There is a zero coupon bond that sells for $4,329.17 and has a par value of $10,000. If the bond has 16 years to maturity, what is the yield to maturity? Assume semiannual compounding.

To calculate the **yield to maturity (YTM)** of a zero-coupon bond, we can use the following formula:

\[
P = \dfrac{F}{(1 + r)^n}
\]

Where:
- $ P $ = Present value (current price of the bond) = $4,329.17
- $ F $ = Face value (par value of the bond) = $10,000
- $ r $ = Yield per period (which is the semiannual yield to maturity, the value we need to find)
- $ n $ = Number of periods (total compounding periods)

Given:
- The bond matures in 16 years, with **semiannual compounding**, so there are $ n = 16 \times 2 = 32 $ periods.

We can rearrange the formula to solve for $ r $:

\[
r = \left( \dfrac{F}{P} \right)^{\frac{1}{n}} - 1
\]

Now, let's substitute the known values into the equation:

\[
r = \left( \dfrac{10,000}{4,329.17} \right)^{\frac{1}{32}} - 1
\]

Let me calculate that.The semiannual yield to maturity (YTM) is approximately $ 2.65\% $ per period.

Since this is the yield for each semiannual period, we can calculate the annual yield to maturity by multiplying it by 2:

\[
\text{Annual YTM} = 2 \times 2.65\% = 5.30\%
\]

Thus, the annual yield to maturity is approximately **5.30%**.

Would you like further clarification or details on the steps?

Here are five related questions to expand on this topic:
1. How does the yield to maturity change with varying bond prices?
2. What is the difference between yield to maturity and current yield?
3. How would a coupon bond differ in yield calculations from a zero-coupon bond?
4. What role does compounding frequency play in bond yield calculations?
5. How does the duration of the bond affect its sensitivity to interest rate changes?

**Tip:** In bond markets, as the price of a bond decreases, its yield to maturity increases, assuming all other factors remain constant.

Find the yield to maturity of a zero-coupon bond selling for $4,329.17 with a par value of $10,000 and 16 years to maturity. Learn how to apply the compound interest formula with semiannual compounding. This finance problem demonstrates key concepts in bond valuation and is ideal for students in undergraduate finance courses.

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