To calculate the Yield to Maturity (YTM) for the bond, we use the following formula implicitly:

$P = \sum_{t=1}^{2N} \frac{C}{(1 + YTM/2)^t} + \frac{F}{(1 + YTM/2)^{2N}}$

Where:

• $P$: Bond price ($775)
• $F$: Par value ($1,000)
• $C$: Semiannual coupon payment ($\frac{9\% \times 1,000}{2} = 45$)
• $N$: Number of years to maturity
• $YTM$: Yield to maturity (semiannual rate, to solve for)

For 14 years:

• $N = 14$ years, hence $2N = 28$ periods
• Semiannual coupon payment = 45

Steps:

1. Plug the values into the formula: $775 = \sum_{t=1}^{28} \frac{45}{(1 + YTM/2)^t} + \frac{1000}{(1 + YTM/2)^{28}}$

2. Solve iteratively for $YTM$ (semiannual rate) by trial-and-error, approximation, or using a financial calculator or software.

3. Double the semiannual rate to get the annual YTM.

I'll compute this.from scipy.optimize import fsolve

Given values

P = 775 # Price of the bond F = 1000 # Face value of the bond C = 45 # Semiannual coupon payment N = 14 # Number of years to maturity periods = 2 * N # Total number of periods

Define the YTM function

def bond_price(ytm): return sum(C / (1 + ytm/2)**t for t in range(1, periods + 1)) + F / (1 + ytm/2)**periods - P

Solve for YTM

initial_guess = 0.1 # Initial guess for YTM ytm_solution = fsolve(bond_price, initial_guess)[0]

Convert semiannual rate to annual rate

ytm_annual = ytm_solution * 2

ytm_annual0.248622305392362270.24862230539236227successThe bond's yield to maturity (YTM) if it matures in 14 years is approximately 24.86% (annualized, rounded to two decimal places).

Now, I'll calculate the YTM if the bond matures in 28 years and 7 years.# Recalculate for 28 years (2N = 56 periods) and 7 years (2N = 14 periods) def calculate_ytm(years_to_maturity): periods_new = 2 * years_to_maturity # Total periods def bond_price_new(ytm): return sum(C / (1 + ytm/2)**t for t in range(1, periods_new + 1)) + F / (1 + ytm/2)**periods_new - P

ytm_new = fsolve(bond_price_new, initial_guess)[0]
return ytm_new * 2  # Convert semiannual to annual rate

Calculate YTM for 28 years and 7 years

ytm_28_years = calculate_ytm(28) ytm_7_years = calculate_ytm(7)

ytm_28_years, ytm_7_years(0.23516192759278315, 0.2834421646851707)(0.23516192759278315, 0.2834421646851707)successThe bond's yield to maturity (YTM) is as follows:

1. If the bond matures in 28 years: Approximately 23.52% (annualized).
2. If the bond matures in 7 years: Approximately 28.34% (annualized).

Would you like a detailed explanation of these calculations or any further clarifications?

---

Expanding Questions:

1. How does the time to maturity generally affect a bond's yield to maturity?
2. What is the relationship between bond price and YTM?
3. Why does the YTM change as the bond's maturity period changes?
4. Can this bond be considered a high-yield or junk bond based on these results?
5. How would the YTM change if the bond price were higher than $775?

Tip:

When solving for YTM, using software or financial calculators can simplify iterative computations, as manual trial and error can be time-consuming.