All interest rates are annual interest rates with semi-annual compounding. All coupon rates are annual rates paid semi-annually. All bonds have a $100 face value. Keep at least 6 decimal digits in all your calculation and answers unless specified otherwise.

Problem 3 (9 points): The price of a risky 10-year zero-coupon bond depends on two parameters: risk-free 10-year spot rate (10) (to simplify notations, denote (10) by r), and the risk factor s and it is given by (,)=$1,000,000(1+0.5+2)20. Today r=0.05 and s=0.12, so, (,)=$1,000,000(1+0.50.05+20.050.12)20=$483,531.61. You want to estimate the change in the bond price using first-order Taylor series approximation, i.e., write the change in price P as a linear function of the change in interest rate r and risk factor s as P=A*r+B*s, where A and B are some constants. a) (5 points) Find A and B. Round to the nearest integer number. b) (2 points) Using the original price function, (,)=$1,000,000(1+0.5+2)20, if the risk factor increases by 0.01, by how much the interest rate should change to keep the price unchanged? c) (2 points) Using the Taylor series approximation, if the risk factor increases by 0.01, how much the interest rate should change to keep the price unchanged?

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