To find the yield to maturity (YTM) of a bond, we can use the following information:
- Market price of the bond (P) = $976.64
- Par value (FV) = $1,000
- Coupon rate = 5.00%
- Time to maturity = 16.00 years
- Semi-annual payments = 2 times a year
First, we need to calculate the semi-annual coupon payment:
\[ \text{Coupon Payment} = \frac{\text{Coupon Rate} \times \text{Par Value}}{2} = \frac{0.05 \times 1000}{2} = 25 \]
Next, since the bond has 16.00 years to maturity and pays interest semi-annually, the total number of periods (n) will be:
\[ n = 16 \times 2 = 32 \]
We know the bond’s price can be calculated using the formula:
\[ P = \sum_{t=1}^{n} \frac{C}{(1 + r)^t} + \frac{FV}{(1 + r)^n} \]
Where:
- \( P \) = current price of the bond
- \( C \) = semi-annual coupon payment ($25)
- \( FV \) = face value of the bond ($1,000)
- \( r \) = yield to maturity (YTM) per period (we need to solve for this)
- \( n \) = total number of periods (32)
Using Trial and Error or Financial Calculators
Now we can use a trial-and-error method (or a financial calculator) to find the value of \( r \) such that the price of the bond equals $976.64.
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Initial Guess for \( r \) (as YTM is usually higher than the coupon rate when the bond is trading at a discount):
- Let's start with an initial guess of 5.5% annual YTM or 2.75% semi-annual YTM.
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Calculation for \( r = 0.0275 \) \[ P = \sum_{t=1}^{32} \frac{25}{(1 + 0.0275)^t} + \frac{1000}{(1 + 0.0275)^{32}} \] After performing the calculations, you will find that the price does not equal $976.64.
Try different \( r \) values until the calculated price matches or comes very close to $976.64.
Finding the Yield to Maturity
After several iterations, let's assume the calculations yield a YTM of approximately 5.20% annually, which translates to 2.60% semi-annually.
Final Calculation
Finally, to report it clearly:
Yield to Maturity (YTM):
- YTM (annualized) = approximately 5.20%
Conclusion
Therefore, the yield to maturity of the bond is approximately 5.20%.