Three years ago Messy House Painting issued a 20-year bond with a $1,000 maturity value and a 4 percent coupon rate of interest. Interest is paid semiannually. The bond is currently selling for $826.
What is the bond's yield to maturity? Do not round intermediate calculations. Round your answer to one decimal place.
%
If the bond can be called in four years for a redemption price of $1,029, what is the bond's yield to call? Do not round intermediate calculations. Round your answer to one decimal place.
%

To calculate both the **Yield to Maturity (YTM)** and **Yield to Call (YTC)**, we need to solve for the interest rate in two different situations: one where the bond is held to maturity, and another where the bond is called after a certain number of years.

### Given Data:
- Face value of the bond: $ F = 1,000 $
- Coupon rate: $ 4\% $ per year (semiannual payments)
- Time since issuance: 3 years ago
- Remaining maturity: $ 20 - 3 = 17 $ years (34 semiannual periods)
- Selling price: $ P = 826 $
- Coupon payments (semiannual): $ \frac{4\% \times 1,000}{2} = 20 $ 
- Call price: $ 1,029 $
- Time to call: 4 years (8 semiannual periods)

### 1. Yield to Maturity (YTM)

The bond's YTM can be found by solving the following equation for $ r $ (the semiannual yield):

\[
P = \sum_{t=1}^{34} \frac{C}{(1 + r)^t} + \frac{F}{(1 + r)^{34}}
\]

Where:
- $ P $ is the current bond price (\$826),
- $ C $ is the semiannual coupon payment (\$20),
- $ F $ is the face value (\$1,000),
- $ t $ is the number of periods (34 periods),
- $ r $ is the semiannual interest rate (which we'll find).

The formula can be simplified using numerical methods to find $ r $, and then the annual yield to maturity is $ 2 \times r $.

### 2. Yield to Call (YTC)

The YTC can be found by solving the following equation for $ r $ (the semiannual yield):

\[
P = \sum_{t=1}^{8} \frac{C}{(1 + r)^t} + \frac{F_{call}}{(1 + r)^8}
\]

Where:
- $ P $ is the current bond price (\$826),
- $ C $ is the semiannual coupon payment (\$20),
- $ F_{call} $ is the call price (\$1,029),
- $ t $ is the number of periods until call (8 periods),
- $ r $ is the semiannual interest rate (which we'll find).

The annual yield to call is $ 2 \times r $.

Now, let's calculate both the YTM and YTC.The bond's yield to maturity (YTM) is **5.6%**, and the yield to call (YTC) is **10.0%**.

Would you like more details on these calculations or have any questions? Here are five related questions to explore further:

1. What factors affect the yield to maturity of a bond?
2. How does the yield to call differ from the yield to maturity?
3. Why might a bond issuer choose to call a bond before maturity?
4. What is the relationship between bond price and interest rates?
5. How does the time to maturity impact the sensitivity of bond prices to interest rate changes?

**Tip:** The YTM is the interest rate that equates the present value of future cash flows to the bond’s current price.

Three years ago Messy House Painting issued a 20-year bond with a $1,000 maturity value and a 4 percent coupon rate of interest. Interest is paid semiannually. The bond is currently selling for $826. What is the bond's yield to maturity? Do not round intermediate calculations. Round your answer to one decimal place. % If the bond can be called in four years for a redemption price of $1,029, what is the bond's yield to call? Do not round intermediate calculations. Round your answer to one decimal place. %

Calculate the yield to maturity and yield to call for a 20-year bond issued by Messy House Painting. Learn how to determine these financial metrics using the present value of cash flows. Includes detailed breakdowns of formulas for both YTM and YTC with real-world data points. Suitable for undergraduate finance students.

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