Question

A couple apply for a $225,000 thirty year home loan; the interest rate on the loan will be 4.5%.

     The bank obtains the following information:

         Stable gross monthly income of the couple       $5,600

         Annual homeowners insurance premium             2,400

         Annual real estate taxes                                           2,900

         Monthly auto loan payment                                       350

         Monthly Visa                                                                    45

         Monthly student loan payment                                  145

         Child support                                                                      0

         Annual auto insurance premium                             1,200

         Projected monthly utilities                                           495

         Monthly P and I on the proposed loan                           ?

      Calculate: (1) The monthly P and I payment;

                        (2) loan amortization for the first three months;

                        (3) the monthly housing payment and total monthly obligations (in dollars);

                        (4) calculate the front-end and back-end ratios.

                        Explain with formulas!

Answer 1

In the given question let us assume that the rate of interest of 4.5% is per annum compounded monthly since it is not clearly mentioned about that. Following are also given in question :-

1. Present value of Loan is $225,000
2. Period of Loan is 30 years
3. Stable gross monthly income of the couple is $5,600
4. Annual homeowners insurance premium is $2,400
5. Annual real estate taxes is $2,900
6. Monthly auto loan payment is $350
7. Monthly Visa is $45
8. Monthly student loan payment $145
9. Annual auto insurance premium $1,200
10. Projected monthly utilities $495

Answer – 1

Rate of Interest per annum is 4.5% or 0.045 in decimal, therefore if we divide it with 12 it comes to 0.375% per month or 0.00375 in decimal, hence period of 30 years will be denominated as 360 months

The following formula will apply for calculating Equated Monthly Installment (EMI) :-

EMI = Present Value of Loan / Sum of Discounted Factor

Where Present Value of Loan is given as $225,000, and

Sum of Discounted Factor = ∑ 1/(1+0.00375)↑360 (From 1 to 360)

                                                    = 197.6201

On putting the above two figures in the EMI formula, we get

EMI = $225000/197.6201

        = $1138.55 per month

The above EMI calculated shall comprise of Principal & Interest that is different for every month of loan period and shall be calculated as below mentioned amortization table (for first year only) –

Month	Opening Principal   (A)	Monthly EMI           (B)	Interest [A*0.00375] ( C )	Principal (D)	Closing Principal (A+C-B)
1	225,000.00	1,138.55	843.75	294.80	224,705.20
2	224,705.20	1,138.55	842.64	295.91	224,409.29
3	224,409.29	1,138.55	841.53	297.02	224,112.28
4	224,112.28	1,138.55	840.42	298.13	223,814.15
5	223,814.15	1,138.55	839.30	299.25	223,514.90
6	223,514.90	1,138.55	838.18	300.37	223,214.53
7	223,214.53	1,138.55	837.05	301.50	222,913.04
8	222,913.04	1,138.55	835.92	302.63	222,610.41
9	222,610.41	1,138.55	834.79	303.76	222,306.65
10	222,306.65	1,138.55	833.65	304.90	222,001.75
11	222,001.75	1,138.55	832.51	306.04	221,695.71
12	221,695.71	1,138.55	831.36	307.19	221,388.52

Answer – 2

Loan amortization for the first three months are solved in the above table

Answer – 3

Monthly housing payment shall be the EMI of the housing loan that is calculated in Answer – 1 as $1138.55

Total monthly obligation is calculated as below :-

Particulars	Description	Amount ($)
Monthly housing payment	EMI	1,138.55
Monthly Homeowners Insurance premium	$2400 / 12	200.00
Monthly real estate taxes	$2900 / 12	241.67
Monthly auto loan payment		350.00
Monthly Visa		45.00
Monthly student loan payment		145.00
Monthly auto insurance premium	$1200 / 12	100.00
Monthly utilities		495.00
TOTAL		2,715.22

Answer – 4

The front-end ratio is also known as the mortgage-to-income ratio and can be calculated as below :-

Front-end ratio = Monthly housing payment / Monthly income

Monthly Income is given as $5600 and monthly housing payment is calculated in Answer – 1 above as $1138.55, putting these two figures in above formula, we get –

Front-end ratio = 1138.55/5600

                         = 20.33%

The back-end ratio is also known as the debt-to-income ratio and can be calculated as below :-

Back-end ratio = Total monthly debt / Monthly income

Total monthly debt shall includes mortgages payment (principal, interest, taxes, and insurance), credit card payments, child support, and other loan payments.

Hence, Total monthly debt in the given question will be $1138.55 + $200 + $241.67 + $350 + $145 + $100 totaled as $2175.22 and Monthly Income is given as $5600, putting these two figures in above formula, we get –

Back-end ratio = 2175.22/5600

                         = 38.84%