Bootstrapping and IRR discounting:

Suppose the current one-year euro swap rate y₀[0, 1] is 1.47%, and the two-year and three-year swap rates are 2.05% and 2.38% respectively. Euro swap rates are quoted with annual payments and 30/360 daycount (thus α = 1).

(a) Use bootstrapping to calculate Z (0, 1), Z (0, 2) and Z (0, 3), and obtain P₀[0, 3], the present value of a three-year annuity paying €1 per year.

(b) Recall that the (annually compounded) zero rate for maturity T is the rate r such that Z(0, T) = Calculate the one-year, two-year and three-year zero rates, and compare them to the swap rates. In general, for upward sloping yield curves, that is, when y₀[0, T₂] > y₀[0, T₁] for T₂ > T₁, will zero rates be higher or lower than swap rates?

(c) An approximate short-cut sometimes used on trading desks to calculate the present value of, say, a three-year annuity is to discount each payment at the three-year swap rate. In other words, it is assumed that Z (0, n) =    or n = 1, 2, 3. This is often called IRR (internal rate of return) discounting. Calculate the error in valuing the annuity in (a) this way.

(d) Calculate the price of a three-year fixed rate bond of notional e1 and annual coupons of 2.38% using the ZCB prices calculated in (a), and verify this equals the price obtained via IRR discounting at a rate of 2.38%.

(e) The value of a swap with fixed rate K (see Question 1(a)) can be thought of as an annuity of amount y₀[0, T_n] - K for period 0 to Tn. In yet another Wall Street quirk, euro swaps embedded in certain contracts are occasionally valued for cash settlement using IRR discounting at the current swap rate y₀[0, T_n], rather than correct valuation using ZCBs. The logic for this originally was to reduce disagreements between banks on cash settlement of swaps. The swap rate is easily observed and IRR discounting is then a deterministic formula, whilst the bootstrapping undertaken in (a) was deemed too complicated. Suppose all euro swaps suddenly moved to this type of valuation. Given your answers to Question 2(a) to (c), comment briefly on whether you would expect euro swap rates to rise, fall or stay the same.