Question

You will be paying off a mortgage of $250,000 over the next 25 years. You have signed a loan agreement with Me-Bank to secure a fixed rate of 5.00%. The mortgage loan is compounded semi-annually.

a) What are the monthly payments?

b) How much will your payments be over the first five years?

c) What is the amount of principal that you pay off with the first payment?

d) How much principal remains to be paid after the first five years?

e) How much will remain to be paid off after you have made your payment at the end of the 15th year?

Redo a) to e) if the mortgage loan is compounded annually.

Answer 1

Step 1

A loan amortization table is a schedule of payments to pay back the loan in installments. It includes the bifurcation of installments, interest paid, principal paid in each installment over the life of the loan.

A loan amortization table is made on the basic concepts of

Interets = Remaining balance ×interest rateRemaining balance = Principal amount - principal paid in installmentPrincipal paid in the installment =Installment - interets paid in the installment.

Step 2

To answer the questions above we need to calculate a loan amortization table as the interest rate is compounded semi-annually we need to calculate the effective rate for a year.

The formula for the effective rate : [1+nominal ratenumber of compounding periods]componding periods−11+nominal ratenumber of compounding periodscomponding periods-1

According to the question,

Nominal rate: 5.00%

Number of compounding periods: 2

By putting them in the formula

Effective rate =[1+0.052]2−1Effcetive rate =[1.025]2−1 Effcetive rate =5.0625%Effective rate =1+0.0522-1Effcetive rate =[1.025]2-1 Effcetive rate =5.0625%

Installment of the loan is calculated from the formula

Installment amount =P×r×(1+r)n(1+r)n−1where,P =Principal amountr = rate of interestn =number of periodsInstallment amount =P×r×(1+r)n(1+r)n-1where,P =Principal amountr = rate of interestn =number of periods

In the question,

P = $250,000

r = 5.0625%

n = 25

Installment amount =$250,000×0.50625×(1+0.050625)25(1+0.050625)25−1$250,000×0.50625×(1+0.050625)25(1+0.050625)25-1

Installment amount=$250,000×0.50625×3.4372.437Installment amount =17,849.39Installment amount=$250,000×0.50625×3.4372.437Installment amount =17,849.39

Step 3 Loan amortization table

YEAR	BALANCE	INTEREST	INSTALLMENT	PRINCIPAL PAID	NEW BALANCE
0	$2,50,000	0	0	0	$2,50,000
1	$2,50,000	$12,656.25	$17,849.39	$5,193.14	$2,44,806.86
2	$2,44,807	$12,393.35	$17,849.39	$5,456.04	$2,39,350.82
3	$2,39,351	$12,117.14	$17,849.39	$5,732.25	$2,33,618.56
4	$2,33,619	$11,826.94	$17,849.39	$6,022.45	$2,27,596.11
5	$2,27,596	$11,522.05	$17,849.39	$6,327.34	$2,21,268.78
6	$2,21,269	$11,201.73	$17,849.39	$6,647.66	$2,14,621.12
7	$2,14,621	$10,865.19	$17,849.39	$6,984.20	$2,07,636.92
8	$2,07,637	$10,511.62	$17,849.39	$7,337.77	$2,00,299.15
9	$2,00,299	$10,140.14	$17,849.39	$7,709.25	$1,92,589.90
10	$1,92,590	$9,749.86	$17,849.39	$8,099.53	$1,84,490.38
11	$1,84,490	$9,339.83	$17,849.39	$8,509.56	$1,75,980.81
12	$1,75,981	$8,909.03	$17,849.39	$8,940.36	$1,67,040.45
13	$1,67,040	$8,456.42	$17,849.39	$9,392.97	$1,57,647.49
14	$1,57,647	$7,980.90	$17,849.39	$9,868.49	$1,47,779.00
15	$1,47,779	$7,481.31	$17,849.39	$10,368.08	$1,37,410.92
16	$1,37,411	$6,956.43	$17,849.39	$10,892.96	$1,26,517.96
17	$1,26,518	$6,404.97	$17,849.39	$11,444.42	$1,15,073.54
18	$1,15,074	$5,825.60	$17,849.39	$12,023.79	$1,03,049.75
19	$1,03,050	$5,216.89	$17,849.39	$12,632.50	$90,417.25
20	$90,417	$4,577.37	$17,849.39	$13,272.02	$77,145.24
21	$77,145	$3,905.48	$17,849.39	$13,943.91	$63,201.32
22	$63,201	$3,199.57	$17,849.39	$14,649.82	$48,551.50
23	$48,552	$2,457.92	$17,849.39	$15,391.47	$33,160.03
24	$33,160	$1,678.73	$17,849.39	$16,170.66	$16,989.37
25	$16,989	$860.09	$17,849.39	$16,989.30	$0.06

Step 4

(a) Monthly payments will be $17,849.39

(b) Total amount paid over the first five years will be $89,246.95

(c) The amount of principal that you pay off with the first payment $5,193.14

(d) the principal remains to be paid after the first five years will be [Total principal amount - principal paid till five years]

i.e [$250,000-$5,193.14] = $244,806.86

(e) Amount remaining to be paid off after making payment at the end of the 15th year will be [Total of installment amount payable till 25th year - total installment amount paid till 15th year] i.e. [$446,234.75-$267,740.85] =$178,493.9