Question

A couple must decide between a blue house and a red house. To purchase the red house they must tae a $155000 mortgage to be repaid over 20 years, interest rate is given as j2 = 7.8%. To purchase the blue house, they must take a mortgage to be repaid over 25 years at interest rate given as j12 - 8%. Both of those two mortgages have semi-annual payments and the semi annual mortgage payments are equal.

(a) Determine the amount of each semi-annual payment if they wish to purchase the red house?

(b) Determine the value of the mortgage on the blue house?

(c) If this couple wants to purchase the red house, and instead of a mortgage, they intend to pay $X, $2X, $3X, $4X at year 3, year 6, year 9, and year 12 to totally cover those $155000, determine X?

Answer 1

(a)

Monthly payment	=	[P × R × (1+R)^N ] / [(1+R)^N -1]
Using the formula:
Loan amount	P	$                                                          155,000
Rate of interest per period:
Annual rate of interest		7.800%
Frequency of payment	=	Once in 6 month period
Numer of payments in a year	=	12/6 =	2
Rate of interest per period	R	0.078 /2 =	3.9000%
Total number of payments:
Frequency of payment	=	Once in 6 month period
Number of years of loan repayment	=	20
Total number of payments	N	20 × 2 =	40
Period payment using the formula	=	[ 155000 × 0.039 × (1+0.039)^40] / [(1+0.039 ^40 -1]
Monthly payment	=	$                                                         7,714.99

Semi-annual payment is $7,714.99

(b)

Value of mortgage is present value of semi-annual payment sof $7,714.99 for 25 years. Value is:

a	Present value of annuity=	P* [ [1- (1+r)-n ]/r ]
P=	Periodic payment	7,714.99
r=	Rate of interest per period
	Annual interest	8.00%
	Number of payments per year	2
	Interest rate per period	0.08/2=
	Interest rate per period	4.000%
n=	number of periods:
	Number of years	25
	Periods per year	2
	number of payments	50
	Present value of annuity=	7714.99* [ (1- (1+0.04)^-50)/0.04 ]
	Present value of annuity=	165,734.84

Amount of mortgage is $165,734.84

(c)

Total of present value factors at 7.8% (compounded semi-annually) for the given years is:

Year	Factor	× times X	Product
3	0.637217	1	0.637217
6	0.406046	2	0.812091
9	0.258739	3	0.776218
12	0.164873	4	0.659492
Total			2.885018

Amount of payment = $1550,000/ 2.885018 = $53,725.83

Answer is X = $53,725.83

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