A 4% coupon bond has a par value of $1,000 and a yield-to-maturity of 6%. You purchase the bond when it has exactly 13 years remaining until maturity. You hold the bond for 6 months, collect the coupon payment, and then sell the bond immediately. If the bond's yield-to-maturity is 9% when you sell it, what is your percentage return over this 6-month holding period? Enter your answer as a decimal and show 4 decimal places. For example, if your answer is 6.25%, enter .0625.

To solve this, we can break it down into the following steps:

1. **Calculate the purchase price of the bond** using the initial yield-to-maturity (YTM) of 6%. 
2. **Calculate the selling price** of the bond after 6 months using the new YTM of 9%.
3. **Determine the coupon payment** collected during the holding period.
4. **Calculate the percentage return** over the 6-month holding period.

### Step 1: Calculate the Purchase Price of the Bond

The bond has a 4% coupon rate, which means the annual coupon payment is:

\[
\text{Coupon Payment} = 0.04 \times 1000 = 40 \, \text{dollars per year}.
\]

Since the bond pays semiannual coupons, the semiannual coupon payment is:

\[
\text{Semiannual Coupon Payment} = \frac{40}{2} = 20 \, \text{dollars}.
\]

Now, we calculate the purchase price using the bond price formula, where $P_0$ is the present value of the bond:

\[
P_0 = \sum \left( \frac{C}{(1 + r)^t} \right) + \frac{F}{(1 + r)^T},
\]

where:
- $C$ is the semiannual coupon payment,
- $F$ is the par value of the bond,
- $r$ is the semiannual yield to maturity (YTM),
- $t$ is the number of periods for each coupon payment,
- $T$ is the total number of periods until maturity.

The bond has 13 years remaining, so there are $13 \times 2 = 26$ periods. The semiannual YTM is $6\% / 2 = 3\%$.

Now let's calculate the purchase price $P_0$.

### Step 2: Calculate the Selling Price of the Bond

The yield-to-maturity changes to 9% after 6 months. So, the new semiannual YTM is $9\% / 2 = 4.5\%$. There are $25$ periods remaining at this point (since 6 months have passed).

Let's calculate the selling price.

### Step 3: Determine the Coupon Payment Collected

In the first 6 months, you collect one semiannual coupon payment of $20.

### Step 4: Calculate the Percentage Return

Finally, the percentage return over the 6-month holding period is calculated as:

\[
\text{Return} = \frac{\text{Coupon Payment} + \text{Selling Price} - \text{Purchase Price}}{\text{Purchase Price}}.
\]

Let me now compute these values.1. The **purchase price** of the bond is approximately \$821.2316.
2. The **selling price** of the bond after 6 months, when the yield-to-maturity is 9%, is approximately \$629.2948.
3. The **percentage return** over the 6-month holding period is approximately $-0.2094$ or $-20.94\%$.

This negative return occurs because the bond's price dropped significantly when the yield-to-maturity increased to 9%.

Would you like more details on any part of this calculation?

Here are five related questions:
1. How does an increase in YTM affect bond prices?
2. What happens to the return if the bond is held until maturity?
3. How is the duration of a bond related to interest rate risk?
4. Can coupon reinvestment improve the total return on a bond investment?
5. How does the coupon rate compare to the YTM in determining bond prices?

**Tip:** The longer a bond's duration, the more sensitive its price is to changes in interest rates.

Learn how to calculate the percentage return of a 4% coupon bond over a 6-month holding period with detailed steps. This problem covers essential concepts like bond pricing, yield-to-maturity (YTM), and the time value of money. Suitable for undergraduate finance students.

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