Question

A 50000$ mortgage is to be repaid by means of monthly payments, at the beginning of each month, for 20 years. If the nominal interest rate is 12% convertible monthly, (a) Find the monthly payment (b) Suppose now an extra payment of 1000$ is made at the end of each year. Determine the monthly payment (This problem needs to be solved using the concept of annuities)..

Answer 1

Solution a
	PV of annuity for making pthly payment
	P = PMT x (((1-(1 + r) ^- n)) / i)
	Where:
	P = the present value of an annuity stream	$     50,000
	PMT = the dollar amount of each annuity payment	PMT
	r = the effective interest rate (also known as the discount rate)	12.68%	(1+12%/12)^12)-1)
	i=nominal Interest rate	12%
	n = the number of periods in which payments will be made	20
	PV of annuity=	PMT x (((1-(1 + r) ^- n)) / i)
	50000=	PMT x (((1-(1 + 12.68%) ^-20)) / 12%)
	Annual payment=	50000/(((1-(1 + 12.68%) ^-20)) / 12%)
	Annual payment=	$ 6,606.81
	Monthly payment=	$     550.57

Solution b:

First we would compute the PV of annual payments
PV of annuity for making pthly payment
P = PMT x (((1-(1 + r) ^- n)) / r)
Where:
P = the present value of an annuity stream
PMT = the dollar amount of each annuity payment	$       1,000
r = the effective interest rate (also known as the discount rate)	12.68%	(1+12%/12)^12)-1)
n = the number of periods in which payments will be made	20
PV of annuity=	PMT x (((1-(1 + r) ^- n)) / r)
PV of annuity=	1000*(((1-(1+12.68%) ^-20)) /12.68%)
PV of annuity=	$ 7,161.00
Initial loan balance	$50,000.00
Remaining loan balance	$42,839.00	50000-7161
PV of annuity for making pthly payment
P = PMT x (((1-(1 + r) ^- n)) / i)
Where:
P = the present value of an annuity stream	$     42,839
PMT = the dollar amount of each annuity payment	PMT
r = the effective interest rate (also known as the discount rate)	12.68%	(1+12%/12)^12)-1)
i=nominal Interest rate	12%
n = the number of periods in which payments will be made	20
PV of annuity=	PMT x (((1-(1 + r) ^- n)) / i)
42839=	PMT x (((1-(1 + 12.68%) ^-20)) / 12%)
Annual payment=	42839/(((1-(1 + 12.68%) ^-20)) / 12%)
Annual payment=	$ 5,660.59
Monthly payment=	$     471.72