(a) Suppose a, b, n are integer numbers and n >0.Prove that (a+b)%n= (a%n+b%n)%n.

(b) Leta, b, n be integer numbers and n >0. Suppose a%n≥b%n. Prove that a%n−b%n= (a−b)%n.

(c) Consider the following claim: Claim 1. Suppose a, b, n are integer numbers and n >0.Then a%n−b%n= (a−b)%n. Show that this claim is not correct. (Hint: To prove that this claim is not correct you need to find an example of integer values a, b, n, where n >0,but a%n−b%n ≠ (a−b)%n.)

(d) Let a, b, n be integer numbers and n >0.Suppose a%n=b%n. Prove that a%n−b%n = (a−b)%n= (b−a)%n.

2. Suppose that Alex attempts to execute a bond market-neutral trading strategy. He establishes a long position in an 8% annual coupon payment 6-year IBM bond and a short position in a 5-year (Treasury) Zero. Suppose that the IBM bond sells at par, whereas the Zero sells at a YTM of 7%.

a. If Alex wants to purchase 100 IBM bonds, how many Zeros must he sell short to finance this long position? (Hint: the long position and the short position should have the same value)

b. Quantify the interest rate risk in Alex's long-short portfolio. Specifically, using bond durations, estimate how much his overall portfolio would change in value if all rates go up by 1%. (Hint: use modified durations to estimate the value changes of both long and short positions, since the YTM are different for these two bonds)

c. What would the new interest rate risk be if the YTMs of both bonds go up by 5%?