Question

1. Matt and Lindsay borrowed $220,000 at 4% interest using a fixed rate mortgage with a maturity of 20 years.  Answer the following questions about their loan.

1. What is the monthly payment necessary to amortize this loan?

2. If the loan required annual payments instead of monthly, what would the annual payment be?

3. Multiply the answer in part (a) by 12.  Why does this amount not equal the answer in part b?

Answer 1

Following are given in the question:

Mortgage borrowed = $220,000

Interest rate per annum = 4%

Maturity = 20 years or 240 months (20 years *12 months in a year)

a. Monthly payment necessary to amortise the loan

This is also called EMI (equated monthly instalments). This can be calculated by using the following formula:

Monthly payment = P * r * (1 + r)^n/(((1 + r)^n) - 1)

Where P = Principal amount (mortgage borrowed)

r = Interest rate of the loan (in months)

n = tenure (maturity) of the loan (in months)

Applying the values in the above formula, = $220,000*(4%/12)*(1+(4%/12))^240/(((1+(4%/12))^240)-1 = $1,333.16

(In the above formula, interest of 4% is divided by 12 to arrive at the monthly interest rate so as to arrive the monthly payments).

This can also be computed in excel using the formula =PMT(rate,nper,-pv)

where rate is the interest rate per month

nper is the period in months

pv is the value of the loan

Thus, using excel, monthly payments =PMT(4%/12,20*12,-220000) = $1,333.16

b. annual payment necessary to amortise the loan

Annual payment = P * r * (1 + r)^n/(((1 + r)^n) - 1)

Where P = Principal amount (mortgage borrowed)

r = Interest rate of the loan (per annum)

n = tenure (maturity) of the loan (in years)

Applying the values in the above formula, = $220,000*(4%)*(1+(4%))^20/(((1+(4%))^20)-1 = $16,187.99

This can also be computed in excel using the formula =PMT(rate,nper,-pv)

where rate is the interest rate per annum

nper is the period in years

pv is the value of the loan

Thus, using excel, monthly payments =PMT(4%,20,-220000) = $16,187.99

c. Answer in part (a) by 12

Monthly payment as above = $1,333.16 *12 = $15,997.88

Yearly payment as above = $16,187.99

Thus, monthly payment * 12 is not equal to yearly payment ($15997.99 is not equal to $16,187.99)

This is because when monthly payments are made, one also pays the principal along with the interest. Thus in month 1, monthly payment will include interest on principal outstanding and also a portion of principal repayment. Thus in month 2 , interest will be calculated on the principal outstanding minus the portion of principal paid in month 1 (reduced principal). However, in annual payment, though one pays both interest and principal, interest payment is calculated for the whole of the mortgage amount for the entire 12 months. Thus, the actual interest payable in annual payment will be higher than monthly payment. This is explained using the following table:

Monthly payments of year 1:

Month	Mortgage outstanding (A)	Monthly payment (B)	Interest (C) = (A)*4%/12	Principal re-payment (D)=(B)-(C)	closing mortgage outstanding (E)=(A)-(D)
1	220,000	1,333	733	600	219,400
2	219,400	1,333	731	602	218,798
3	218,798	1,333	729	604	218,195
4	218,195	1,333	727	606	217,589
5	217,589	1,333	725	608	216,981
6	216,981	1,333	723	610	216,371
7	216,371	1,333	721	612	215,759
8	215,759	1,333	719	614	215,145
9	215,145	1,333	717	616	214,529
10	214,529	1,333	715	618	213,911
11	213,911	1,333	713	620	213,291
12	213,291	1,333	711	622	212,669
	Total (sum)	15,998	8,667	7,331

Yearly payment of year 1:

Year	Mortgage outstanding (A)	Yearly payment (B)	Interest (C) = (A)*4%	Principal re-payment (D)=(B)-(C)	closing mortgage outstanding (E)=(A)-(D)
1	220,000	16,188	8,800	7,388	212,612

Thus, in monthly payments, total interest paid is $8,667 for the outstanding mortgage of $220,000 , effective interest rate is 3.94% (220000/8667) as against 4% paid in annual payments for $8,800.