Harrison Inc. has issued a zero/ coupon bond with par value of $1000. The bond pays no coupons until the end of the 6th year, at which point it pays 10% annual coupons. What is the issue (initial) price of this bond if it has a 20 year maturity and yield to maturity of 6%?
To determine the issue price of the zero-coupon bond, we need to use the present value formula and calculate the present value of the future cash flows.
In this case, we have a zero-coupon bond, which means there are no coupons until the 6th year. From that point on, it pays 10% annual coupons. Thus, the cash flows can be divided into two parts:
1. No coupons until the end of the 5th year.
2. Annual coupons of 10% starting from the 6th year until the maturity of the bond at the end of the 20th year.
Let's break down the calculation step-by-step:
Step 1: Calculate the present value of the future cash flows from the 6th year until the 20th year.
Since the coupons start from the 6th year, this portion of the bond is a regular annuity. We can use the annuity formula to calculate its present value:
PV = C * [1 - (1 + r)^(-n)] / r
Where:
PV = Present Value
C = Coupon payments (10% of $1000, i.e., $100)
r = Yield to maturity rate (6% or 0.06)
n = Number of periods (20 - 6 = 14 years)
Substituting the values into the formula, we get:
PV = $100 * [1 - (1 + 0.06)^(-14)] / 0.06
Step 2: Calculate the present value of the par value at the maturity (20th year).
Since this is the final cash flow, its present value is simply the par value of $1000 discounted back to the present:
PV = $1000 / (1 + r)^n
n = Number of periods (20 years) and r = Yield to maturity rate (6% or 0.06).
Step 3: Calculate the present value of the no-coupon period (from year 1 to year 5).
This period represents the initial price of the bond, which is the present value of the par value at the maturity (20th year).
PV = $1000 / (1 + r)^n
n = Number of periods (5 years) and r = Yield to maturity rate (6% or 0.06).
Step 4: Sum up the present values from all three steps to find the issue price of the bond.
Issue Price = Present Value of Step 1 + Present Value of Step 2 + Present Value of Step 3
Calculating each step individually and summing them up will yield the issue price of the zero-coupon bond.