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 $k$ th payment satisfies

\[M(k)=(1+r) M(k-1)-B\]

for $k=0,1, \ldots$ This equation has the solution

\[M(k)=(1+r)^{k} M-\left[\frac{(1+r)^{k}-1}{r}\right] B \text {. }\]

Let us suppose that the mortgage has $n$ periods and $B$ is chosen so that $M(n)=0$; namely,

\[B=\frac{r(1+r)^{n} M}{(1+r)^{n}-1}\]

The $k$ th payment has an interest component of

\[I(k)=r M(k-1)\]

and a principal component of

\[P(k)=B-r M(k-1) .\]

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

(b) Find $V$ in terms of $r, n, M$ only.

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

(d) What is the value of $V$ as $n \rightarrow \infty$ ?

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