In this question all interest rates are assumed to be continuously compounded. (a) Show that the instantaneous forward rates f* (T) are given in terms of the T-maturity risk-free rates, r(0, T), by f* (T) = r(0, T) + T partial differential r (0, T)/partial differential T (b) Given that the yield curve is r(0, T) = 5% + (1 - e^(-t/4) x 3%, find the instantaneous forward rates, f* (T). (c) If the yield curve is changed to r(0, r) = 5% + (1 - e^(-r/4) + sin(500T)/500 x 3%, find the corresponding instantaneous forward rates, f* (T). (d) Plot and compare the yield curves represented by r(0, T) and f(0, T) for O lessthanorequalto T lessthanorequalto 10. (e) Plot and compare the instantaneous forward rates f* (0, T) and f* (0, T) for 0 lessthanorequalto T lessthanorequalto 10. (f) Based on the results of (d) and (e), what can you say about the potential effects of small perturbations (or errors) in the yield curve on the (instantaneous) forward curve? Note also that in sin(t) in (c), t should be given in radians, not degrees.]