I missed one day of class and now i cannot figure this out... I thought that price and YTM relied on each other and now I have a question asking to solve for each?

Question

I missed one day of class and now i cannot figure this out... I thought that price and YTM relied on each other and now I have a question asking to solve for each?

An 11% semiannual bond matures in 7 years. The bond has a face value of $1,000 and a current yield of 10.76%. What are the bonds price and YTM?

so.... PMT=55
N=14
FV=$1,000

but I either have to know the PV (price) to get the YTM (I/Y) or the YTM to get the price...

Answer 1

To solve for price (PV) and yield to maturity (YTM), you can use a financial calculator or spreadsheet software like Microsoft Excel. If you don't have access to those tools, you can use trial and error by trying different interest rates until you find the solution.

Let's start with finding the bond's price (PV):

1. Identify the given variables:
- Coupon payment (PMT) = $55 (11% of $1,000 face value, paid semiannually)
- Number of periods (N) = 14 (since there are 7 years and two coupon payments per year)
- Face value (FV) = $1,000

2. Calculate the bond's price using the present value formula:
PV = PMT * [1 - (1 + r)^(-N)] / r + FV / (1 + r)^N

Here, r represents the yield to maturity (YTM), which is the interest rate. Since we don't know the YTM yet, we'll use a starting guess, e.g., 5%.

3. Substitute the values into the formula and solve for PV:
PV = $55 * [1 - (1 + 5%)^(-14)] / 5% + $1,000 / (1 + 5%)^14

4. Compute the resulting value of PV using a calculator or spreadsheet software. In this case, the PV would be approximately $958.44.

Now, let's move on to finding the bond's yield to maturity (YTM):

1. Rearrange the present value formula to solve for YTM:
PV = PMT * [1 - (1 + YTM)^(-N)] / YTM + FV / (1 + YTM)^N

2. Use the previously calculated price (PV = $958.44) instead of the unknown PV.

3. Guess a reasonable YTM estimate, e.g., 5%. Substitute the values into the formula.

4. Solve the equation to find the resulting YTM using a calculator or spreadsheet software. In this case, the YTM would be approximately 11.78%.

This YTM is slightly higher than the given current yield of 10.76%, indicating that the bond is selling at a discount (PV < face value). You can continue adjusting the YTM guess and re-solving until you find a closer approximation.

Remember, using a financial calculator or spreadsheet software will provide a more accurate and efficient way to solve for price and YTM.