Loan Amortization
4.4 Mortgages: Formula Approach
A. Mortgage Definition and Types
A mortgage is a loan that is issued by a financial institution to a borrower to purchase real estate properties (business or residential properties) at a specific interest rate for a specific time period. The mortgage amortization period is the length of time it takes to pay off the entire amount borrowed based on the original mortgage contract. A mortgage term is the length of time for which the mortgage agreement (a specific interest rate) will be in effect. Mortgage agreements are generally negotiated several times during the amortization period. Mortgage agreements are generally negotiated several times during the amortization period. The interest rate, the size of the payment, and payment and compounding frequencies can change each term. Figure 4.4.1 depicts an example of the amortization period and typical renewal terms for a 25-year mortgage.
Figure 4.4.1 An example of the amortization period and typical renewal terms for a 25-year mortgage
Financial institutions usually offer two types of mortgages: fixed-rate mortgages and variable-rate mortgages. Fixed-rate mortgages have a fixed interest rate for a specific period of time (mortgage term). Once the term is completed, the mortgage is renewed. The new term may have a different interest rate, size of the payment, and payment frequency. Variable-rate Mortgages are the ones in which the interest rate fluctuates depending on changes in the money market. This rate change is usually related to the lender’s prime rate.
Both mortgage types can be either open or closed. Closed Mortgages have a specific term and borrowers would have to pay a penalty for prepaying a large amount, closing the loan, or transferring to another lender. In contrast, open Mortgages allow borrowers to prepay any amount at any time, close, or transfer from one lender to another without being penalized. Because of the flexibility, interest rates of open mortgages are usually higher than those of closed mortgages. Figure 4.4.2 summarizes the types of mortgages.
Figure 4.4.2 Types of Mortgages
B. Payment Amounts and the Outstanding Principal Balance on Mortgages
Calculations of payment amounts and the outstanding principal balance on mortgages are similar to what was discussed previously for all loan types. The next example illustrates.
John purchased a house for $350,000. He made a down payment of 15% of the value of the house and received a mortgage for the rest of the amount amortized over 20 years. He negotiated a fixed interest rate of 3.8% compounded semi-annually for a 3-year term. a) Calculate the end-of-quarter payment size. b) Find the outstanding principal balance at the end of the 3-year term. c) Calculate the new end-of-quarter payment amount if the mortgage was renewed for another 3 years at 2.5% compounded semi-annually.
Show/Hide Solution
Given information:
- Interest is compounded semi-annually so [asciimath]C//Y = 2[/asciimath]
- Payments are made at the end of every quarter so [asciimath]P//Y = 4[/asciimath]
[asciimath]C//Y != P//Y[/asciimath] [asciimath]=>[/asciimath] Ordinary General Annuity
- Amortization period: [asciimath]t = 20[/asciimath] years
- Number of payments in the amortization period: [asciimath]N =P//Y*t=4 (20) =80[/asciimath]
- Nominal interest rate for Term 1:[asciimath]I//Y_1 = 3.8%[/asciimath]
- Periodic interest rate for Term 1: [asciimath]i = (I//Y_1)/(C//Y) = (3.8%)/2=1.9%[/asciimath]
- Nominal interest rate for Term 2: [asciimath]I//Y_2 = 2.5%[/asciimath]
- Periodic interest rate for Term 2: [asciimath]i = (I//Y_2)/(C//Y) = (2.5%)/2=1.25%[/asciimath]
- Purchase price [asciimath]= $350,000[/asciimath]
- Down payment [asciimath]= 15%(350,000) = $52,500[/asciimath]
a) Next, we use the PV and other given values to compute the size of the payments for the first 3-year term. We first need to find the interest rate per payment period [asciimath]i_2[/asciimath]:
[asciimath]i_2=(1+i)^((C//Y)/(P//Y) )-1[/asciimath]
[asciimath]i_2=(1+0.019)^(2/4)-1[/asciimath]
[asciimath]=0.00945529...[/asciimath]
Substituting the values into Formula 3.10b, we obtain
[asciimath]PMT=(PV* i_2)/(1-(1+i_2)^(-N))[/asciimath]
[asciimath]PMT=(297,500(0.00945529... ))/(1-(1+0.00945529... )^(-80))[/asciimath]
[asciimath]=5317.616...[/asciimath]
[asciimath]~~$5317.62[/asciimath]
The size of the quarterly payments is $5317.62.
b) We use Formula 4.1 to calculate the balance at the end of the first term with the focal date in the 3rd year. To find the future value of the original loan, we use the future value formula of compound interest (Formula 2.4a), and to calculate the future value of the payments, we apply the future value formula of the ordinary general annuity (Formula 3.6a). Since the annuity is general, the number of compounding periods and payments in the term are not equal. therefore, we need to calculate them separately.
Number of payments in Term 1: [asciimath]N_p=P//Y *"Number of years in Term 1"[/asciimath][asciimath]= 4(3) = 12[/asciimath]
Number of compounding periods in Term 1: [asciimath]N_c=C//Y *"Number of years in Term 1"[/asciimath] [asciimath]= 2(3) = 6[/asciimath]
[asciimath]BAL_("Focal date")=FV_("Original loan")-[/asciimath] [asciimath]FV_("PMTs made up to focal date")[/asciimath]
[asciimath]BAL_("Focal date")=PV(1+i)^(N_c)-[/asciimath] [asciimath]PMT[((1+i_2)^(N_p)-1)/i_2][/asciimath]
[asciimath]=297,500(1+0.019)^6-[/asciimath] [asciimath]5317.62[((1+0.00945529... )^12-1)/(0.00945529...) ][/asciimath]
[asciimath]~~333,067.36-67,236.75[/asciimath]
[asciimath]=$265,830.61[/asciimath]
c) The renewed term will have a new PMT as the nominal interest rate has reduced. To compute the new PMT, we use the outstanding balance at the end of the first term as the new PV, the remaining number of years as the new amortization period, and the new interest rate [asciimath]I//Y_2[/asciimath].
The new amortization period equals the original amortization period minus the number of years in the first term of the mortgage.
- New amortization period: [asciimath]t_2[/asciimath] [asciimath]= 20 \ "years" - 3 \ "years" = 17 \ "years"[/asciimath]
- [asciimath]N = P//Y (t_2)=4(17)=68[/asciimath]
- [asciimath]i = 1.25%[/asciimath]
- The present value is the balance at the end of the first term: [asciimath]PV= $265,830.61[/asciimath]
Since the interest rate has changed, we need to recalculate the interest rate per payment period [asciimath]i_2[/asciimath]:
[asciimath]i_2=(1+i)^((C//Y)/(P//Y) )-1[/asciimath]
[asciimath]i_2=(1+0.0125)^(2/4)-1[/asciimath]
[asciimath]=0.00623058...[/asciimath]
Substituting the values into Formula 3.10b, we obtain
[asciimath]PMT=(PV* i_2)/(1-(1+i_2)^(-N))[/asciimath]
[asciimath]PMT=(265,830.61(0.00623058... ))/(1-(1+0.00623058... )^(-68))[/asciimath]
[asciimath]=4807.704...[/asciimath]
[asciimath]~~$4807.70[/asciimath]
Therefore, the size of the quarterly payments in the renewed term is $4807.70.
The above figure includes a bar chart displaying the interest and principal portions of both the payments already made and those projected until the end of the amortization period. Also, it features line charts depicting the outstanding principal balance for Term 1 (covering the first 12 payments) and Term 2 (encompassing payments 13 to 24). It is noteworthy that due to the lower interest rate in the second term, the size of the payments and the interest portions are smaller compared to those in the first term. The slider tool allows you to zoom in and focus on a specific period within the chart.
Try an Example
Section 4.4 Exercises
- Sheniqua purchased a house for $930,000. He made a down payment of 16% of the value of the house and received a mortgage for the rest of the amount amortized over 25 years. She negotiated a fixed interest rate of 3.56% compounded semi-annually for a 5-year term. a) Calculate the size of month-end payments. b) Calculate the principal balance at the end of the 5-year term. c) Calculate the size of month-end payments if the mortgage was renewed for another 5-year term at a fixed rate of 2.97% compounded semi-annually.
Show/Hide Answer
a) PMT = $3,925.08
b) BAL = $674,757.75
c) New PMT = $3,725.93
- Joey purchased a house for $1,770,000. He made a down payment of 15% of the value of the house and received a mortgage for the rest of the amount amortized over 25 years. He negotiated a fixed interest rate of 3.2% compounded semi-annually for a 4-year term. a) Calculate the size of month-end payments. b) Calculate the principal balance at the end of the 5-year term. c) Calculate the size of month-end payments if the mortgage was renewed for another 4-year term at a fixed rate of 2.01% compounded semi-annually.
Show/Hide Answer
a) PMT = $7,275.27
b) BAL = $1,336,349.88
c) New PMT = $6,499.72
- Arman bought an apartment for $850,000. He made a down payment of 26% of the value of the apartment and received a mortgage for the rest of the amount amortized over 25 years. He negotiated a fixed interest rate of 3.96% compounded semi-annually for a 7-year term. a) What was the size of month-end payments? b) Calculate the principal balance at the end of the 7-year term. c) What is the size of month-end payments if the mortgage is renewed for another 7-year term at a fixed rate of 3.9% compounded semi-annually?
Show/Hide Answer
a) PMT = $3,295.04
b) BAL = $509,698.20
c) New PMT = $3,279.57
