#5 Non-Standard Interest Rate Swaps - Roller Coaster Interest Rate Swaps At the current date to, assume the existence of a term structure of simple and continuously compounded default- free spot interest rates, denoted by {R(to, t)}tto and {r(to, t)}>to respectively, with associated capitalisation and discount factors {C(to, t)}>to and {Z(to, t)}>to respectively. Consider a Roller Coaster Interest Rate Swap (RCIRS) with fixed simple swap rate K, initiated at date to. expiring at date tn, with reset dates {to, t1, t2,...,tn-1}, referencing floating rates {R(ti-1, ti); 1 i n}, varying nominals {N; 1 in} and with payment dates {t1, t2.t3.....tn}, i.e. a different nominal amount applies to each payment period. At each t {t1, t2, t3,..., t}, the following cash flows occur: floating cash flow: NR(ti-1, ti)Ti fixed cash flow: NKT, where Ti = (ti-ti-1)/365 for i = {1,2,..., n}. (i) Provide one real-world scenario where a RCIRS may be used, and preferred to a standard IRS. (ii) Derive an expression for the initial fair value of the RCIRS at date to. Use no-arbitrage arguments to justify the "replacement" of future floating rates with their forward counterparts, where necessary. (iii) Derive an expression for the fair RCIRS rate K within the primary market context. (iv) Derive a strategy to replicate the RCIRS with standard and forward-starting IRSs. The following NACC swap zero curve prevails in the South African market on to = 13-Jun-2014: 2 r(to,t) = -0.0001 -to` 365 + 0.0025 +0.055, 365 for t to. This curve gives you access to the complete set of discount factors at the current date to. (v) Compute the fair RCIRS rate for a roller coaster payer swap with semi-annual payments in arrears, over the next 2 years. The initial notional is R80,000, the second exchange of payments is based on a notional of R100,000 and the final two exchange of payments is based on a notional of R40,000. The floating leg references 6-month JIBAR (simple, ACT/365). Generate all reset and payment dates according to standard South African swap conventions.