Final answer:
The yield to maturity (YTM) for the PBJ Corporation bond with a coupon rate of 5.5%, a $1,000 face value, and a market price of $950 on January 1, 2012, is estimated to be approximately 6.49%. This estimate is obtained using the approximation formula that takes into account the coupon payments, the difference between the face value and the market price, and the number of years until maturity.
Step-by-step explanation:
To calculate the yield to maturity (YTM) for a bond, we must consider the total returns the investor receives from the bond, which include both the interest payments and any capital gains or losses. In this case, the PBJ Corporation bond has a coupon rate of 5.5%, with interest payments made semiannually, and the face value of the bond is $1,000. On January 1, 2012, the bond is priced at $950, which is below its face value.
The YTM calculation requires solving for the discount rate that equates the present value of all future cash flows (interest payments plus the redemption of the face value at maturity) to the current market price of the bond. Since the bond has 5.5% coupon rate and the par value is $1,000, the semiannual interest payment is $27.50 (5.5% of $1,000 divided by 2). There are 18 periods remaining until maturity (9 years x 2).
The YTM calculation is an iterative process and is frequently done using financial calculators or spreadsheet software. However, for estimate purposes, we can use the following approximation formula:
YTM ≈ [C + ((F - P) / n)] / [(F + P) / 2]
Where:
C = annual coupon payment ($55)
F = face value of the bond ($1,000)
P = price of the bond ($950)
n = number of years until maturity (9)
Plugging in these values into the formula gives us:
YTM ≈ [$55 + (($1,000 - $950) / 9)] / [($1,000 + $950) / 2] ≈ 6.49%
The YTM is thus approximately 6.49%, which is a more accurate reflection of the bond's profitability than the coupon rate alone because it takes into account the current market price, the coupon payments, and the gain or loss on the bond at maturity.