Two years ago, Gamma Inc. sold a $250 million bond issue to finance the purchase of new jet airliners. These bonds were issued in $1,000 denominations with an original maturity of 14 years and a coupon rate of 12% with interest paid semiannually. Determine the value today of one of these bonds to an investor who requires a 14 percent return on these securities. Explain why the bond has a value that is not equal to the par value.

Face value = $1,000

Coupon rate = 12%
Frequency of coupon payment = Semiannual
Coupon payment = $1,000*12%*1/2 = $60
Time to maturity now = 14 – 2 = 12 years
Required rate of return = 14%
Value of bond today = $60*PVIFA14%/2, 12*2 + $1,000*PVIF14%/2, 12*2
= $60*PVIFA7%, 24 + $1,000*PVIF7%, 24
= $60*11.46933 + $1,000*0.19715
= $688.16 + 197.15
= $885.31
Since Coupon rate ≠ Required rate of return, bond value is not equal to par value.
Coupon rate < Required rate of return, Bond value < Par value

Face value = $1,000

Coupon rate = 12%
Frequency of coupon payment = Semiannual
Coupon payment = $1,000*12%*1/2 = $60
Time to maturity now = 14 – 2 = 12 years
Required rate of return = 14%
Value of bond today = $60*PVIFA14%/2, 12*2 + $1,000*PVIF14%/2, 12*2
= $60*PVIFA7%, 24 + $1,000*PVIF7%, 24
= $60*11.46933 + $1,000*0.19715
= $688.16 + 197.15
= $885.31
Since Coupon rate ≠ Required rate of return, bond value is not equal to par value.
Coupon rate < Required rate of return, Bond value < Par value

To determine the value of the bond today, we need to calculate the present value of its future cash flows.

Step 1: Calculate the number of periods remaining for the bond. Since the original maturity of the bond was 14 years and two years have passed, there are now 12 years remaining (14 - 2 = 12). As interest is paid semiannually, there are a total of 24 payment periods (12 years × 2 = 24).

Step 2: Determine the present value of the bond's coupon payments. The bond has a coupon rate of 12%, which is distributed semiannually. The formula to calculate the present value of the coupon payments is:

PV = C * [1 - (1 + r)^(-n)] / r

Where PV is the present value, C is the coupon payment, r is the required return, and n is the number of periods.

Considering the bond's coupon payment of $1,000 (since it is a $1,000 denomination bond) and the required return of 14% (0.14) for the investor, we can calculate the present value of the coupon payments:

PV_coupon = $1,000 * [1 - (1 + 0.14/2)^(-24)] / (0.14/2)

Step 3: Determine the present value of the bond's face value (par value) at maturity. As the bond will be redeemed at its par value ($1,000), discounted by the required return and the remaining number of periods, we can calculate the present value of the face value:

PV_facevalue = $1,000 / (1 + 0.14/2)^24

Step 4: Calculate the total present value of the bond by adding the present value of the coupon payments (PV_coupon) and the present value of the face value (PV_facevalue):

Total PV = PV_coupon + PV_facevalue

The resulting value represents the value today of one of the bonds issued by Gamma Inc. to an investor who requires a 14% return on these securities.

Now, let's address why the bond has a value that is not equal to the par value. The value of a bond is influenced by several factors, including market interest rates, coupon rate, and time to maturity. When market interest rates rise, the value of existing bonds, with lower coupon rates, decreases. Conversely, when market interest rates fall, the value of existing bonds generally increases. In this case, since the investor requires a 14% return but the bond's coupon rate is 12%, the bond's value will be lower than its par value to compensate for the difference in required return.