.66 + $276.70 = $1,060.75

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