Assume that you are considering the purchase of a 30-year, noncallable bond with an annual coupon rate of 8.5%. The bond has a face value of $1,000, and it makes semiannual interest payments. If you require an 7.4% yield to maturity on this investment, what is the maximum price you should be willing to pay for the bond?
To calculate the maximum price you should be willing to pay for the bond, you can use the present value formula for a bond:
PV = ∑(C/(1+r)^t) + F/(1+r)^n
Where:
PV = Present value or maximum price
C = Coupon payment per period
r = Yield to maturity rate per period
t = Number of periods
F = Face value of the bond
n = Number of periods (total number of semi-annual payments)
Let's break the problem down step-by-step:
Step 1: Calculate the coupon payment per period (C)
Since the annual coupon rate is 8.5% and the bond makes semiannual payments, the coupon payment per period is (8.5% * $1,000) / 2 = $42.50.
Step 2: Calculate the yield to maturity rate per period (r)
The yield to maturity rate is given as 7.4%, and since the bond makes semiannual payments, divide it by 2 to get the yield to maturity rate per period, which is 3.7%.
Step 3: Calculate the number of periods (t)
Since it's a 30-year bond and it makes semiannual payments, there will be 30 years * 2 = 60 periods.
Step 4: Calculate the present value or maximum price (PV)
Using the present value formula mentioned earlier, we can substitute the values we know into the equation:
PV = ∑(C/(1+r)^t) + F/(1+r)^n
PV = (∑($42.50/(1+0.037)^t)) + $1,000/(1+0.037)^60
To evaluate the sum, we need to calculate the present value of each coupon payment individually and sum them up. The formula to calculate the present value of a single coupon payment is:
Present value of a single payment = C/(1+r)^t
Now, let's calculate the present value for each coupon payment from t = 1 to t = 60:
Present value of first coupon payment = $42.50/(1+0.037)^1
Present value of second coupon payment = $42.50/(1+0.037)^2
...
Present value of sixtieth coupon payment = $42.50/(1+0.037)^60
Once we have calculated all the present values for the coupon payments, we can sum them up and add the present value of the face value:
PV = Present value of the coupon payments + Present value of the face value
Finally, you can now calculate the maximum price (PV) using the formula and the values determined above.
To calculate the maximum price you should be willing to pay for the bond, you need to determine the present value of the bond's future cash flows.
Step 1: Determine the number of periods. Since it is a 30-year bond with semiannual interest payments, there will be a total of 60 periods (30 years * 2 periods per year).
Step 2: Determine the coupon payment. The annual coupon rate is 8.5%, so the semiannual coupon payment would be $1,000 * 8.5% / 2 = $42.50.
Step 3: Determine the discount rate. Since you require an annual yield to maturity of 7.4%, the semiannual discount rate would be 7.4% / 2 = 3.7%.
Step 4: Calculate the present value of the coupon payments. Since the bond makes semiannual interest payments, it can be treated as an ordinary annuity. The formula for the present value of an annuity is:
PV = C * [1 - (1 + r)^(-n)] / r
Where:
PV = Present value of the annuity (maximum price)
C = Coupon payment per period ($42.50)
r = Discount rate per period (3.7%)
n = Number of periods (60)
PV = $42.50 * [1 - (1 + 0.037) ^ (-60)] / 0.037
PV ≈ $42.50 * (1 - 0.194584) / 0.037
PV ≈ $42.50 * 0.805416 / 0.037
PV ≈ $918.59
Therefore, the maximum price you should be willing to pay for the bond is approximately $918.59.