
Assume that the​ six-month Treasury spot rate is 1.6 % ​APR, and the​ one-year rate is 2.0 % ​APR, both compounded semiannually. What is the price of a​ one-year ​$1000 par Treasury bond with 2.0 % ​coupons?

To find the price of a one-year $1000 par Treasury bond with 2.0% coupons, given that the 6-month Treasury spot rate is 1.6% APR and the one-year rate is 2.0% APR (both compounded semiannually), we can follow these steps:

### Step-by-Step Breakdown:

1. **Coupon payments and par value:**
   - The bond has a coupon rate of 2.0% per annum, but coupons are paid semiannually.
   - Therefore, each coupon payment is $ \frac{2.0\%}{2} = 1.0\% $ of the par value ($1000).
   - The bond makes two coupon payments of $ 1000 \times 1.0\% = 10 $ dollars.
   - The bond also pays back the face value of $1000 at maturity (one year).

2. **Discounting the cash flows:**
   The price of the bond is the present value (PV) of its cash flows, discounted using the semiannual rates.

   - First coupon is paid in 6 months.
   - Second coupon + par value ($1000) is paid in one year.

3. **Semiannual discount rates:**
   Since the rates are given as APR compounded semiannually, we first convert them to semiannual rates.
   - 6-month rate: $ \frac{1.6\%}{2} = 0.8\% $ semiannual rate.
   - 1-year rate: $ \frac{2.0\%}{2} = 1.0\% $ semiannual rate.

4. **Present Value of the First Coupon Payment:**
   - The first $10 coupon is paid in 6 months, so we discount it using the 6-month semiannual rate of 0.8%.
   \[
   PV_1 = \frac{10}{(1 + 0.008)} = \frac{10}{1.008} \approx 9.92
   \]

5. **Present Value of the Second Coupon and Par Value:**
   - The second $10 coupon and the par value ($1000) are paid in one year, so we discount them using the one-year semiannual rate of 1.0%.
   \[
   PV_2 = \frac{1010}{(1 + 0.01)^2} = \frac{1010}{1.0201} \approx 989.22
   \]

6. **Total Price of the Bond:**
   - The total price is the sum of the present values of both coupon payments and the par value.
   \[
   \text{Price} = PV_1 + PV_2 = 9.92 + 989.22 = 999.14
   \]

### Final Answer:
The price of the one-year $1000 par Treasury bond is approximately **$999.14**.

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

### Follow-up Questions:
1. How do you calculate the present value of future cash flows in bond pricing?
2. What is the significance of using spot rates in bond pricing?
3. Can the bond price be higher or lower than the par value? Why?
4. How does compounding frequency affect bond pricing?
5. What is the relationship between bond prices and interest rates?

### Tip:
For bond pricing, always ensure you use the correct discount rates and adjust for the compounding frequency!

Assume that the​ six-month Treasury spot rate is 1.6% APR, and the​ one-year rate is 2.0% APR, both compounded semiannually. What is the price of a​ one-year ​$1000 par Treasury bond with 2.0% ​coupons?

Learn how to price a one-year $1000 par Treasury bond with a 2.0% coupon rate. The problem uses spot rates of 1.6% APR for six months and 2.0% APR for one year, compounded semiannually. This step-by-step guide walks you through calculating the present value of cash flows and determining the bond price.

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