time2

问题如下：

An analyst on the fixed-income desk of an investment bank is calculating the price of a 3-year zero-coupon bond with a face value of JPY 1,000,000. The analyst determines that investors currently demand an annual risk premium of 25 bps when they purchase the bond as compensation for interest rate volatility risk. The 1-year spot interest rate tree over the 3-year period is given below:

What is the price of the 3-year zero-coupon bond?

选项：

A.

JPY 935,500

B.

JPY 937,800

C.

JPY 940,100

D.

JPY 942,400

解释：

A is correct. The risk premium of 25 bps should be added to each year of duration risk of the zero-coupon bond. Since in this question the interest rate only changes once per year there are two years of duration risk in a 3-year zero-coupon bond (The return over 1 year is known as being 2.00% and is not uncertain). This will adjust the interest rate tree upward at each future node as shown below:

So, using a terminal price of 1 at time 3 and discounting using the adjusted interest rate tree, the following price tree can be derived:

The prices at each node are calculated the same way, for example the price at the upper node is found in the following manner: (1 * 0.5 + 1 * 0.5) / (1 + 0.032) = 0.9690, with the prices at all other nodes calculated using the same method. This results in the price at time 0 being (0.9477 * 0.5 + 0.9607 * 0.5) / (1+ 0.02) = 0.9355.

So, the price of a zero-coupon bond with a face value of JPY 1,000,000 should be JPY 935,500.

B is incorrect. This answer choice adds a risk premium only once at the nodes in time 2, rather than the necessary 2 times. This results in a price at Time 0 or 0.9378.

C is incorrect. This answer choice does not add a risk premium to time 1 or time 2 but only adds a risk premium to the final discounting period, where it should not be applied. This results in a price at Time 0 of 0.9401.

D is incorrect. This answer choice forgets that the risk premium needs to be added to the interest rate tree given in the question stem. This results in a price at Time 0 of 0.9424.