.Provide solutions to the following questions.

RST Plc is a stock exchange listed manufacturing company. Following a decision made by the board to expand its manufacturing facilities, the company needs to raise additional finance amounting to $120 million by way of a five year corporate bond issue. The proposed corporate bond will have a par value of $100 and will be redeemed at this par value at the end of year five. Existing funds were raised by way of a ten year bond which matures in three years' time. The existing bond has a coupon rate of 4.75% and a nominal value of $100. The bond will be redeemed at this nominal value three years' time from now and coupon payments are made annually. There will be sufficient funds to redeem the existing bond. The issue will significantly change the company's capital structure and as such, the credit rating will fall from the current AAA to A. the company's treasurer has been advised that this is still within the investment grade. Five government bonds are in issue, bond 1, bond 2, bond 3, bond 4 and bond 5. Each bond has a par value of $100 and is redeemable at the par value upon maturity. Coupon payments on each bond are made annually.

The following additional information is available in respect of each bond:

Bond Maturity term Annual coupon rate Price

Bond 1 1 year 3.25% $99.90

Bond 2 2 years 3.75% $98.75

Bond 3 3 years 3.85% $97.80

Bond 4 4 years 4.15% $96.50

Bond 5 5 years 4.20% $96.10

In addition, the following table showing the credit spreads applicable to the sector in which RST Plc operates has been obtained from a credit rating agency:

Credit spreads in basis points

Credit Rating 1 year 2 years 3 years 4 years 5 years

AAA 20 30 40 50 60

AA 45 55 64 76 82

A 52 62 73 85 96

The following proposals have been made in respect of the proposed bond issue:

Proposal A

Issue the proposed corporate bond with a fixed annual coupon rate of 6%, with the first coupon payment being made at the end of year 1.

Proposal B

Issue the proposed corporate bond with an annual fixed coupon rate of 4% from year 1 to year 3 and a fixed annual coupon rate of 7% from year 4 to year 5.

Proposal C

Issue the proposed corporate bond at an annual fixed coupon rate but such that the issue price will be equal to the bond's par value of $100.

Proposal D

Issue the proposed corporate bond with a variable annual coupon rate based on the Bank Base rate so that the annual coupons will be Bank Base Rate + 40 basis points.

Required:

(a) Calculate the percentage decrease in the market value of the existing bond arising from the decrease in credit rating from AAA to A.

(b) Calculate whether the proposed bond would be issued at a discount or at a premium if the terms of issue were:

(i) Those in proposal A

(ii) Those in proposal B

(c) Calculate what the fixed annual coupon rate would be if the proposed bond was issued based on the terms of proposal C.

(d) Explain why a company may consider issuing a bond based on the terms stated under proposal B.

(e) Discuss the problems that are likely to be faced by the company if the proposed bond was issued based on the terms of proposal D.

8. Find a limit point for the set {-1 : ne ( ). Bolzano 9. Let A = {-1 : ne N}. a) Find A in R. b) Explain why A is a compact subset of H. 10. Let X be a compact topological space and let 7 - {F1, F2, F3, ...} be a class of closed sets such that (1 7 = 0. Prove that there is a number N such that Fin FinFin ... FN = 0. [Hint: De Morgan] 11. Prove The Maximum Theorem: Let f: [a, b] > IR be continuous, then f(x) obtains a maximum value for some x e [a, b]. [Hint: There is some [c, d] containing f([a, b]). ] 12. Let K be a compact subset of a Hausdorff space X and let pe K'. Prove that there exists disjoint open sets G and H of X such that peG and K C H. [Hint: See the proof of Theorem 2.] 13. Let A and B be disjoint compact sets in a Hausdorff space X. Prove that there are disjoint open subsets G and H of X such that A c G and B C H. [Hint: Use Exercise 12.]Problem 3: (a) Find s(2), defined and rapidly decaying on R, such that Sf(x) = [ s(x - y) f(y) dy = (s* f) (x), (5) the convolution of s with f, is the unique solution of the boundary value problem: (-02 + 1) u(2) = f(x), lim u(x) = 0. (6) r=100 (b) Prove that S is a bounded linear operator on L?(R). (c) Prove that S : L'(R) - L'(R) is not a compact operator. Hint: Review the definition of a compact operator. Let f E L'(R) be a fixed function and consider the sequence {fn}21, where f(x) = f(x -n). (d) Determine the resolvent set and spectrum of S. Hint: Let Fly]($) = foe ing(x)dx be the Fourier transform of f on R, and recall that F : L' (R) - L"(R) is defined as a bounded linear operator. Use the Fourier transform to explore the solvability of Su = Au + F for Fe L'(R)