5. (Instantaneous rates o) Let s(1). 000, denote a spot rate curve; that is, the present value of a dollar to be received at time is ex For <2, let (11, 12) be the forward rate between 1 and 2 implied by the given spot rate curve

(a) Find an expression for f (h. 12). 10

(b) Let (1) lim, Show that (1) f(r.) We can call (1) the instantaneous interest rate at time f s(t) + s'(r)t

(c) Suppose an amount .vo is invested in a bank account at r=0 which pays the instan- taneous rate of interest (r) at all r (compounded). Then the bank balance x() will satisfy dx(r)/dr=r()x(t). Find an expression for x(1) [Hint. Recall in general that ydz +zdy d(yz). =