Consider the following cash flow stream:

Year $123456$

CF $250200140180240150$

a$)$ Given the current interest rate of $12\%,$ calculate the duration for this cash flow stream. How do you interpret the concept "duration"? How is it related to another concept "immunization"?

b$)$ Suppose that you own this stream of cash flows as an investment in fixed$-$income securities$.$ How might you hedge this investment against interest rate risk? Illustrate that your hedging strategy does work by calculating the value of your investment at a particular future time point for a few interest rate scenarios, such as the initial interest rate of $12\%,$ the interest rate rises to $14\%$ or falls to $9\%.$ Does your investment have more or less similar value?