Matt purchased a 20-year par value bond with an annual coupon rate of 8% compounded semiannually for a price of 1722.25. The coupons begin paying 6-months after the bond is purchased, and the bond can be called at par value X on any coupon date starting at the end of year 15, after the coupon is paid. The price guarantees that Matt will receive a nominal annual yield rate of at least 6% compounded semiannually. Note: The price guarantees that the yield will be a minimum of 6%. Find X (a) 1460 (b) 1440 (c) 1420 (d) 1400 (e) 1380

As the coupon rate of 8% is greater than the yield to maturity (YTM) of 6% annually, the bond is selling at a premium. Hence, the bond will be called at the earliest i.e. 15 years.

Coupon = Call Price * Semi-annual coupon rate = X * [0.08 / 2] = X * 0.04

Yield to call = 6% annually = 3% semi-annually

Time = 15 years * 2 = 30

We know that,

Current Price of bond = Coupon * [1 - (1 + YTC)-call date] / YTC + Call Price / (1 + YTC)call date

1,722.25 = [X * 0.04] * [1 - (1 + 0.03)-30] / 0.03 + [X / (1 + 0.03)30]

1,722.25 = [X * 0.04] * 19.60 + [X * 0.41]

1,722.25 = X * [(0.04 * 19.60) + 0.41]

1,722.25 = X * 1.194

X = 1,722.25 / 1.194
X=$ 1,442.42 \approx $ 1,440