1. Consider a multi-factor key rate model with key rates being YTM on 2, 5, 10, and 30-year par ponds. In lectures, we have shown that the price p(y) of a T-year par bond, where y is the bonds YTM, satisfies . Assume today the YTM for 2, 5, 10, and 30 year par bonds are 3%, 5%, 8%, and 12% respectively. Assume also that all bonds have a $100 face value. All interest rates are annual interest rates with semi-annual compounding. All coupon rates are annual rates paid semi-annually.

1. Find KR012, KR015, KR0110, and KR0130 for 2, 5, 8, 10, 15, and 30-year par bonds. Please, show your calculations and enter your answers in the table below. Keep at least 4 decimal digits.

	Maturity of a par bond
	2 years	5 years	8 years	10 years	15 years	30 years
KR01
KR01
KR01
KR01

2. You have a portfolio of 1000 of 15-year par bonds which you want to hedge using 2, 5, 10, and 30-year par bonds. How many of each of these bonds you should buy or sell to construct the best hedge? Round your answer to the nearest integer number of bonds
3. Assume you still have a portfolio of 1000 of 15-year par bonds but 10-year bonds are not available now. You still want to hedge your position using 2, 5, 8, and 30-year par bonds. Can you hedge all of your key rate risks? If yes, find how many bonds of each type you need to buy or sell (round your answer to the nearest integer). If no, explain why.