Assume the current yield curve shows that the spot rates for six​ months, one​ year, and one and a half years are 1​%, 1.1​% and 1.3​%, ​respectively, all quoted as​ semi-annually compounded APRs. What is the price of a ​$1000 ​par, 4.75​% coupon bond maturing in one and a half years​ (the next coupon is exactly six months from​ now)?

To calculate the price of the bond, we need to discount the future cash flows to the present using the spot rates. The bond will pay coupons semi-annually and will return the principal at the end of one and a half years.

Here are the steps:

1. Calculate the semi-annual coupon payment: The bond has a 4.75% annual coupon rate, so the semi-annual coupon payment is 4.75%/2 = 2.375% of the par value. So, the semi-annual coupon payment is $1000 * 2.375$23.75.

2. Discount the first coupon payment: The first coupon payment is six months from now, so we use the six-month spot rate to discount it. The present value of the first coupon payment is $23.75 / (1 + 1$23.51.

3. Discount the second coupon payment: The second coupon payment is one year from now, so we use the one-year spot rate to discount it. The present value of the second coupon payment is $23.75 / (1 + 1.1$23.26.

4. Discount the third cash flow: The third cash flow is the final coupon payment plus the return of the principal, which is one and a half years from now. We use the one and a half year spot rate to discount it. The present value of the third cash flow is ($23.75 +$1000) / (1 + 1.3%/2)^3 = $1001.37.

5. Add up the present values: The price of the bond is the sum of the present values of the future cash flows. So, the price of the bond is $23.51 +$23.26 + $1001.37 =$1048.14.

So, the price of the $1000 par, 4.75$1048.14.