Final answer:
The price today of a two-year default-free security with a face value of $1,000 and a 3% coupon rate is $981.23, calculated using the present value of the coupon payments and the face value at the given zero-coupon yields.
Step-by-step explanation:
To calculate the price today of a two-year default-free security with a face value of $1,000 and an annual coupon rate of 3%, we use the present value formula for both the coupon payments and the face value. The annual coupon payment is $30 (3% of $1,000). Given the zero-coupon yields of 4.0% for the first year and 4.0% for the second year, the present value (PV) of the coupons and the face value are calculated as follows:
- PV of first coupon = $30 / (1 + 0.04) = $28.85
- PV of second coupon and face value = ($30 + $1,000) / (1 + 0.04)^2 = $952.38
The sum of these values gives us the price of the security today, which is:
Price today = $28.85 + $952.38 = $981.23