A couple purchased a house and signed a mortgage contract for $110,000 to be paid in “every second week “ installments over 25 years, at 3.0 %. The contract stipulates that after 3 years the mortgage will be renegotiated at the new prevailing rate of interest. Calculate:
a) The every second week payment for the initial 3-year period
b) The outstanding principal after 3 years.
c) The new payment ( now, once a month )after the 3 years, at 4.5 %
NOTE: mortgages rates in Canada are always compounded twice a year.
first we must find the rate for the "every second week" period.
let that rate be i
(1+i)^26 = 1.015 , (semi-annual rate is 1.5%)
1+i = 1.015^(1/26) = 1.000572803
i = .000572803 ( I stored the full decimal in memory of calculator)
then if P is the payment
110000 = P[ 1 - 1.000572803^-650 ]/.000572803
P = 202.73
b) This continues for 3 years, or 78 payments
Balance after 3 years
= 110000(1.000572803)^78 - 202.73[ 1.000572803^78 - 1]/.000572803
= 115024.61 - 16167.01
= 98857.60
c) Now we have to find the equivalent monthly rate for 4.5%
let it be j
(1+j)^6 = 1.0225
j = .00371532
new montly payment M , 22 years left or 264 payments
98857.60 = M [1 - 1.00371532^-264]/.00371532
M = 588.30 per month
I suggest you check my "arithmetic" on this one.
Totally missed the part that our rates are assumed to be compounded semi-annually !!
So we first have to find the equilavalent semi-annual rate equal to 3% compounded annually.
let that rate be j
(1+j)^2 = 1.03
1+j = √1.03
j = .014889157 (I stored that in calculator memory)
Now we must find the rate for the "every second week" period.
let that rate be i
(1+i)^26 = 1.014889157 , (semi-annual rate is 1.014889157 %)
same steps as above ...
i = .0005686
then if P is the payment
110000 = P[ 1 - 1.0005686^-650 ]/.0005686
P = 202.47
b) This continues for 3 years, or 78 payments
Balance after 3 years
= 110000(1.0005686)^78 - 202.73[ 1.0005686^78 - 1]/.0005686
= 114986.94 - 16143.68
= 98843.26 ---> balance after 3 years (78 payments)
c) Equivalent rate compounded semi-annual ...
let that rate be k
(1+k)^2 = 1.045
k = .0022252415
Now we have to find the equivalent monthly rate for the above rate
let it be j
(1+j)^6 = 1.0022252415
j = .003674809
new montly payment M , 22 years left or 264 payments
98857.60 = M [1 - 1.003674809^-264]/.003674809
M = 585.57 per month
This is the problem when "cutting and pasting"
It is so easy to miss a change that should be made
2nd last line should say
98843.26 = M [1 - 1.003674809^-264]/.003674809
To calculate the mortgage payments, outstanding principal, and new payment, we can use the formula for calculating mortgage payments. The formula is:
P = (r * PV) / (1 - (1 + r)^(-n))
Where:
P is the payment per period,
r is the interest rate per period,
PV is the present value (loan amount), and
n is the total number of periods.
Let's proceed with calculating each part of the problem:
a) The every second week payment for the initial 3-year period:
Since the mortgage payments are made every second week, we need to adjust the interest rate per period and the number of periods.
Interest rate per period:
The annual interest rate is 3.0%. To find the rate per period, we divide it by the number of periods in a year (which is 52 weeks, considering biweekly payments). Thus, the rate per period would be (3.0% / 52) = 0.0577.
Number of periods:
Since the mortgage is paid over 25 years and payments are made every second week, the total number of periods would be (25 * 52) = 1,300.
Using these values in the formula, we can calculate the every second week payment for the initial 3-year period.
P = (0.0577 * 110,000) / (1 - (1 + 0.0577)^(-1,300))
Using a calculator, evaluate the right-hand side of the equation to find P, which will give you the every second week payment for the initial 3-year period.
b) The outstanding principal after 3 years:
To find the outstanding principal after 3 years, we need to calculate the remaining balance on the loan. It can be calculated by considering the mortgage payments made during the initial 3 years.
First, calculate the number of every second week payments made in 3 years:
Number of payments per year = 52 weeks.
Number of payments in 3 years = (52 payments/year * 3 years) = 156 payments.
Using the number of payments made, we can calculate the remaining balance on the loan after 3 years.
Remaining balance = (P * (1 - (1 + r)^-n)) / r
where,
P is the every second week mortgage payment calculated in part (a),
r is the interest rate per period, and
n is the number of payments made (156 payments).
Substitute the relevant values into the formula and calculate the remaining balance to find the outstanding principal after 3 years.
c) The new payment (now, once a month) after the 3 years, at 4.5%:
After 3 years, the mortgage will be renegotiated at the new prevailing interest rate of 4.5%. Since the new payment will be made once a month, the interest rate per period and the number of periods need to be adjusted accordingly.
Interest rate per period:
The annual interest rate is 4.5%. To find the rate per period, we divide it by the number of periods in a year (which is 12 months). Thus, the rate per period would be (4.5% / 12) = 0.375.
Number of periods:
The remaining mortgage term is 25 years - 3 years = 22 years.
Since payments will be made monthly, the total number of periods would be (22 * 12) = 264.
Using these values and the remaining balance from part (b), we can calculate the new payment using the mortgage payment formula.
P = (0.375 * remaining balance) / (1 - (1 + 0.375)^(-264))
Plug in the values and calculate P to find the new monthly payment after the 3 years, at 4.5%.