You find a zero coupon bond with a par value of $10,000 and 19 years to maturity. The yield to maturity on this bond is 4.7 percent. Assume semiannual compounding periods. What is the dollar price of the bond?

To calculate the dollar price of a zero-coupon bond, we use the following formula:

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

Where:
- $ P $ = Present value (the price of the bond)
- $ F $ = Face value (par value) of the bond
- $ r $ = Yield per period (YTM divided by number of compounding periods per year)
- $ n $ = Total number of periods (years to maturity multiplied by compounding periods per year)

Given:
- Par value $ F = 10,000 $
- Yield to maturity (YTM) $ = 4.7\% $ per year
- Semiannual compounding: 2 periods per year
- Years to maturity $ = 19 $

### Step-by-step:

1. **Calculate the yield per period:**
   \[
   r = \frac{4.7\%}{2} = \frac{0.047}{2} = 0.0235
   \]

2. **Calculate the total number of periods:**
   \[
   n = 19 \times 2 = 38
   \]

3. **Calculate the price of the bond:**
   \[
   P = \frac{10,000}{(1 + 0.0235)^{38}}
   \]
   \[
   P = \frac{10,000}{(1.0235)^{38}}
   \]
   \[
   P \approx \frac{10,000}{2.4137} = 4143.48
   \]

Thus, the **price of the bond** is approximately **$4,143.48**.

Would you like further explanation on any step, or more details?

Here are five related questions:
1. How would the price change if the yield to maturity were higher?
2. What if the bond compounded annually instead of semiannually?
3. How do you calculate the price for a bond with coupon payments?
4. What is the effect of a shorter time to maturity on bond price?
5. How can you calculate the effective yield for this bond?

**Tip:** The longer the time to maturity or the higher the yield, the lower the price of a zero-coupon bond.

Learn how to calculate the price of a zero-coupon bond with a $10,000 par value, 19 years to maturity, and a 4.7% yield to maturity (YTM) using semiannual compounding. This problem covers key financial mathematics concepts, including bond pricing and compound interest, with a step-by-step solution.

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