Q7 thxSince may need reference to q6, q6 is also included

6. a. Write a function to approximate the change of an arbitrary function f() with respect to a small change in r, say ox, via its Taylor's series expansion: For any infinitely differentiable function f (.), one can write f(x+Ax) = f(I)+ dx )- dj (2) 1d2f(x) + 2 dx2 2 ( 4x) 24 . .. 4 1 d" f ( I . ) n! dan (x) "+ 1 dntif() (n+1)! dan+1 (Ax)"+1 where & E (x, r + Ax). Comment on the differences between the true and the approximation values as a function of the number of terms included in the expansion. b. Using Taylor's series expansion, or otherwise, justify the following approximation: P(y) - P(y + dy) ~ -D(by), and P(y + by) ~ P(y)(1 - D*(by) } where D and D* denote the dollar duration and the modified duration for a bond P(y) with yield y. You should also argue why the bond price can be infinitely differentiable with respect to y if you make use of the result in (a) to support your claim. Illustrate the approximation via a figure with R and/or Python. 7. We consider the following zero-coupon curve: Maturity (years) 2 3 4 S Zero-coupon rate (%) 4.00 4.25 4.75 4.90 5.00 a. What is the yield to maturity (YTM) of a 5 year bond with 6% coupon rate and annual coupon payment? You are required to solve this problem using R and/or Python via Newton-Raphson/bisection method. b. Suppose that the zero-coupon curve increases uniformly by o.25% immediately after the bond specified in (a) is purchased. What should be the new price and the new YTM of this bond? Using the results obtained in Q6 or otherwise, calculate also the unrealised profit/loss due to this sudden change in the zero-coupon curve