Assume the current Treasury yield curve shows that the spot rates for six months, one yearand one and a half years are 1.0% 1.1%and 1.3, all quoted as semiannually compounded APRS. What is the price of a $1,000 par4.5% coupon bond maturing in one and a half years (the next coupon is exactly six months from now)?

To calculate the price of a bond, we need to discount the bond’s future cash flows (both coupon payments and face value) using the appropriate discount factors derived from the spot rates.

### Given:
- **Face Value (Par Value)**: $1,000
- **Coupon Rate**: 4.5% per annum (semi-annually compounded)
- **Spot Rates**:
  - $ r_{0.5} = 1.0\% $ (six months)
  - $ r_{1.0} = 1.1\% $ (one year)
  - $ r_{1.5} = 1.3\% $ (one and a half years)
- **Time to maturity**: 1.5 years
- **Coupon Payments**: The bond pays a coupon of 4.5% on a semi-annual basis, which means:
  \[
  \text{Coupon Payment} = \frac{4.5\% \times 1,000}{2} = 22.5 \text{ dollars every six months.}
  \]

### Step-by-Step Solution:
The bond will have the following cash flows:
1. **At 6 months**: $22.50 (first coupon payment)
2. **At 12 months**: $22.50 (second coupon payment)
3. **At 18 months**: $22.50 (third coupon payment) + $1,000 (face value)

Next, we discount each of these cash flows using the spot rates provided.

#### 1. Present value of the coupon at 6 months:
\[
PV_{0.5} = \frac{22.50}{\left(1 + \frac{1.0\%}{2}\right)^1} = \frac{22.50}{1.005} = 22.387 \, \text{(approximately)}
\]

#### 2. Present value of the coupon at 12 months:
\[
PV_{1.0} = \frac{22.50}{\left(1 + \frac{1.1\%}{2}\right)^2} = \frac{22.50}{1.011025} = 22.249 \, \text{(approximately)}
\]

#### 3. Present value of the coupon and face value at 18 months:
\[
PV_{1.5} = \frac{1,022.50}{\left(1 + \frac{1.3\%}{2}\right)^3} = \frac{1,022.50}{1.019604} = 1,002.18 \, \text{(approximately)}
\]

### Total Bond Price:
Now, summing up all the present values:
\[
\text{Bond Price} = PV_{0.5} + PV_{1.0} + PV_{1.5}
\]
\[
\text{Bond Price} = 22.387 + 22.249 + 1,002.18 = 1,046.816
\]

Thus, the price of the bond is approximately **$1,046.82**.

Would you like further details or have any questions?

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Here are 5 related questions to expand your understanding:
1. How would the price change if the coupon rate was lower or higher?
2. What happens to bond prices when spot rates increase?
3. Can you calculate the yield to maturity (YTM) based on this bond's price?
4. What is the relationship between bond prices and market interest rates?
5. How would the price be affected if the bond paid coupons quarterly instead of semiannually?

### Tip:
When spot rates increase, the price of a bond decreases, since future cash flows are discounted more heavily.

Assume the current Treasury yield curve shows that the spot rates for six months, one year and one and a half years are 1.0%, 1.1%, and 1.3%, all quoted as semiannually compounded APRs. What is the price of a $1,000 par 4.5% coupon bond maturing in one and a half years (the next coupon is exactly six months from now)?

Learn how to calculate the price of a $1,000 par value bond with a 4.5% coupon rate, maturing in 1.5 years. This detailed guide covers spot rates, discounting cash flows, and bond pricing theory. Suitable for undergraduate finance or economics students.

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