14. (Mortgage division) Often a mortgage payment stream is divided into a principal payment stream and an interest payment stream, and the two streams are sold separately We shall examine the component values Consider a standard mortgage of initial value M = M(0) with equal periodic payments of amount B If the interest rate used is r per period, then the mortgage principal after the kth payment satisfies M(k) (1+r) M(k-1) - B for 0, 1, This equation has the solution M(k) (1+1) M. (1+ B r Let us suppose that the mortgage has a periods and B is chosen so that M (n) = 0; namely, B = 1(1+1)"M (1+r)-1 The kth payment has an interest component of and a principal component of 1(k) M(k-1) 1 P(k) B M (k-1). =

(a) Find the present value V (at rate ) of the principal payment stream in terms of B.1.n. M

(b) Find in terms of,, M only

(c) What is the present value W of the interest payment stream?

(d) What is the value of V as 00?

(e) Which stream do you think has the larger duration-principal or interest?