To calculate the Yield to Maturity (YTM) of a bond, we typically use the following formula for a bond that pays interest annually:

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

Where:

• $P$ = Price of the bond
• $C$ = Coupon payment (Annual coupon rate * Face value)
• $F$ = Face value of the bond (typically $1,000)
• $n$ = Number of years to maturity
• $YTM$ = Yield to Maturity (what we're solving for)

The YTM is the discount rate that equates the present value of all future cash flows (coupons and face value) to the current price of the bond.

To calculate YTM accurately, it is generally easier to use a financial calculator or an iterative method, as it involves solving a polynomial equation. However, we can estimate it using the following approximation for bonds with similar maturities and coupon rates:

$YTM \approx \frac{C + \frac{F - P}{n}}{\frac{F + P}{2}}$

Where:

• $C$ = Coupon payment
• $F$ = Face value
• $P$ = Price of the bond
• $n$ = Years to maturity

Now, let's go through each of the given bonds.

Bond 1:

• Par Value: $1,000
• Coupon Rate: 8%
• Years to Maturity: 30
• Price: $600

Coupon payment: $0.08 \times 1,000 = 80$

Using the YTM approximation formula:

$YTM \approx \frac{80 + \frac{1,000 - 600}{30}}{\frac{1,000 + 600}{2}}$

$YTM \approx \frac{80 + \frac{400}{30}}{800} \approx \frac{80 + 13.33}{800} \approx \frac{93.33}{800} = 0.1167 = 11.67\%$

Bond 2:

• Par Value: $1,000
• Coupon Rate: 7%
• Years to Maturity: 30
• Price: $790

Coupon payment: $0.07 \times 1,000 = 70$

$YTM \approx \frac{70 + \frac{1,000 - 790}{30}}{\frac{1,000 + 790}{2}}$

$YTM \approx \frac{70 + \frac{210}{30}}{895} \approx \frac{70 + 7}{895} \approx \frac{77}{895} = 0.0860 = 8.60\%$

Bond 3:

• Par Value: $5,000
• Coupon Rate: 9%
• Years to Maturity: 15
• Price: $6,450

Coupon payment: $0.09 \times 5,000 = 450$

$YTM \approx \frac{450 + \frac{5,000 - 6,450}{15}}{\frac{5,000 + 6,450}{2}}$

$YTM \approx \frac{450 + \frac{-1,450}{15}}{5,725} \approx \frac{450 - 96.67}{5,725} \approx \frac{353.33}{5,725} = 0.0618 = 6.18\%$

Bond 4:

• Par Value: $5,000
• Coupon Rate: 11%
• Years to Maturity: 20
• Price: $5,000

Coupon payment: $0.11 \times 5,000 = 550$

$YTM \approx \frac{550 + \frac{5,000 - 5,000}{20}}{\frac{5,000 + 5,000}{2}}$

$YTM \approx \frac{550 + 0}{5,000} \approx \frac{550}{5,000} = 0.11 = 11\%$

Summary of Yields to Maturity (YTM):

1. Bond 1 (8%, 30 years, $600 price): YTM ≈ 11.67%
2. Bond 2 (7%, 30 years, $790 price): YTM ≈ 8.60%
3. Bond 3 (9%, 15 years, $6,450 price): YTM ≈ 6.18%
4. Bond 4 (11%, 20 years, $5,000 price): YTM ≈ 11%

Would you like more details on any of these calculations or further clarification on how YTM works?

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Follow-up questions:

1. How does the coupon rate affect the YTM of a bond?
2. What happens to a bond's YTM when its price falls below par value?
3. How can you use a financial calculator or Excel to find the exact YTM?
4. What role does the time to maturity play in determining the YTM?
5. What are the differences between current yield and YTM?

Tip:

When calculating YTM, remember that it's an iterative process, and the approximation formula can give a good estimate, but the exact YTM may require the use of a financial calculator or software that solves the bond price equation.