SOLUTION: . Calculate the fair present values of the following bonds, all of which pay interest semiannually, have a face value of $1,000, have 12 years remaining to maturity, and have a req

Question 1170652: . Calculate the fair present values of the following bonds, all of which pay interest semiannually, have a face value of $1,000, have 12 years remaining to maturity, and have a required rate of return of 10 percent.
a. The bond has a 6 percent coupon rate.
b. The bond has a 8 percent coupon rate.
c. The bond has a 10 percent coupon rate.
d. What do your answers to parts
(a) through (c) say about the relation between coupon rates and present values?

Answer by CPhill(1959) (Show Source): You can put this solution on YOUR website!
Let's calculate the fair present values of these bonds.
**Understanding the Concepts**
* **Face Value (FV):** $1,000 (the amount paid at maturity)
* **Years to Maturity:** 12 years
* **Semiannual Payments:** Payments are made twice a year.
* **Required Rate of Return (r):** 10% per year (5% semiannually)
* **Coupon Rate:** The annual interest rate paid on the face value.
* **Coupon Payment (PMT):** (Coupon Rate * Face Value) / 2
**Formulas**
* **Semiannual Required Rate (i):** r / 2
* **Number of Periods (n):** Years to Maturity * 2
* **Present Value (PV) of Bond:**
* PV = (PMT * [1 - (1 + i)^-n] / i) + (FV / (1 + i)^n)
**Calculations**
**a) 6% Coupon Rate**
* Coupon Payment (PMT): (0.06 * $1,000) / 2 = $30
* Semiannual Required Rate (i): 0.10 / 2 = 0.05
* Number of Periods (n): 12 * 2 = 24
* PV = (30 * [1 - (1 + 0.05)^-24] / 0.05) + (1000 / (1 + 0.05)^24)
* PV = (30 * [1 - 0.3094216] / 0.05) + (1000 / 3.225099)
* PV = (30 * 0.6905784 / 0.05) + 310.0695
* PV = (30 * 13.811568) + 310.0695
* PV = 414.34704 + 310.0695
* PV ≈ $724.42
**b) 8% Coupon Rate**
* Coupon Payment (PMT): (0.08 * $1,000) / 2 = $40
* Semiannual Required Rate (i): 0.10 / 2 = 0.05
* Number of Periods (n): 12 * 2 = 24
* PV = (40 * [1 - (1 + 0.05)^-24] / 0.05) + (1000 / (1 + 0.05)^24)
* PV = (40 * [1 - 0.3094216] / 0.05) + (1000 / 3.225099)
* PV = (40 * 0.6905784 / 0.05) + 310.0695
* PV = (40 * 13.811568) + 310.0695
* PV = 552.46272 + 310.0695
* PV ≈ $862.53
**c) 10% Coupon Rate**
* Coupon Payment (PMT): (0.10 * $1,000) / 2 = $50
* Semiannual Required Rate (i): 0.10 / 2 = 0.05
* Number of Periods (n): 12 * 2 = 24
* PV = (50 * [1 - (1 + 0.05)^-24] / 0.05) + (1000 / (1 + 0.05)^24)
* PV = (50 * [1 - 0.3094216] / 0.05) + (1000 / 3.225099)
* PV = (50 * 0.6905784 / 0.05) + 310.0695
* PV = (50 * 13.811568) + 310.0695
* PV = 690.5784 + 310.0695
* PV ≈ $1,000.65
**d) Relation between Coupon Rates and Present Values**
* When the coupon rate is less than the required rate of return (6% and 8%), the bond's present value is less than its face value. This is because the bond pays less interest than what the market demands.
* When the coupon rate is equal to the required rate of return (10%), the bond's present value is approximately equal to its face value.
* In essence, the higher the coupon rate relative to the required rate of return, the higher the present value of the bond. Conversely, the lower the coupon rate, the lower the present value.

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