Derivatives- continuation.

For a Eurozone government bond yield curve, let P() be the price and y() the yield of a zero coupon bond of length , where r = 1, 2, 3 ... years. Further, let /() be the one- year forward rate from / - 1 to f, and g() be the yield of a par coupon bond of maturity / years. All yields are annually compounded. (Note: The par coupon yield is the annual coupon on a bond priced at par. You may assume in this question that we are not interested in intermediate points along the curve./ (i) Derive a formula for P() given y(). [1] (ii) Derive formulae for /() and g() in terms of P() for : = 1, 2, 3 ... . [2] (iii) Prove that /(1) = g(1) = y(1) and that the slope of /() is approximately twice that of y(f) at / = 1. [2] [Hint: Let Af -1(2) -A(1) and Av =)(2) -y(1)] Maturity (years) Spor yield 2.7% 3.2% U AWN- 3.5% 3.6% 3.6% (iv) (a) Using your formulae in (i) and (ii), calculate the values of the forward rates and par yields given the above table of spot yields, and hence verify numerically the result in (iii). (b) Sketch the three curves on a single graph. (c) Comment on the similarities between the par and zero curves. [7] (v) (a) Use your curve to value a EUR 100 million 5-year interest rate swap, paying 4% fixed rate annually and receiving floating rate annually. (b) Suggest why the true market value of such a swap is likely to be different