A mortgage payment stream is usually divided into a principal payment cash- flow

Question

A mortgage payment stream is usually divided into a principal payment cash- flow and an interest payment cash-flow. Consider a standard mortgage of initial value M(0) with equal monthly payments of amount B. The interest rate used is r per month. We shall denote by M(k) the mortgage principal or balance after the kth payment, by I(k) the interest component of the kth payment and by P(k) the principal component of the kth payment.

C) Find the present value V (at the rate r) of the principal payment cash-flow m in terms of B, r, n, M(0) and in terms of r, n, M(0) only. Other parts of this question have been answered but I just need a complete answer to C and how it is worked out

Explanation / Answer

1. present value V (at the rate r) of the principal payment cash-flow m in terms of B, r, n, M(0):

V = [n(B - rM(0)]/1+r

Explanation: n*B = total payments (principal+interest), r*M0 = amount of total interest of the mortgage

Hence n(B - rM(0) = principal payments towards the mortgage.

Present value is obtained by dividing the principal payments with the present value discount factor i.e 1+r

2. In terms of r, n, M(0) only:

For this we will substitute for B. The equation thus obtained is:

V = rnM(0)/[(1+r)*(1+r)^n - 1]

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