5. Consider three zero-coupon bonds with 2, 10, and 30 years to maturity and with required yields 4%, 7%, and 9%, respectively.

a. Calculate the price and modified duration of the three bonds using annual compounding.

b. How can a trader use convexity to set up a profitable trade in case she expects the yield curve to move in a parallel way? (Hint: assume a short position of 100 10 year bonds and then solve the system of two linear equations in two unknowns)

c. Now assume that the yield curve does not shift in a parallel way but instead flattens for the shorter maturities. More precisely assume y2 increases 1%, y10 decreases 1% and y30 remains unchanged. What are the trade positions values and the trade profit in this case?