Consider a multi-factor key rate model with key rates being YTM on 2, 5, 10, and 30-year par ponds. Assume also that currently YTM on 2, 5, 10, and 30-year bonds are 4%, 5%, 6%, and 7% respectively. Assume all bonds have a $100 face value. Keep at least 6 decimal digits.

a) (2 points) Find KR012 for 2-year par bond; KR015 for 5-year par bond; KR0110 for 10-year par bond; and KR0130 for 30-year par bond

b) (1 point) Find all key rates (KR012, KR015, KR0110, and KR0130) for a 15-year par bond if YTM for this bond is 6.25%

c) (1 point) Find all key rates (KR012, KR015, KR0110, and KR0130) for a 20-year par bond if YTM for this bond is 6.5%

d) (1 point) Find all key rates (KR012, KR015, KR0110, and KR0130) for an 8-year par bond if YTM for this bond is 5.6%

e) (1 point) Find all key rates (KR012, KR015, KR0110, and KR0130) for a 6-year par bond if YTM for this bond is 5.2%

f) (6 points) You would like to hedge your portfolio that consists of a thousand 6-year par bonds using 2, 8, 15, and 20 bonds. How would you do it? Please, specify the number of each bonds (number of bonds for every type of bonds) you should buy or sell (note you may end up using only some of these bonds). Use the KR01 rates you found earlier and make sure to write the system of equations you want to solve before trying to solve it.

All interest rates are annual interest rates with semi-annual compounding. All coupon rates are annual rates paid semi-annually.