There is a bond that has a quoted price of 95.859 and a par value of \( \$ 2,000 \). The coupon rate is 6.57 percent and the bond matures in 15 years. If the bond makes semiannual coupon payments, what is the YTM of the bond?

We are given that the bond’s: • Quoted price = 95.859 (which is 95.859% of par) • Par value = $2,000 • Coupon rate = 6.57% (annual) • Maturity = 15 years • Coupons are paid semiannually Step 1. Find the cash flows. Since the quoted price is in percent of par, the market price is   Price = (95.859/100) × $2,000 = $1,917.18 The annual coupon is 6.57% of $2,000, hence   Annual coupon = 0.0657 × 2,000 = $131.40 Since coupons are paid twice a year, the semiannual coupon is   Coupon per period = 131.40⁄2 = $65.70 And the number of periods is   n = 15 × 2 = 30 Step 2. Write the bond valuation formula. Let r be the semiannual yield (i.e. the yield per period). The present value (PV) of the bond is given by the sum of the coupon payments plus the redemption amount:   Price = [Coupon × (1 – (1 + r)^(–n)) / r] + [Par × (1 + r)^(–n)] Substitute in the numbers:   1,917.18 = 65.70 × [1 – (1 + r)^(–30)]⁄r + 2,000 × (1 + r)^(–30) Step 3. Solve for r. This equation does not have a closed‐form solution for r so we must solve it numerically. (You can use a financial calculator, Excel’s RATE function, or any numerical method.) One typically tries various guesses for r. For example, let’s try r = 3.50% per half‐year (0.035):   (1 + 0.035)^(30) ≈ 2.805  → (1 + r)^(–30) ≈ 1/2.805 = 0.3563 Then the annuity factor is   [1 – 0.3563]⁄0.035 = 0.6437⁄0.035 ≈ 18.410 Coupons’ PV = 65.70×18.410 ≈ 1,209.54 Redemption PV = 2,000×0.3563 ≈ 712.60 Total PV = 1,209.54 + 712.60 ≈ 1,922.14 This is slightly above the market price of 1,917.18. A slightly higher r will give a slightly lower price. Trying r ≈ 3.51% per period (0.0351) yields a price very close to 1,917.18. (A more exact numerical solution will show that the semiannual yield r is approximately 3.51%.) Step 4. Convert the semiannual yield to an annual yield. Bonds with semiannual compounding are often quoted as an annual nominal yield equal to 2r. Therefore, the annual yield to maturity (YTM) is   YTM ≈ 2 × 3.51% = 7.02% per annum Thus, the bond’s yield to maturity is approximately 7.02% (annual nominal yield compounded semiannually).