Crossfade Corp. has a bond with apar value of $$ \$ 2,000 $$ that sells for $$ \$ 1,948.34 $$. The bond has a coupon rate of 6.54 percent and matures in 14 years. If the bond makes semiannual coupon payments, what is the YTM of the bond? a. $$ 6.49 \% $$ b. $$ 6.83 \% $$ c. $$ 3.41 \% $$ d. $$ 5.12 \% $$ e. $$ 6.15 \% $$

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We are given the following information about the bond:

• Face (par) value = $2,000  
• Current price = $1,948.34  
• Annual coupon rate = 6.54% → Annual coupon payment = 0.0654 × $2,000 = $130.80  
• Since the bond pays semiannual coupons, each coupon payment = $130.80/2 = $65.40  
• Number of years until maturity = 14 years → Total number of periods = 14 × 2 = 28  
• Let i be the semiannual yield (in decimal form). Then the yield to maturity (YTM) on an annual basis (nominal rate compounded semiannually) is 2i.

The bond’s price is the present value of all future coupon payments plus the present value of the face value at maturity. This relationship can be written as:
  Price = Coupon × [1 – (1 + i)^(-n)]/i + Face Value × (1 + i)^(-n)
where n = 28.

Plugging in the numbers:
  1,948.34 = 65.40 × [1 – (1 + i)^(-28)]/i + 2,000 × (1 + i)^(-28)

We need to solve for i.

Rather than solving this equation algebraically, we can use trial and error (or a financial calculator). Let’s check the candidate answer associated with an annual yield of 6.83%. Since the yield is compounded semiannually, the corresponding semiannual yield is:
  i = 6.83%/2 = 3.415% or 0.03415 (in decimal).

Step 1. Compute (1 + i)^28:
  (1 + 0.03415)^28 ≈ exp(28 × ln(1.03415))
  First, ln(1.03415) ≈ 0.0336, so:
  28 × 0.0336 ≈ 0.9408 
  exp(0.9408) ≈ 2.562
Thus, (1 + i)^28 ≈ 2.562.

Step 2. Compute the discount factor:
  (1 + i)^(-28) = 1/2.562 ≈ 0.390

Step 3. Calculate the present value of the coupon payments:
  Annuity factor = [1 – 0.390] / 0.03415 ≈ 0.610 / 0.03415 ≈ 17.86
  Present Value of Coupons = 65.40 × 17.86 ≈ 1,168

Step 4. Calculate the present value of the par value:
  PV of Face Value = 2,000 × 0.390 ≈ 780

Step 5. Add the two present values:
  Total PV ≈ 1,168 + 780 = 1,948

This computed price is virtually identical to the given market price of $1,948.34, confirming that our semiannual yield of about 3.415% is correct. Consequently, the annual yield (YTM) is:
  YTM = 2 × 3.415% ≈ 6.83%

Thus, the correct answer is 6.83%, which corresponds to option b.
The YTM of the bond is 6.83%, which is option b.

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