Financial Applications-Mortgages

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Mortgages

The largest investment most people ever make is buying a house. Since the price of many houses in urban areas exceeds $400 000, it is usually necessary to borrow a large sum of money. Money borrowed with property as security is called a mortgage.

Amortization Period

Mortgage payments to repay the loan are made, usually monthly, over an extended period of time (15, 20, 25, 35) years are called the amortization period. However, since interest rates can vary widely, a mortgage agreement is usually for a short term. For example, a $300 000 (25-year mortgage) may be for 3 years at 13%, after which time it must be renegotiated at the rates then current depends on the economy situation.

3 Factors

The monthly payment is determined by three factors:The amount of the mortgage (Principal)The interest rateThe amortization period

Rule of 27

You can afford approximate 27% of your monthly income.

Laws Canadian law requires that mortgage

interest rates be compounded semi-annually. For example, a mortgage rate of 8% means “8 % compounded semi-annually.” Since the payments are usually made monthly, the payment interval is not the same as the compounding period. We must allow for this discrepancy when we do calculations that involve mortgages interest rates.

Equivalent RateTwo interest rates are said to be

equivalent if they yield the same amounts at the end of one year, or at the end of any number of years. The nominal rate of interest is the annual rate. 21

21 11 nn ii 21

2

2

1

1 11nn

n

r

n

r

n: is the compounding

period

Equivalent Rate Converting a semi-annual rate to a monthly

rateTo convert a mortgage rate of 8% compounded

semi-annually to a monthly rate, consider these time diagrams.

Example 1: Equivalent Rate Convert a mortgage rate of 8% / a, compounded

semi-annually to a monthly rate?122

121

2

08.01

r

12

104.1 6

1 r

104.112

6

1

r

i

006558197.012

r

i

078698364.0r

Therefore 8%/a compounded semi-annually is equivalent to 7.87%/a compounded monthly.

Example 2: MortgageVictor is buying a house for $196 500. He makes a down

payment of 25% of the price and negotiates a mortgage at 7.5%, amortized over 25 years, for the balance of the price.

a) How much is Victor’s mortgage?

25.0196500$

49125$196500$ 147375$

Down Payment:

Mortgage:

49125$

House: $196500Down payment: $ 49125Mortgage: $147375

b) Convert a mortgage rate of 7.5% / a, compounded semi-annually to a monthly rate.

122

121

2

075.01

r

12

10375.1 6

1 r

10375.112

6

1

r

i

006154524.012

r

i

073854286.0r

Therefore 7.5%/a compounded semi-annually is equivalent to 7.39%/a compounded monthly.

21

21 11 nn ii

c) How much are Victor’s monthly payment?

i

iRPV

n

11

006154524.0i??$R 3001225 n

006154524.0

006154524.011147375

300R

13.1078$

147375$PV

300006154524.11

006154524.0147375

R

Therefore Victor’s monthly payment is $1078.13.House: $196500

Down payment: $ 49125Mortgage: $147375

d) About what amount of interest (cost of borrow) is paid over the 25 years?

176064$

323439$30013.1078$

147375$323439$

Total payment – Mortgage = Total Interest (Cost of Borrow)

Total Interest (Cost of Borrow) is $176064

for a mortgage of $147375.

House: $196500Down payment: $ 49125Mortgage: $147375Monthly payment: $1078.13

e) What is the total value Victor paid for the house?

372564$

49125$323439$

Total Value = Total payment + Down Payment

Total payment to the house is $372564

for a house of $196500.House: $196500Down payment: $ 49125Mortgage: $147375Monthly payment: $1078.13Interest Paid: $176064

f) Approximately when will Victor start to pay back the principal?

176064$

13.1078$176064$

Cost of borrow:

Therefore, you can imagine the first 13years and 7 months of the mortgage time (amortization period) are used for paying interest.

Monthly payment: 13.1078$

months 304.163months 7 and years 13

It is just a general idea, the exact value will be calculated by using the amortization table since there is a portion of monthly payment going to the principal and a part going to the interest.

House: $196500Down payment: $ 49125Mortgage: $147375Monthly payment: $1078.13Interest Paid: $176064

g) What is Victor’s annual income in order to afford to this mortgage? (Rule of 27)

27.013.1078$

89.47916$

Monthly payment: 13.1078$

07.3993$

It is just a general idea for the bank to determine whether they should approve the mortgage or not.

Monthly income:

Annual income: 1207.3993$

House: $196500Down payment: $ 49125Mortgage: $147375Monthly payment: $1078.13Interest Paid: $176064

h) How much is still owing on the mortgage after the first 5 years?

After 5 years, 60 payments (out of 300) had been made, 240 periods remaining.

i

iRPV

n

11

006154524.0

006154524.01113.1078$ 240PV 15.135002$

Monthly payment: $1078.13

After 5 years: $1078.13 x 60 = $64687.80 had been paid. The outstanding of the mortgage after 5 years is $135002.15 Principal Paid in 5 years:

85.12372$15.135002$147375$ Interest Paid during the first 5 years:

95.52314$85.12372$80.64687$

Amortization Table

Payment: $1078.13 i = 0.006154524

Payment #

Balance Payment Interest Paid Principal Paid Outstanding Balance

0 - - - - $147375

1 $147375 $1078.13 $907.02$147375 x i

$171.11$1078.13-$907.02

$147203.89$147375-$171.11

2 $1078.13 $905.97$147203.89 x i

$172.16$1078.13-$905.97

$147031.73$147203.89-$172.16

3 $1078.13 $904.91$147031.73 x i

$173.22$1078.13-$904.91

$146858.51$147031.73-$173.22

... ... ... ... ...

180 $1078.13 $564.98$91799.14 x i

$513.15$1078.13-$564.98

$91285.99$91799.14-$513.15

... ... ...

300 $1078.13 $1071.54$1078.13-$6.59

$0