Final answer:
To find the purchase price of the bond with a 5% semiannual coupon and a 7-year maturity redeemable at $1030 at a yield rate of 6% compounded semiannually, calculate the present value of the bond's future cash flows, which include all coupon payments and the redemption value, discounted at the yield rate.
Step-by-step explanation:
To calculate the purchase price of a bond at a different yield to its coupon rate, we need to use the concept of present value. The purchase price is determined by the present value of the bond's future cash flows, which include the semiannual coupon payments and the redemption value at maturity.
The bond in question offers a 5% semiannual coupon on a $1000 face value, so each coupon payment will be $1000 * 0.05 / 2 = $25. Since the bond matures in 7 years and pays semiannually, there will be a total of 14 coupon payments. The redemption value is $1030, which will be paid out at the end of the 7th year.
To calculate the bond's purchase price, use the formula for the present value of an annuity to find the present value of the coupon payments and add the present value of the redemption amount.
The yield rate given is 6% compounded semiannually, which we divide by 2 to get the period rate: 6% / 2 = 3%. Now, we use this period rate to discount the coupon payments and the redemption amount back to present value terms.
Using financial calculators or spreadsheet software makes this calculation easier, but it can also be done by hand using the present value formula for each payment and summing them up.
The purchase price calculation involves finding the present value of each $25 coupon payment using the formula PV = C / (1 + r)^n, where C is cash flow, r is the period rate, and n is each period number. Then, also calculate the present value of the terminal redemption value of $1030 using the same method. Summing these present values gives you the bond's purchase price at the yield rate of 6% compounded semiannually.
While this problem requires some calculations, understanding the relationship between coupon rate, market interest rate, and bond price is essential in finance. As interest rates fluctuate, the market prices of existing bonds will adjust accordingly to offer yields comparable to new issues.