Suppose you borrow M = $100, 000 from a bank as a fixed-rate mortgage with a fiv

Question

Suppose you borrow M = $100, 000 from a bank as a fixed-rate mortgage with a five year term (N = 60 monthly payments) and monthly interest rate r = 1%. The minimum value for the monthly payments m is decided such that the present value of the monthly payments is equal to the principal loan.

(a) Assuming continuous compounding and continuous payment, write an expression for the present value of all the payments.

(b) Use the expression in part (a) to compute the minimum monthly payment mc (use e 0.6 0.55 in your computation).

(c) Assuming discrete compounding and discrete payment, write an expression for the present value of all the payments (use summation notation).

(d) It is known that ( N k=1 ) (1/q)^k = (1 /(q 1)) ( 1 q ^(N) ) . Apply this to the expression in part (c) and compute the payment amount md (use 1.01^(60) .55 in your calculation).

Explanation / Answer

a) Present Value of an Annuity with Continuous Compounding

PV=PMT / (er 1) * [11/e rt  ]

b) Using e-0.6 = 0.55,

100,000 = PMT / (e0.01 - 1) * [1-e(-0.01*60)]

100,000 = PMT / (e0.01 - 1) * [1-0.55]

100,000 = PMT / (1.01005 - 1) * 0.45

PMT = 100,000 * 0.01005 / 0.45

= $ 2233.33

Monthly payment = $ 2,233.33 for 60 months

c) Present Value of an Annuity with discrete Compounding

PV=PMT / i * [ 11/(1+i) n ]

d) using 1.01-60 = 0.55,

100,000 = PMT / 0.01 * [ 1 - 1/(1+0.01) 60]

100,000 = PMT / 0.01 * [ 1 - 0.55]
PMT = 100,000 * 0.01 / 0.45 = $ 2,222

Monthly payment = $ 2,222 for 60 months

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